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Time-homogeneous parabolic Anderson model

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Time-homogeneous parabolic Anderson model

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Content Time-homogeneous Parabolic Anderson Model by Hyun-Jung Kim A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) August 2018 Copyright 2018 Hyun-Jung Kim Acknowledgments First and foremost, I would like to express my sincere and deep gratitude to my advisor Prof. Lototsky for his tremendous academic support, continuous motivation, valuable opportunity, and insightful advice on my Ph.D. research. His professional guidance helped me in all the time of research and writing of this thesis. Moreover, thank you for encouraging me to participate in several conferences and summer schools and to meet various people related to my research area. I would also like to thank you for always being available to give extra information and feedback about my performance of research and presentations. I could not have imagined having a better advisor and mentor for my Ph.D. study and life. Besides my advisor, I would like to take a moment to deeply thank the rest of my thesis committee: Prof. Mikulevicius and Prof. Ghanem for letting my defense be an enjoyable moment and for their brilliant comments and challenging questions, which allow my thesis to get improved. This experience certainly strengthens my mathematical logics and skills. My sincere thanks also goes to Prof. Kukavica for his constant faith in my research work. Whenever I ask him to review my articles and write reference letters, he always generously spends and shares his time with very kind support and comments. I also would like to express my special thanks to my family for their endless love, kindness and support they have shown during the past years. It denitely has taken me to nalize ii this thesis. Words cannot express how grateful I am to my parents: Sungtae Kim and Jungja Han. Last but not the least, I thank my friends: Joonho Back, Najung Kim, Dongwook Kim, Jeeyun Baik, Seungjong Lee, Jiyeon Park, Eunjung Noh, Fanhui Xu, Blair Poon and Shiqi Chen for encouraging and supporting me spiritually throughout my life. Especially, thank you all for your love, sharing happy moments, and giving me a ride. iii Table of Contents Acknowledgments ii Abstract vi Chapter 1 Introduction 1 Chapter 2 Stochastic Integrals 6 2.1 The Wiener Integral 7 2.2 The It^ o Stochastic Integral 9 2.3 The Stratonovich Stochastic Integral 14 2.4 The Wick-It^ o-Skorokhod Stochastic Integral 16 Chapter 3 Stochastic PAM with Additive Noise 26 Chapter 4 Stochastic PAM in the It^ o Interpretation 31 4.1 An Adapted It^ o Solution 32 Chapter 5 Stochastic PAM in Stratonovich Interpretation I 38 5.1 The Classical Geometric Rough Path Solution 41 5.2 The Generalized Geometric Rough Path Solution 48 5.3 The Fundamental Solution 52 Chapter 6 Stochastic PAM in Stratonovich Interpretation II 54 6.1 The Schauder Estimates 54 6.2 The Mild Solution and its Regularity 60 Chapter 7 Stochastic PAM in the Wick-It^ o-Skorokhod Interpretation 69 7.1 The Chaos Spaces 70 7.2 The Chaos Solution 78 7.3 Basic Regularity of the Chaos Solution 85 7.4 Time Regularity of the Chaos Solution 92 7.5 Space Regularity of the Chaos Solution 97 7.6 The Fundamental Chaos Solution 102 iv Chapter 8 Concluding Remarks 106 8.1 Conclusion 106 8.2 Further Directions 107 Index 115 Bibliography 116 v Abstract The linear stochastic heat equation is often the starting point in the analysis of various physical phenomena involving randomness. The noise in the equation can enter additively or multiplicatively. In the basic physical setting, additive noise corresponds to random heat source; mathematical ramications of this model is that many properties of the solution can be studied without using stochastic calculus. Multiplicative noise, which physically corre- sponds to random potential, always requires stochastic integration from the very beginning of the analysis. The noise, which is assumed Gaussian, can act either in space only, or in time only, or in both space and time. Multiplicative time-homogeneous (or space-only) white noise presents special interest from both mathematical and physical points of view, and will be the emphasis of the discussion. Unlike time-dependent models that have been well-studied using the It^ o theory, time-homogeneous models have yet to receive the attention they deserve. In the time- homogeneous case, one cannot typically use standard It^ o techniques to dene an (adapted) solution. It means that the absence of time variable in the noise leads to a big dierence and diculty in the analysis. We focus on the stochastic parabolic Anderson model driven by multiplicative time- homogeneous Gaussian random potential. The objective of the current work is to develop vi new methods to construct a solution and investigate analytic properties of the solution for the model in one space dimension. The Stratonovich interpretation and the Wick-It^ o-Skorokhod interpretation are mainly discussed. vii Chapter 1 Introduction Consider the equation in the form of @u(t;x) @t = u(t;x) +u(t;x) _ W (x); t> 0; x2R d : (1.1) In physics and related applications, equations of the type (1.1) can represent either the Schr odinger equation (after making time imaginary) or the (continuum) parabolic Anderson model (PAM). In both cases, the potential _ W is often random, to model complexity and/or incomplete knowledge of the corresponding Hamiltonian. It naturally appears in homoge- nization problems for PDEs driven by highly oscillating stationary random elds. It is one of the simple stochastic partial dierential equations (SPDEs) that one might wish to solve easily, but it turns out that it is not straightforward. The notation _ W (x) in this paper stands for the formal derivative of a Brownian sheet W (x) (called white noise), and the equation (1.1) may be written as @u(t;x) @t = u(t;x) +u(t;x) @ d @x 1 @x d W (x): (1.2) 1 When the time variable is present in _ W , the productu _ W can often be interpreted as the It^ o stochastic integral; with only space parameter, such an interpretation is not possible and (1) the Stratonovich integral and (2) the Wick-It^ o-Skorokhod integral become available options. Depending on the integral, the equation (1.2) is classied as follows: (1) PAM with usual pathwise product in Stratonovich interpretation: @u(t;x) @t = u(t;x) +u(t;x) @ d @x 1 @x d W (x); (1.3) (2) PAM with Wick product in Wick-It^ o-Skorokhod interpretation: @u(t;x) @t = u(t;x) +u(t;x) @ d @x 1 @x d W (x): (1.4) As the idea of Stratonovich integral (known as the method of regularization) suggests, the equation (1.3) is equivalently interpreted as a pathwise limit of approximated equations @u " (t;x) @t = u " (t;x) +u " (t;x) @ d @x 1 @x d W " (x); (1.5) where W " are smooth approximations of a sample path of W for "> 0. The tradition when studying (stochastic) partial dierential equations is that solutions inherit properties from the coecients. There exists very irregular spatial coecient _ W in (1.2), and the noise term denitely aects the spatial regularity of the solution. If d 2, the usual pathwise product of u @ d @x 1 @x d W is classically not well-dened, and constructing a solution of (1.3) presents a major challenge in the standard theory of 2 stochastic partial dierential equations. Indeed, it is known that d-dimensional Brownian sheet W has regularity 1=2" in each parameter for any " > 0. Therefore, @ d @x 1 @x d W can be understood to have regularityd=2" in total. Then, the solution u is expected to have regularity 2d=2" so that the sum of the regularity of u and @ d @x 1 @x d W is strictly less than 0. Hence, unfortunately, classical integration theory cannot be applied to the product u @ d @x 1 @x d W , and advanced techniques are inevitably required. When d = 2 and on the whole space R 2 , the paper [6] introduces a particular renor- malization procedure and constructs a solution to (1.3) by subtracting a divergent constant from the equation. On a torus ofR 2 , a solution of (1.3) is constructed independently using paracontrolled distributions in [4] and using the theory of regularity structures in [5]. When d = 3, using the theory of regularity structures, the paper [8] carries out the construction of (1.3) on the whole space R 3 . An alternative construction of solution to (1.3) on a torus of R 3 is also established in [7]. It turns out that for the model (1.3) when d = 1, there are several ways to dene a solution in the Stratonovich sense, and the regularity of solution is worth attention. In [12], the Feynman-Kac solution for (1.3) in the Stratonovich sense is introduced. The paper proves that the Feynman-Kac solution is almost H older 3/4 continuous in time and almost H older 1/2 continuous in space. Here, \almost" H older continuity of order means H older continuity of any order less than . However, the standard parabolic theory implies that the spatial regularity can be improved. Indeed, consider the additive model @u(t;x) @t = @ 2 u(t;x) @x 2 + _ W (x) (1.6) 3 as a reference for optimal regularity. It is known that the explicit solution of (1.6) is almost H older 3/4 continuous in time and almost H older 3/2 continuous in space. On the other hand, the Wick-It^ o-Skorokhod interpretation (1.4) draws attention from several authors. For example, the paper [28] by Uemura when d = 1, and the paper [11] by Hu when d < 4, dene the chaos solution using multiple It^ o-Wiener integrals. Also, the paper [28] gives some regularity results: The chaos solution is almost H older 1/2 continuous in both time and space: One can expect the spatial regularity to get improved. Also, the paper [12] investigates the large time behavior of p-th moment of either the Stratonovich solution or the Wick-It^ o-Skorokhod solution, called intermittency: for allx and p 2, Eju(t;x)j p e p 3 t 3 ; for large t> 0: Earlier than the paper [12], the paper [11] has already found the same result of the Wick- It^ o-Skorokhod solution for the case p = 2. The objectives of this paper are to dene a solution in the Stratonovich interpretation or in the Wick-It^ o-Skorokhod interpretation of @u(t;x) @t = @ 2 u(t;x) @x 2 +u(t;x) _ W (x); t> 0; 0<x<; u(t; 0) =u(t;) = 0; u(0;x) =u 0 (x); (1.7) to establish optimal space-time regularity of each solution in line with the deterministic PDE theory. That is, each solution is almost H older 3=4 continuous in time and almost H older 3=2 continuous in space; 4 to make a connection between the Stratonovich solution and the Wick-It^ o-Skorokhod solution. In Chapter 2, we introduce several types of stochastic integrals. Chapter 3 discusses the stochastic PAM with additive noise as a reference model. Chapter 4 devotes the It^ o inter- pretation for (1.7). Chapter 5 and Chapter 6 pay attention to Stratonovich interpretations for (1.7) in two dierent ways. Chapter 7 investigates the Wick-It^ o-Skorokhod interpretation for (1.7). In Chapter 8, we give conclusions and suggest further directions of research. Throughout the paper, we often use the following notations: f t (t;x) = @f(t;x) @t ; f x (t;x) = @f(t;x) @x ; f xx (t;x) = @ 2 f(t;x) @x 2 ; 5 Chapter 2 Stochastic Integrals This section addresses an issue which came up in the analysis of mild solutions to the stochastic parabolic Anderson model driven by multiplicative time-homogeneous Gaussian white noise ind space dimension. We emphasize that dening a stochastic integral is a quite necessary and important task in order to understand our analysis and results. One of our main goals is to construct a solution of u t (t;x) =u xx (t;x) +u(t;x) _ W (x); t> 0; x2GR d u(0;x) =u 0 (x); (2.1) where _ W (x) represents time-homogeneous (or space-only) Gaussian white noise. Note that a domain GR d can be either bounded or unbounded. If G is bounded, we add boundary conditions as well. 6 We will say that a random eldu(t;x) is a mild solution of (2.1) with additional conditions if it satises the Duhamel formula for all t> 0, x2G, u(t;x) = Z G P(t;x;y)u 0 (y)dy + Z t 0 Z G P(ts;x;y)u(s;y) _ W (y)dyds; almost surely (2.2) whereP denotes the corresponding heat kernel. Throughout the paper, we use the notation P for the Gaussian heat kernel; P D for the Dirichlet heat kernel; P N for the Neumann heat kernel. The product of u and _ W or the stochastic integral of u with respect to _ W is not clearly dened at this moment. In order to make sense of the mild solution, we need to give a meaning to the stochastic integral Z t 0 Z G P(ts;x;y)u(s;y) _ W (y)dyds (2.3) appearing in (2.2). 2.1 The Wiener Integral For deterministic ingrands f, R fdW is known as the Wiener integral and is often denoted by _ W (f). 7 Denition 1. Let _ W be Gaussian white noise onR d with control measure . Then, n _ W (f); f2L 2 (R d ;) o is a Gaussian system such that E h _ W (f) i = 0 and E h _ W (f) _ W (g) i = Z R d f(x)g(x)(dx). This collection is referred to as the isonormal Gaussian process on (or over) the Hilbert space L 2 (R d ;). Note that the denition provides _ W (f) for all square-integrable functions at once (as opposed to dening _ W (f) for each f individually). Since, for every measurable set AR d with (A) < 1, the indicator function 1 A is square-integrable, and then the isonormal Gaussian process onL 2 (R d ;) becomes an extension of the Gaussian white noise onR d with control measure . When the control measure is non-atomic, the linear span of indicator functions 1 A with(A)<1 is dense inL 2 (R d ;), and then the Gaussian white noise onR d with control measure extends by continuity to an isonormal Gaussian process onL 2 (R d ;), making the two objects essentially equivalent. More generally, if H is a separable Hilbert space with inner product (;) H , then an isonormal Gaussian process on (or over) H is a Gaussian system _ W (f); f 2H, such that E h _ W (f) i = 0 andE h _ W (f) _ W (g) i = (f;g) H : This construction makes it possible to recover many familiar objects from stochastic analysis. For example, if H =L 2 ([0;T ]), then W (t) = _ W 1 [0;t] ; 0tT is a standard Brownian 8 motion. Similarly, ifH =L 2 ([0;T ] [0;L]), thenW (t;x) = _ W 1 [0;t][0;x] is a two-parameter Brownian motion, or Brownian sheet. Given an isonormal Gaussian process and an H-valued random element f, denition of _ W (f), which should correspond to a stochastic integral, is, in general, not clear and requires additional assumptions, both about the Hilbert space H and the random element f. 2.2 The It^ o Stochastic Integral Denition 2. LetF be a -eld. We say thatF =fF x g x0 is a ltration if (1) For any x,F x F is a -eld, and (2) For any x<y,F x F y . Denition 3. We say that the process = (x) isF-adapted, and we write 2F, if (x) isF x -measurable for all x 0. Let ( ;F;F;P) be a ltered probability space, whereF =fF x g 0x is a ltration. Given our interest in space-only noise, we denote the parameter as space variable x and restrict it to a bounded interval [0;] for simplicity. We denoteL 2 (F) as the space ofF-adapted square integrable random processes. Denition 4. We say that 2 L 2 (F) is an elementary process if there exists a partition 0 = x 0 < < x n = such that (x) = (x i ) for all x2 [x i ;x t+1 ), i = 0; ;n 1. We denote it as 2L 0 2 (F). 9 From now on, we assume thatF :=F W is the (forward) ltration generated by Brownian motion W . That is, F W =fF x g x0 whereF x is the -algebra generated byfW (y)g 0yx . Then, for 2L 0 2 (F), we dene the It^ o stochastic integral in a pathwise manner: Z x 0 (y;!)dW (y;!) := X i: x i <x (x i ) [W (x i+1 ^x;!)W (x i ;!)]; 0x: (2.4) Proposition 5. Let 2L 0 2 (F) and denote M(x) := R x 0 (y)dW (y). Then, (1) M is continuous a.s. (2) For any 1 ; 2 2 L 0 2 (F) and any constants a 1 ;a 2 2R, we have a 1 1 +a 2 2 2 L 0 2 (F) and R x 0 [a 1 1 (y) +a 2 2 (y)]dW (y) =a 1 R x 0 1 (y)dW (y) +a 2 R x 0 2 (y)dW (y). (3) M is anF-martingale. In particular,E [M(x)] = 0. (4) M2 L 0 2 (F), and N(x) := (M(x)) 2 R x 0 j(y)j 2 dy is an F-martingale. In particular, E jM(x)j 2 =E R x 0 j(y)j 2 dy . Proof. (1) and (2) are clear. For (3), it is enough to show that, for any i, M(x) =E M(x i+1 ) F x ; x i xx i+1 : Since (x i )2F x i F x , E M(x i+1 )M(x) F x = E (x i )(W (x i+1 )W (x)) F x = (x i )E W (x i+1 )W (x) F x = (x i )E [W (x i+1 )W (x)] = 0: 10 Thus, M is anF-martingale. For (4), let's prove thatN(x) = (M(x)) 2 R x 0 j(y)j 2 dy is a martingale. Similarly to (3), it suces to show that N(x) =E N(x i+1 ) F x ; x i xx i+1 : Since N(x i+1 )N(x) = M 2 (x i+1 )M 2 (x) 2 (x i )(x i+1 x) = (M(x i+1 )M(x)) 2 + 2M(x) (M(x i+1 )M(x)) (x i ) 2 (x i+1 x) = (x i ) 2 (W (x i+1 )W (x)) 2 (x i+1 x) + 2M(x)(x i ) (W (x i+1 )W (x)); we have E N(x i+1 )N(x) F x = 2 (x i )E (W (x i+1 )W (x)) 2 (x i+1 x) F x + 2M(x)(x i )E [W (x i+1 )W (x)] = 0: We now extend the \It^ o stochastic integration" to all processes in L 2 (F). Lemma 6. For any 2L 2 (F), there exist n 2L 0 2 (F) such that lim n!1 k n k 2 2 := lim n!1 E Z 0 j n (y) (y)j 2 dy = 0: 11 Proof. (Case 1) Assume that is continuous and bounded. Then, for each n, dene n (x) := n1 X i=0 (x i )1 [x i ;x i+1 ) ; where x i = i n , i = 0;:::;n. By the Dominated convergence theorem, the statement is true. (Case 2) We only assume that is bounded, or sayjjC for someC > 0. Let's mollify by convolution with a mollier: " := " . The function is innitely dierentiable and compactly supported with the total mass 1 and " (x) := " 1 x " for all x2 [0;]. Then, clearly,j " jC and " is continuous. Also, we can easily check that lim "!0 Z 0 j " (x) (x)j 2 dx = 0: By the Dominated convergence theorem again, we get lim "!0 k " k 2 ; a.s. Therefore, for eachn, there exists" n such thatk "n k 2 1 2n . By the Case 1, there exists n 2L 0 2 (F) such thatk ^ "n n k 2 1 2n . Thus, we have k n k 2 1 n : 12 (Case 3) For general , for each n, denote ^ n := (n)_ ^n. Then, clearly, ^ n ! andj n jjj. By the Dominated convergence theorem, lim n!1 k ^ n k 2 = 0: Also, sincej ^ n j n, by the Case 2, there exists n 2 L 0 2 (F) such thatk ^ n n k 2 1 n . Hence, k n k 2 k ^ n k 2 +k n ^ n k 2 k ^ n k 2 + 1 n ! 0; as n!1: For the above n 2L 0 2 (F), we deneM n (x) := R x 0 n (y)dW (y). By applying the Doob's maximum inequality, we get kM n M m k 1;2 := sup 0x EjM n (x)M m (x)j 2 1=2 2k n m k 2 ! 0; n;m!1: Then, there exists a unique process M2F such that lim n!1 kM n Mk 1;2 = 0: (2.5) The processM does not depend on the choices of n . Therefore, we deneM as the stochastic integral of against W . In other words, for each x2 [0;], Z x 0 (y)dW (y) := lim n!1 Z x 0 n (y)dW (y) 13 in the sense of (2.5). Remark 7. Proposition 5 still holds true for any 2L 2 (F) and M(x) = R x 0 (y)dW (y). Theorem 8. (Burkholder-Davis-Gundy (BDG) inequality) For any p> 0, let M(x) = R x 0 (y)dW (y) with 2L 2 (F) and M (x) := sup 0yx jM(y)j: Then, there exist universal constants 0<c p <C p such that c p E " Z x 0 j(y)j 2 dy p=2 # E [jM (x)j p ]C p E " Z x 0 j(y)j 2 dy p=2 # : Proof. See Theorem 23.12 on p. 443 of [13]. 2.3 The Stratonovich Stochastic Integral We next dene Stratonovich stochastic integral Z x 0 (y;!)dW (y;!) in a pathwise manner by the limit in mean square of X i: x i <x x i + (x i+1 ^x) 2 ;! [W (x i+1 ^x;!)W (x i ;!)] (2.6) 14 as the mesh of the partition 0 =x 0 <x 1 <<x n = tends to 0, if it exists. Equivalently, we may dene Z x 0 (y;!)dW (y;!); if it exists, by the limit in mean square of Z x 0 (y;!)dW " (y;!); where W " are smooth approximation of W . The It^ o integral is widely used, but in some cases, the Stratonovich integral is more convenient since the Stratonovich stochastic integral obeys the nice classical chain rule; cf. [30]. Proposition 9. (Stochastic chain rule) The Stratonovich integral satises the chain rule of classical Calculus: For any f2C 1 (0;), Z x 0 f 0 (W (y))dW (y) =f (W (x))f (W (0)): The following shows a connection between It^ o and Stratonovich integrals. Proposition 10. (It^ o-Stratonovich correction term) For any f2C 1 (0;), the following identity holds: Z x 0 f (W (y))dW (y) = Z x 0 f (W (y))dW (y) + 1 2 Z x 0 f 0 (W (y))dy: 15 2.4 The Wick-It^ o-Skorokhod Stochastic Integral There is another way of analyzing the stochastic integral R (x)dW (x), which has been developed in [10], [11] and [24]. The stochastic integral can be understood by introducing Wick product. Using the Wick product, we interpret the stochastic integral as Z (x)dW (x) := Z (x) _ W (x)dx: We start this section with the denition of the Hermite polynomials H n (x). Denition 11. The Hermite polynomials H n (x) are dened by H n (x) := (1) n e 1 2 x 2 d n dx n e 1 2 x 2 ; n = 1; 2;:::: In fact, we may understand H n (x) via the equality: F (x;t) :=e xt t 2 2 = 1 X n=0 H n (x) n! t n : (2.7) In particular, the rst ve Hermite polynomials are H 0 (x) = 1; H 1 (x) =x; H 2 (x) =x 2 1; H 3 (x) =x 3 3x; H 4 (x) =x 4 6x 2 + 3; H 5 (x) =x 5 10x 3 + 15x and etc. Since @F @x =tF , we notice that H 0 n (x) =nH n1 (x) for any n 1. 16 Proposition 12. Let be a standard Gaussian random variable. Then, for all n 1, EH n () = 0: Moreover, for all n;m 1, E (H n ()H m ()) = 8 > > < > > : n! if n =m 0 if n6=m: Proof. It immediately follows from (2.7). See Lemma 1.3 on p.2 of [24]. We note that n H n ()= p n!; n 0 o is an orthonormal basis inL 2 (), the space of square- integrable random variables that are measurable with respect to the -algebra generated by . Then, we have the chaos expansion of 2L 2 () as follows: = 1 X n=0 n n! H n (); with n =E (H n ())2R. Our next goal is to extend the chaos expansion from one dimensional spaceL 2 () further to innite dimensional space L 2 _ W ;U , whereL 2 _ W ;U is the space of square-integrable random variables that are measurable with respect to the -algebra generated by the col- lection of i.i.d standard Gaussian random variables k ;k 1. Here,U is a separable Hilbert space. For example,U =L 2 ([0;]). 17 Remark 13. For an orthonormal basisfu k ;k 1g inU, k can be generated by k = _ W (u k ); k 1: We now discuss the denition of a multi-index. Denition 14. A multi-index = ( 1 ; 2 ;::: ) is an ordered collection of non-negative integers j , j 1 such that P j1 j <1. In fact, P j1 j <1 is equivalent to the condition that all but nitely many of j are zero. We can check that the set of all multi-indices is countable. Throughout the paper, we use the following notations for simplicity: M = the collection of all multi-indices; (0) = (0; 0;::: ) = the zero multi-index;"(k) = (0;:::; 0; 1; 0;::: ) = the unit multi-index with the only one 1 at position k; , j j for all j 1; + = ( 1 + 1 ; 2 + 2 ;::: ); = (max ( 1 1 ; 0); max ( 2 2 ; 0);::: ); jj = P j1 j , ! = Q j1 j !; (k) ="(k); x = Q j1 x j j , wherex = (x 1 ;x 2 ;::: ) is a sequence of real numbers; (2N) = Q 1 j=1 (2j) j . 18 We dene H () := Y j1 H j ( j ); where := ( 1 ; 2 ;::: ) and j 's are standard normal variables. Then, by Cameron-Martin theorem,fH ()g 2M forms an orthogonal basis in L 2 () and EjH ()j 2 =!: We also consider a stochastic exponential function on [0;]. LetU be a dense subset of L 2 (0;) andf k (x)g k1 be an orthonormal basis in L 2 (0;). For h 2 U with h k = R 0 h(x) k (x)dx, we dene E(h) = exp 1 X k=1 h k k 1 2 1 X k=1 h 2 k ! : Or equivalently, E(h) = X 2M h ! H (); whereh = (h 1 ;h 2 ;::: ). Theorem 15. (Cameron-Martin theorem [1]) The collectionfH ();2Mg is the orthogonal basis of L 2 _ W ;U . The basis is called the Cameron-Martin basis. 19 As a result of the Cameron-Martin theorem, we have the Wiener-It^ o chaos expansion theorem. Theorem 16. (The Wiener-It^ o chaos expansion theorem) Any 2L 2 _ W ;U has a unique representation: = X 2M ! H (); where =E (H ())2R. Also, we haveE ( 2 ) = P 2M ( ) 2 ! <1. 2.4.1 The Wick Product and the Hida Spaces Let's now dene the Wick product of two elements ; of L 2 _ W ;U . Denition 17. The Wick product of two elements such that = X 2M ! H ()2L 2 _ W ;U ; = X 2M ! H ()2L 2 _ W ;U is dened by := X ; !! H + (): In particular, we have H n ()H m () =H n+m (); n;m 0: (2.8) 20 But, white noise is not an element of L 2 _ W ;U . Thus, we want to extend the space L 2 _ W ;U further to a bigger space so that a new space contains white noise. We introduce the Hida test function spaceQ and the Hida distribution spaceQ . Denition 18. (a) The Hida test function spaceQ consists of all = X 2M c H ()2L 2 _ W ;U with c 2R such that sup c 2 !(2N) k <1; for all k2N: (b) The Hida distribution spaceQ consists of all formal expansions = X 2M d H () with d 2R such that sup d 2 !(2N) q <1; for some q2N: Then, we have the following proposition. Proposition 19. For any 1<p<1, QL p _ W ;U Q : 21 Proof. See Corollary 2.3.8 on p. 40 of [10]. Next, we show thatQ contains (singular) white noise. We have the explicit formula for Brownian motion W on GR: W (x) = 1 X k=1 Z x 0 m k (y)dy k ; (2.9) where m k 's form an orthonormal basis in L 2 (G). By formally taking the derivative of (2.9) with respect to x, we may write white noise as _ W (x) = 1 X k=1 m k (x) k : In particular, we may write space-only white noise onR as _ W (x) = 1 X k=1 m k (x) k = 1 X k=1 m k (x)H "(k) (); where m k (x) := 2 k k! p 1 2 e 1 2 x 2 h k (x); and h k (x) := (1) k e x 2 d k dx k e x 2 is the physicists' Hermite function. Proposition 20. Let _ W be white noise on GR. Then, _ W (x)2Q for each x2GR. 22 Proof. It is enough to show the statement forR. We see that sup k m 2 k (x)(2N) q 1 X k=1 m 2 k (x)(2k) q <1 for any q2N since the normalized Hermite functions satisfy the boundjm k (x)j 0:816 for all k and all real x inR (e.g. p.260 in [29] ), completing the proof. Now, we extend the denition of the Wick product as follows. Denition 21. The Wick product of two elements such that = X 2M c H ()2Q ; = X 2M d H ()2Q is dened by := X ; c d H + (): Remark 22. For any random process = P 2M ! H ()2L 2 _ W ;U , we have (x) _ W (x) = X ;k (x)m k (x) ! H +"(k) (): Proposition 23. (Closedness under Wick product) (a) If ; 2Q, then 2Q; (b) If ; 2Q , then 2Q . Proof. See Lemma 2.4.4 on p. 46 of [10]. 23 It is clear that if (x)2Q ; x2G, then (x) _ W (x) exists inQ for eachx2G. Next, we show that R (x) _ W (x)dx exists inQ as well. Consider integrals with values inQ . Denition 24. We say a random process (x) :G!Q isQ -integrable if Z G E (f(x))dx<1 for all f2Q: Then, theQ -integral of (x), denoted by R G (x)dx, is the (unique)Q -element such that E f Z G (x)dx = Z R E (f(x))dx; f2Q: Lemma 25. Suppose that (x)2Q has the chaos expansion (x) = X c (x)H (); where X kc k 2 L 1 (G) !(2N) q <1; for some q2N: Then, (x) isQ -integrable and Z G (x)dx = X Z G c (x)dxH (): Proof. See Lemma 2.5.6 on p. 54-55 of [10]. 24 Theorem 26. Assume that for each x2G, (x) = X c (x)H ()2Q ; where sup kc k 2 L 1 (G) !(2N) q <1; for some q2N: Then, (x) _ W (x) isQ -integrable and Z G (x) _ W (x)dx = X ;k Z G c (x)m k (x)dxH +"(k) (): Proof. See Lemma 2.5.6 on p. 55-56 of [10]. Proposition 27. Suppose that is an F-adapted and L 2 (F)-bounded stochastic process. Then (x) is both Wick-It^ o-Skorokhod integrable and It^ o integrable. Moreover, the two inte- grals coincide: Z G (x) _ W (x) = Z G (x)dW (x): Proof. See Proposition 2.5.4 on p. 53-54 of [10]. 25 Chapter 3 Stochastic PAM with Additive Noise The objective of this chapter is to establish the bench-mark space-time regularity result for (1.7) by considering the corresponding equation with additive noise: U t =U xx + _ W (x); t> 0; x2 (0;); U(0;x) = 0; U(t; 0) =U(0;) = 0: (3.1) By the variation of parameters formula, the solution of (3.1) is U(t;x) = Z t 0 Z 0 P D (s;x;y)dW (y)ds: Then, U(t;x) = 2 X k1 k 2 1e k 2 t ) sin(kx) k ; (3.2) U x (t;x) = 2 X k1 k 1 1e k 2 t cos(kx) k ; (3.3) 26 where k = Z 0 sin(kx)dW (x); k 1; are independent Gaussian random variables with zero mean. In particular, the series on the right-hand sides of (3.2) and (3.3) converge with probability one for every t > 0 and x2 [0;]. We start with an overview of the parabolic H older space. Given a locally compact metric space X with the distance function , denote byC(X) the space of real-valued continuous functions onX. For 2 (0; 1), denote byC (X) the space of H older continuous real-valued functions onX, that is, functions F satisfying sup a;b:(a;b)>0 jF (a)F (b)j (a;b) <1; C (X) is a Banach algebra with norm kFk C (X) = sup a jF (a)j + sup a;b:(a;b)>0 jF (a)F (b)j (a;b) : For a positive integerk, the spaceC k+ (X) consists of real-valued functions that arek times continuously dierentiable, and the derivative of order k is inC (X). Of special interest isX = (0;T )G, GR d ; with parabolic distance (t;x); (s;y) = p jtsj +jxyj: 27 To emphasize the presence of both time and space variables, the corresponding notations becomeC =2; (0;T )G andC k+(=2);2k+ (0;T )G . Also, kvk C (1+)=2;1+ (0;T )G = sup t;x jv(t;x)j +kv x k C =2; (0;T )G + sup t6=s;x jv(t;x)v(s;x)j jtsj (1+)=2 : We also write f2C (x 1 ;x 2 ), or f is almost H older continuous, if f2C " (x 1 ;x 2 ) for every "2 (0;). The main tool for establishing H older regularity of random processes is the Kolmogorov continuity criterion: Theorem 28. Let T be a positive real number and X =X(t), a real-valued random process on [0;T ]. If there exist numbers C > 0, p> 1, and qp such that, for all t;s2 [0;T ], EjX(t)X(s)j q Cjtsj p ; then there exists a modication ofX with sample trajectories that are almost H older (p1)=q continuous. Proof. See, for example Karatzas and Shreve [14, Theorem 2.2.8]. We now apply Theorem 28 to the solution of equation (3.1). 28 Theorem 29. The random eld U =U(t;x) dened in (3.2) satises U(;x)2C 3=4 (0;T ); x2 [0;]; T > 0; (3.4) U x (;x)2C 1=4 (0;T ); x2 [0;]; T > 0; (3.5) U(t;)2C 3=2 (0;); t> 0: (3.6) Proof. For every t > 0 and x;y 2 [0;], the random variables U(t;x) and U x (t;x) are Gaussian, so that, by Theorem 28, statements (3.4), (3.5), and (3.6) will follow from EjU(t +h;x)U(t;x)j 2 C(")h 3=2" ; "2 (0; 3=2); (3.7) EjU x (t +h;x)U x (t;x)j 2 C(")h 1=2" ; "2 (0; 1=2); (3.8) EjU x (t;x +h)U x (t;x)j 2 C(")h 1" ; "2 (0; 1); (3.9) respectively, if we use p =q=2 with suitable and suciently large q. Using (3.2) and (3.3), and keeping in mind that k ; k 1; are iid Gaussian with mean zero and variance =2, EjU(t +h;x)U(t;x)j 2 = 2 X k1 k 4 e 2k 2 t (1e k 2 h ) 2 sin 2 (kx); (3.10) EjU x (t +h;x)U x (t;x)j 2 = 2 X k1 k 2 e 2k 2 t (1e k 2 h ) 2 cos 2 (kx); (3.11) EjU x (t;x +h)U x (t;x)j 2 = 2 X k1 k 2 (1e k 2 h ) 2 cos(k(x +h)) cos(kx) 2 : (3.12) 29 We also use 1e ; 0< 1; > 0; (3.13) cos ; 0< 1; > 0: (3.14) Then Inequality (3.7) follows from (7.21), (3.10), and (3.13) by taking < 3=4; Inequality (3.8) follows from (7.21), (3.11), and (3.13) with < 1=4; Inequality (3.9) follows from (7.21), (3.12), and (3.14) with < 1=2. 30 Chapter 4 Stochastic PAM in the It ^ o Interpretation Let us try to follow the It^ o stochastic integral to interpret the stochastic integral (2.3) and look for a mild solution of u t (t;x) =u xx (t;x) +u(t;x) _ W (x); t> 0; 0<x<; u(t; 0) =u(t;) = 0; u(0;x) =u 0 (x): (4.1) Assume that there exists a spatially adapted mild solution satisfying the equation almost surely in the It^ o sense u(t;x) = Z 0 P D (t;x;y)u 0 (y)dy + Z t 0 Z 0 P D (ts;x;y)u(s;y)dW (y)ds; (4.2) whereP D (t;x;y) is the Dirichlet heat kernel. However, it does not quite make sense: On the one hand, the left-hand side of (4.2) is in the ltration up to the space x sinceu is spatially adapted. On the other hand, the right-hand side cannot be in the same ltration with the left-hand side because the right-hand side needs to integrateu over the whole spatial domain 31 [0;] in the Duhamel formula (4.2). We therefore introduce a new concept of mild solution to resolve the issue. 4.1 An Adapted It^ o Solution Denition 30. We say that u F is a forwardF-adapted mild solution of (4.1) if u F (t;) isF-adapted for each t> 0, lim t!0 +u F (t;x) =u 0 (x), u F (t; 0) =u F (t;) = 0, u F satises the equation a.s. u F (t;x) = Z 0 P D (t;x;y)u 0 (y)dy +E Z 0 Z t 0 P D (ts;x;y)u F (s;y)dsdW (y) F x ; (4.3) where P D (t;x;y) = 1 X k=1 e k 2 t m k (x)m k (y); m k (x) = r 2 sin(kx), is the Dirichlet heat kernel. Denition 31. Let p> 0 be given. We say that a random process u is L p;1 (F) - bounded if u isF-adapted in space and sup t2[0;T ]; x2[0;] Eju(t;x)j p <1: We say that u is L p;1 (F 0 )- bounded if u is deterministic orF 0 -adapted and sup t2[0;T ]; x2[0;] ju(t;x)j p <1: 32 Theorem 32. Assume that u 0 is L 2;1 (F 0 )- bounded. Then, there exists a unique L 2;1 (F)- bounded mild solution u F of (4.1). Proof. (Uniqueness) Assume that u F and u 0 F are L 2;1 (F)-bounded mild solutions of (4.1). Dene v =u F u 0 F and f(t) = sup r2[0;t];x2[0;] E v(r;x) 2 : Then, we observe that f is non-decreasing and f <1 uniformly in t2 [0;T ] since u F and u 0 F are L 2;1 (F)-bounded. By the It^ o isometry, f(t) = sup r2[0;t]; x2[0;] E v(r;x) 2 = sup r2[0;t]; x2[0;] E Z 0 Z r 0 P D (rs;x;y) [u F (y;s)u 0 F (y;s)]ds 2 dy: By H older inequality, f(t) T sup r2[0;t]; x2[0;] Z 0 Z r 0 (P D ) 2 (rs;x;y)E [u F (y;s)u 0 F (y;s)] 2 dsdy = T sup r2[0;t];x2[0;] Z r 0 Z 0 (P D ) 2 (rs;x;y)E [u F (y;s)u 0 F (y;s)] 2 dyds: Note that for any 0<t<T , P D (t;x;y) C T p 4t e jxyj 2 =4t ; 33 for some C T > 0. Recall that P (t;x) = 1 p 4t e x 2 =4t is the Gaussian heat kernel. Therefore, for any 0<tT , f(t) sup r2[0;t]; x2[0;] C 2 T T Z r 0 Z 0 P 2 (rs;xy)E [u F (y;s)u 0 F (y;s)] 2 dyds sup r2[0;t] C 2 T T p 8 Z r 0 f(s) p rs ds; integrated in y C 2 T T p 8 Z t 0 f(s) p ts ds; since f is increasing: We iterate the argument to have f(t) C 2 T T p 8 2 Z t 0 Z s 0 f(u) p (su)(ts) duds = C 2 T T p 8 2 Z t 0 f(u) Z t u 1 p (su)(ts) dsdu = C 4 T T 2 8 Z t 0 f(u)du: The last equality comes from the fact that the integral overs is equal to. By iterating this process n times, we get f(t) C 4 T T 2 8 n+1 1 n! Z t 0 f(u)(tu) n du f(t) C 4 T T 2 8 n+1 1 n! Z t 0 (tu) n du = f(t)t n+1 C 4 T T 2 8 n+1 1 (n + 1)! f(t) C 4n+4 T T 3n+3 8 n+1 1 (n + 1)! : 34 We know that f(u) <1. Now, we suppose that f(t)6= 0 for some 0 t T . Then, by cancelling the term f(t) in the both sides, we get 1 C 4n+4 T T 3n+3 8 n+1 1 (n + 1)! : If we send n to innity, we see a contradiction. Thus, f(t) = 0 for all 0 t T . Hence, u F u 0 F . (Existence) We will use the xed point argument (or Picard iteration). Let u 0 F (t;x) 0 and u n+1 F (t;x) = R 0 P D (t;x;y)u 0 (y)dy +E h R 0 R t 0 P D (ts;x;y)u n F (s;y)dsdW (y) F x i : We dene u n F (t;x) :=u n+1 F (t;x)u n F (t;x) and then u n+1 F (t;x) =E Z 0 Z t 0 P D (ts;x;y) u n F (s;y)dsdW (y) F x : Denote f n (t) = sup r2[0;t]; x2[0;] E ( u n F (r;x)) 2 : 35 As before, we have f n+1 (t) = sup r2[0;t];x2[0;] E h u n+1 F (r;x) 2 i = sup r2[0;t]; x2[0;] E " E Z 0 Z t 0 P D (rs;x;y) u n F (s;y)dsdW (y) F x 2 # sup r2[0;t]; x2[0;] E " E " Z 0 Z t 0 P D (rs;x;y) u n F (s;y)dsdW (y) 2 F x ## = sup r2[0;t]; x2[0;] E " Z 0 Z t 0 P D (rs;x;y) u n F (s;y)dsdW (y) 2 # : By the It^ o isometry, we deduce that = sup r2[0;t]; x2[0;] E Z 0 Z r 0 P D (rs;x;y)u n F (s;y)ds 2 dy sup r2[0;t]; x2[0;] T Z 0 Z r 0 (P D ) 2 (rs;x;y)E [u n F (s;y)] 2 dsdy sup r2[0;t]; x2[0;] C 2 T T Z 0 Z r 0 P 2 (rs;xy)E [u n F (s;y)] 2 dsdy sup r2[0;t] C 2 T T p 8 Z r 0 f n (s) p rs ds: We next employ the same argument as above. Then, it follows that for any 0<tT , C 2 T T p 8 Z t 0 f n (s) p ts ds C 4 T T 2 8 n+1 1 n! Z t 0 f 0 (u)(tu) n du = f 0 (t) C 4 T T 2 8 n+1 t n+1 (n + 1)! = f 0 (T ) C 4 T T 2 8 n+1 T n+1 (n + 1)! : 36 This goes to 0 asn!1. Thus,u n F is a Cauchy sequence inL 2;1 (F). There exists the xed point by the xed point argument, which is the adapted mild solution of (4.1). Corollary 33. If u 0 is L p;1 (F 0 ) -bounded for some p 2, then the adapted mild solution u F is also L p;1 (F) -bounded on [0;T ] [0;]. Proof. Apply the BDG inequality. Remark 34. Using the backward ltration, we may construct a backward adapted mild so- lution of (4.1). Remark 35. The \adapted" solution is an interesting mathematical object, but it is some- what physically articial. We may interpret the solution of (4.1) as the variation in tempera- ture of a rod with length over time when the initial distribution of heat is given and random things happen irregularly and independently at each point of the rod (i.e., the environment of the rod is random). Then, the adapted solution u F corresponds to \adapted temperature" and the formula for the adapted temperature with the zero temperature at the boundaries is given by u F (t;x) = Z 0 P D (t;x;y)u 0 (y)dy +E Z 0 Z t 0 P D (ts;x;y)u F (s;y)dsdW (y) F x : The equation implies that the temperature at the positionx is in uenced by the whole region of the rod, and we have all information on the environment up to the regionx (orfW (y)g 0yx ). However, for some reason, we are uncertain for the environment from x to , which is not natural. 37 Chapter 5 Stochastic PAM in Stratonovich Interpretation I LetW =W (x) be a continuous function on [0;], and denote by _ W the generalized derivative of W : Z 0 W (x)h 0 (x)dx = Z 0 _ W (x)h(x)dx for every continuously dierentiable h with compact support in (0;). By direct computa- tions, _ W is a generalized function from the closure of L 2 (0;) with respect to the norm khk 2 1 = X k1 h 2 k k 2 ; h k = r 2 Z 0 h(x) sin(kx)dx: The objective of this chapter is investigation of the equation @u(t;x) @t = @ 2 u(t;x) @x 2 +u(t;x) _ W (x); 0<t<T; 0<x<; u(0;x) ='(x); u(t; 0) =u(t;) = 0: (5.1) If _ W were a H older continuous function of order 2 (0; 1), then standard parabolic regularity would imply that the classical solution of (5.1) is H older 2 + in space and 38 H older 1 + (=2) in time; cf. [15, Theorem 10.4.1]. If W is H older , then it might be natural to expect for the solution of (5.1) to lose one derivative in space and \one-half of the derivative" in time, that is, to become H older 1 + in space and H older (1 +)=2 in time. The objective of the paper is to establish this result in a rigorous way. The starting point must be interpretation of the product u _ W , which we do using the ideas from the rough path theory. Here is an outline of the ideas. If h = h(x) is a continuously dierentiable function with h(0) = 0, then, by the funda- mental theorem of calculus, Z x 0 h(s)h 0 (s)ds = h 2 (x) 2 ; (5.2) for a continuous functionW =W (x), the integral R x 0 W (s) _ W (s)ds is, in general, not dened. An extension of the Riemann-Stieltjes construction, due to Young, is possible when W is H older continuous of order bigger than 1=2; then (5.2) continues to hold as long asW (0) = 0. For less regular functions W , the value of Z x 0 W (s) _ W (s)ds = Z x 0 Z s 0 _ W (x 1 )dx 1 _ W (s)ds and possibly higher order iterated integrals Z x 0 Z x 1 0 Z x 2 0 _ W (s)ds _ W (x 2 )dx 2 _ W (x 1 )dx 1 ; 39 etc. must be postulated, which is the subject of the rough path theory. The collection of all such iterated integrals, known as the signature of the rough path W , is encoded in the solution of the ordinary dierential equation dY (x) dx =Y (x) _ W (x); x> 0; (5.3) One particular case of the rough path construction stipulates that the traditional rules of calculus, such as (5.2), continue to hold. That is, given a continuous functionW =W (x); x 0, the solution of the ordinary dierential equation is dened to be Y (x) =Y (0)e W (x)W (0) ; (5.4) in what follows we refer to this as the geometric rough path (GRP) solution of (5.3). The main consequence of (5.4) is that if W (") = W (") (x), " > 0; is a sequence of continuously dierentiable functions such that lim "!0 sup x2(0;L) jW (x)W (") (x)j = 0, then lim "!0 sup x2(0;L) jY (x)Y (") (x)j = 0; where Y (") (x) =Y (0)e W (") (x)W (") (0) ; that is, dY (") (x) dx =Y (") (x) dW (") (x) dx : 40 It is therefore natural to dene a GRP solution of (5.1) as a suitable limit of the solutions of @u (") (t;x) @t = @ 2 u (") (t;x) @x 2 +u (") (t;x) _ W (") (x); 0<t<T; x2 (0;); u (") (t; 0) =u (") (t;) = 0; u (") (0;x) ='(x); (5.5) where'2L 2 (0;) andW (") are absolutely continuous approximations ofW so thatW (") (x) = R x 0 _ W (") (s)ds and _ W " 2L 1 (0;). Without further regularity assumptions on' and _ W (") , equation (5.5) can only be inter- preted in variational form, leading to a generalized solution; under additional assumptions about ' and _ W (") , equation (5.5) can have a classical solution. Accordingly, this chapter denes and investigates the two possible GRP solutions of (5.1), classical (Section 5.1) and generalized (Section 5.2). The last section discusses the fundamental solution of (5.1). Here is the summary of the main notations used in the paper: h 0 = h 0 (x) denotes the usual derivative; _ W denotes the generalized derivative; u t ,u x ,u xx denote the corresponding partial derivatives (classical or generalized); to shorten the notations, the interval (0;) is sometimes denoted by G. 5.1 The Classical Geometric Rough Path Solution Denition 36. The classical solution of equation U t (t;x) =U xx (t;x) +b(t;x)U x (t;x) +c(t;x)U(t;x) +f(t;x); x2 (0;); t2 (0;T ); U(0;x) ='(x); U(t; 0) =U(t;) = 0; (5.6) 41 is a function U =U(t;x) with the following properties: for every t2 (0;T ) and x2 (0;), the function U is continuously dierentiable in t, twice continuously dierentiable in x, and (5.6) holds; for every t2 (0;T ), U(t; 0) =U(t;) = 0; for every x2 [0;], lim t!0+ U(t;x) ='(x). The following result is well known; cf. [15, Theorem 10.4.1]. Proposition 37. If '2C(0;), '(0) = '() = 0, and b;c;f 2C =2; (0;T ) (0;) for some 2 (0; 1), then equation (5.6) has a unique classical solution. If in addition '2 C 2+ (0;), then U2C 1+=2;2+ (0;T ) (0;) . The next result makes it possible to dene the classical geometric rough path solution of equation (5.1). Theorem 38. Assume that, for some ;2 (0; 1), W (") 2C 1+ (0;) for all " > 0, W 2 C (0;), and '2C 1+ (0;). If lim "!0 kWW (") k L1(0;) = 0; then there exists a function u2C (1+)=2;1+ (0;T ) (0;) such that lim "!0 kuu (") k L1 (0;T )(0;) = 0: Note that there is no connection between and in the conditions of the theorem. 42 Denition 39. The function u from Theorem 38 is called the classical GRP (geometric rough path) solution of equation (5.1). Proof of Theorem 38. Letu (") =u (") (t;x) be the classical solution of (5.5). Dene the functions H (") W (x) = exp Z x 0 W (") (y)dy ; v (") (t;x) =u (") (t;x)H (") W (x): By direct computation, @v (") (t;x) @t = @ 2 v (") (t;x) @x 2 2W (") (x) @v (") (t;x) @x + W (") (x) 2 v (") (t;x); t> 0; x2 (0;); v (") (t; 0) =v (") (t;) = 0; v (") (0;x) ='(x)H (") W (x): (5.7) Dene H W (x) = exp Z x 0 W (y)dy (5.8) and let v =v(t;x) be the classical solution of @v(t;x) @t = @ 2 v(t;x) @x 2 2W (x) @v(t;x) @x +W 2 (x)v(t;x); t> 0; x2 (0;); v(t; 0) =v(t;) = 0; v(0;x) ='(x)H W (x): (5.9) Let V (") (t;x) =v(t;x)v (") (t;x). We write the equation for V (") as V (") t = e A W V (") +V (") W 2 +F (") V (") (0;x) ='(x) H W (x)H (") W (x) ; (5.10) 43 where e A W is the operator h7!h 00 2Wh 0 with zero Dirichlet boundary conditions on G = (0;), and F (") (t;x) = 2v (") x (t;x) W (x)W (") (x) +v (") (t;x) W 2 (x) W (") (x) 2 : Denote by e t ; t> 0; the semigroup generated by the operator e A W . Then (8.6) becomes V (") (t;x) = e t [V (") (0;)](x) + Z t 0 e ts [V (") (s;)W 2 ()](x)ds + Z t 0 e ts [F (") (s;)](x)ds: (5.11) The maximum principle for the operator e A W implies k e t hk L1(G) khk L1(G) ; t> 0; (5.12) similarly, sup 0<t<T k( e t h) x k L1(G) C 1 T;kWk L1(G) khk L1(G) +kh x k L1(G) ; (5.13) cf. [15, Sections 8.2, 8.3]. By assumption, there exists a positive number C 0 such that kWk L1(G) C 0 ;kW (") k L1(G) C 0 44 for all "> 0. Dene (") W =kW (") Wk L1(G) : Then (5.7), (5.12), (5.13), and Gronwall's inequality imply sup 0<t<T kv (") x k L1(G) (t) + sup 0<t<T kv (") k L1(G) (t)C 2 k'k L1(G) ; with C 2 depending only on C 0 and T . Also, by the intermediate value theorem, kV (") k L1(G) (0)k'k L1(G) e 2C 0 (") W : Then we deduce from (5.11) that sup 0<t<T kV (") k L1(G) (t)C 3 (") W ; with C 3 depending only on C 0 ;T; and '. It remains to note that u = v H W ; (5.14) juu (") j jV (") j H W + jH W H (") W j H W H (") W jv (") j and then, with C 4 depending only on C 0 ;T and ', sup t2(0;T );x2G ju(t;x)u (") (t;x)jC 4 kWW (") k L1(G) ; (5.15) 45 completing the proof of the theorem. The following is the main result about the classical GRP solution of (5.1). Theorem 40. If W 2C (0;), '2C 1+ (0;), and '(0) = '() = 0, then (5.1) has a unique GRP solution given by u(t;x) = t [H W '](x) H W (x) ; where t is the semigroup of the operator A W :h(x)7!h 00 (x) 2W (x)h 0 (x) +W 2 (x)h(x) with zero boundary conditions on (0;) and H W is from (5.8). The solution belongs to the parabolic H older space C (1+)=2;1+ (0;T ) (0;) and is an innitely dierentiable function oft fort> 0. If, in addition,'(x) = (x)=H W (x) for some 2C 2+ (0;), then u2C 1+(=2);1+ (0;T ) (0;) : Proof. This is a direct consequence of (5.14) and Proposition 37. The solution is a smooth function of t for t > 0 because the coecients of the operator A W do not depend on time (cf. [15, Theorem 8.2.1]), whereasC regularity of the coecients of A W implies that the function t ['](x) cannot be better thanC 2+ in space. Because H W 2C 1+ (0;), a typical initial condition cannot be better thanC 1+ (0;) and similarly, the solution cannot be more regular in space thanC 1+ (0;). If, with a special choice of the initial condition, we ensure 46 that'H W 2C 2+ (0;), then, by Proposition 37, we get better time regularity of the solution near t = 0. Remark 41. If W is a sample trajectory of the standard Brownian motion, then, with probability one, W 2C 1=2 (0;), that it, W is H older continuous of every order less than 1=2. By Theorem 40, with< 1=2, we conclude that, with probability one, a typical classical GRP solution of the corresponding equation (5.1) isC 3=4 in time andC 3=2 in space; with a special choice of the initial condition, it is possible to achieveC 5=4 regularity in time. Similarly, if W =B H is fractional Brownian motion with Hurst parameter H2 (0; 1), then a typical classical GRP solution of (5.1) isC (1+H)=2 in time andC 1+H in space. Inequality (5.15) provides the rate of convergence of the approximate solutions u (") tou; this rate depends on the particular approximation W (") of W . As an example, consider W (") (x) = 1 " Z 0 xy " W (y)dy; where 0; 2C 1 (R); (x) = 0; x = 2 (;); Z 1 0 (x)dx = 1; so that W (") 2C 1 (0;). By direct computation, if W2C (0;), then jW (x)W (") (x)j" sup x6=y jW (x)W (y)j jxyj ; 47 and so sup t;x ju(t;x)u (") (t;x)j ~ C 4 " : With some extra eort, similar arguments show that, for every 2 (0;), kuu (") k C (1+ )=2;1+ (0;T )(0;) C 5 " : 5.2 The Generalized Geometric Rough Path Solution The classical solution of (5.5) might not exist if the initial condition ' is not a continuous function, or if the functions _ W (") are not H older continuous, or if condition'(0) ='() = 0 does not hold. The generalized solution is an extension of the classical solution using the idea of integration by parts. To simplify the notations, we write G = (0;); (g;h) 0 = Z 0 g(x)h(x)dx;khk 2 0 = (h;h) 0 : Next, we need an overview of the Sobolev space on G. The space H 1 0 (G) is the closure of the set of smooth functions with compact support in G with respect to the normkk 1 , where khk 2 1 =khk 2 0 +kh 0 k 2 0 : 48 The space H 1 (G), mentioned in the introduction, is a separable Hilbert space with norm kk 1 and is the dual ofH 1 0 (G) relative to the inner product (;) 0 ; the corresponding duality will be denoted by [;] 0 . Denition 42. Given bounded measurable functionsa;b;c and a generalized functionf from L 2 (0;T );H 1 (G) , the generalized solution of equation v t = (v x +av) x +bv x +cv +f; t> 0; x2G; (5.16) with initial condition vj t=0 = '2 L 2 (G) and zero Dirichlet boundary conditions vj x=0 = vj x= = 0 is an element of L 2 (0;T );H 1 0 (G) such that, for every h2H 1 0 (G) and t2 (0;T ), v;h 0 (t) = ';h 0 Z t 0 v x +av;h 0 0 (s)ds + Z t 0 bv x +cv;h 0 (s)ds + Z t 0 [f;h] 0 (s)ds: (5.17) The following result is well known; cf. [17, Theorem III.4.1]. Proposition 43. Let a =a(t;x); b =b(t;x); c =c(t;x) be bounded measurable functions, '2L 2 (G); f2L 2 (0;T );H 1 (G) : Then equation (5.16) has a unique generalized solution and v2L 2 (0;T );H 1 0 (G) \ C (0;T );L 2 (G) ; sup t2(0;T ) kvk 2 0 (t) + Z T 0 kvk 2 1 (s)dsC k'k 2 0 + Z T 0 kfk 2 1 (s)ds ; 49 with C depending only on T and the L 1 norms of a;b;c. If W (") is continuously dierentiable and u (") is a classical solution of (5.5), then we can multiply both sides by a continuously dierentiable function h with compact support in G and integrate with respect to t;x to get, after integration by parts, u (") ;h 0 (t) = ';h 0 Z t 0 u (") x +u (") W (") ;h 0 0 (s)ds Z t 0 u (") x W (") ;h 0 (s)ds: (5.18) Comparing (5.17) and (5.18), we conclude that u (") is a generalized solution of u (") t = (u (") x +u (") W (") ) x u (") x W (") : (5.19) If we now pass to the limit "! 0 in (5.19) and assume that all the limits exist and all the equalities continue to hold, then we get the equation u t = (u x +uW ) x u x W; (5.20) which is a particular case of (5.16). Before conrming that this passage to the limit is indeed justied, let us use this non-rigorous argument to dene the generalized GRP solution of (5.1). Denition 44. The generalized geometric rough path (GRP) solution of (5.1) is the gener- alized solution of (5.20). The next result is an immediate consequence of Proposition 43. 50 Theorem 45. If W2L 1 (0;) and '2L 2 (0;), then (5.1) has a unique generalized GRP solution u and u2L 2 (0;T );H 1 0 (0;) \ C (0;T );L 2 (0;) ; sup t2(0;T ) kuk 2 0 + Z T 0 kuk 2 1 (s)dsCk'k 2 0 ; with C depending only on T andkWk L1(0;) . Note that generalized GRP solution may exist even when W is not continuous. It remains to conrm that passing to the limit "! 0 in (5.19) is justied and indeed leads to (5.20). Theorem 46. Assume that W (") 2L 1 (G) for all "> 0, W2L 1 (G), and '2L 2 (G). Let u (") be the generalized solution of (5.18) and let u be the generalized solution of (5.20). If lim "!0 kWW (") k L1(G) = 0; then lim "!0 sup t2(0;T ) kuu (") k 2 0 (t) + Z T 0 kuu (") k 2 1 (s)ds = 0: Proof. LetU (") =uu (") . Then (5.19) and (5.20) imply thatU (") is the generalized solution of U (") t = (U (") x +U (") W ) x U (") x W + (u (") u (") x ) (WW (") ); Uj t=0 = 0: 51 Recall that C 0 denotes the common upper bound on the L 1 norms of W and W (") . By Proposition 43, sup t2(0;T ) kU (") k 2 0 (t) + Z T 0 kU (") k 2 1 (s)ds C 6 (T;C 0 )kWW (") k L1(G) Z T 0 ku (") k 2 1 (t)dt; and Z T 0 ku (") k 2 1 (t)dtC 7 (T;C 0 )k'k 2 0 ; completing the proof. Remark 47. By construction, a classical solution of (5.5) is automatically a generalized solution. Then Theorems 40 and 46 imply that a classical GRP solution of (5.1), if exists, coincides with the generalized GRP solution. 5.3 The Fundamental Solution We will now construct the fundamental GRP solution of equation (5.1). Let L : H 1 0 (G)! H 1 (G) be the operator h7! h x +hW x +h x W: Then (5.20) becomes u t =Lu: It is known [27] that, for every W2L 1 (0;), The operator L has discrete pure point spectrum; 52 The eigenvalues k ;k 1; satisfy lim k!1 k k 2 = 1; The corresponding eigenfunctions m k belong to H 1 0 (0;); The collectionfm k ; k 1g can be chosen to form an orthonormal basis in L 2 (0;). Then, using the functions m k in the denition of the generalized solution of (5.20), we conclude by Theorem 45 that u(t;x) = 1 X k=1 e k t (';m k ) 0 m k (x): (5.21) Equality (5.21) suggests calling the function P(t;x;y) = 1 X k=1 e k t m k (x)m k (y) the fundamental GRP solution of (5.1). By Theorem 40, P(t;x;y) =e R x 0 W (s)ds P W (t;x;y)e R y 0 W (s)ds ; whereP W is the fundamental solution of (5.9). 53 Chapter 6 Stochastic PAM in Stratonovich Interpretation II The objective of this chapter is to give a dierent point of view on the equation (5.1): We will dene a solution, called a mild solution in a variational form. At the end, we see that the two solutions dened in Section 5.1 and Section 6.2 ultimately coincide under proper assumptions. 6.1 The Schauder Estimates In this section, we establish a modied version of the Schauder estimates for parabolic type on an interval in some spaces. We start with the H older spaces onR d . Denote by @ @z i the dierentiation operator with respect to z i , and for a multi-index = ( 1 ; ; d ) with i 2N 0 andjj = P d i=1 i <1, denote @ z = @ 1 @z 1 1 @ d @z d d : 54 Let G be a domain inR d . For 0< < 1, let [u] := sup z6=y2G ju(z)u(y)j jzyj : We say that u is H older continuous with H older exponent (or H older continuous) on G if sup z2G ju(z)j + [u] <1: The collection of H older continuous functions on G is denoted byC (G) with the norm kuk :=kuk L1(G) + [u] ; where kuk L1(G) := sup z2G ju(z)j: We say that u is a k times continuously dierentiable function on G if @ z u exists and is continuous for alljjk. The collection of k times continuously dierentiable functions on G such that @ z u2C (G);jj =k is denoted byC k+ (G) with the norm kuk k+ := X jjk sup z2G j@ z u(z)j + X jj=k [@ z u] <1: 55 In particular, due to the presence of time and space variables, we often write C k 1 + 1 ;k 2 + 2 t;x ((0;T )G) with the norm kuk k 1 + 1 ;k 2 + 2 := X nk 1 jjk 2 sup t2(0;T ); x2G j@ n t @ x u(t;x)j + @ k 1 t u 1 + X jj=k 2 [@ x u] 2 <1: We restrict the space domain by G = (0;) and let us denote the Dirichlet heat kernel on (0;) by P D (t;x;y) = 1 X k=1 e k 2 t m k (x)m k (y); m k (x) = r 2 sin(kx); k 1: Dene a convolution ? for a function f by P D ?f (t;x) = Z t 0 Z 0 P D (ts;x;y)f(s;y)dyds: Let 0 < = 2N and T > 0 be given. The classical parabolic type of Schauder's estimate in H older spaces (Theorem 5.2 of Chapter IV in [17]) says that the convolution mapping f7!P D ?f 56 is continuous from C =2; t;x ((0;T ) (0;)) to C 1+ =2;2+ t;x ((0;T ) (0;)). More specically, there exists a Schauder constant C T > 0 such that C T remains bounded as T! 0 and P D ?f 1+ =2;2+ C T kfk =2; : Unfortunately, the classical Schauder constant C T does not give a good estimate for our purpose. We now give a relaxed version of Schauder's estimate in fractional Sobolev spaces instead of in H older spaces. The modied results will be useful for the existence of mild solution in the next section. Denote by H s p the fractional Sobolev space as the collection of all functions f such that () s=2 f Lp(0;) <1; where () s=2 f = 1 (s=2) Z 1 0 (e t I)f dt t 1+ s 2 ; is the gamma function, ande t is the semigroup of Laplacian operator on (0;) with zero boundary conditions. Dene := () 1=2 : Lemma 48. Let h2L p (0;), p 1 and let 0< 2. For any 0<tT , we have Z 0 P D (t;;y)h(y)dy Lp(0;) C(T;;p)t =2 khk Lp(0;) 57 for some C(T;;p)> 0 depending on T , and p. Proof. From [16, Lemma 7.3], it is enough to show that @ t Z 0 P D (t;;y)h(y)dy Lp(0;) C t khk Lp(0;) (6.1) for some C > 0 depending only on p. Indeed, (6.1) follows from the fact @ t e t h Lp(0;) = @ t Z 0 P D (t;;y)h(y)dy Lp(0;) = Z 0 @ t P D (t;;y)h(y)dy Lp(0;) and [2, Lemma 2.5]. Theorem 49. Let 0< < < 1 and T > 0 be given. For p 1, the convolution mapping f7!P D ?f is continuous from L 1 0;T ;H p (0;) to L 1 0;T ;H 2+ p (0;) . In particular, kP D ?fk L1(0;T ;H 2+ p (0;)) CT ( )=2 kfk L1(0;T ;H p (0;)) (6.2) for some constant C > 0 depending only on and . Proof. Let 0< < < 1 be given. By the triangle inequality, we have P D ?f(t;) H 2+ p (0;) = Z t 0 Z 0 P D (ts;;y)f(s;y)dyds H 2+ p (0;) Z t 0 Z 0 P D (ts;;y)f(s;y)dy H 2+ p (0;) ds: 58 For each 0 < s < t T and 0 < < 2, there exists a constant C 0 := C(;T;p) > 0 such that Z 0 P D (ts;;y)f(s;y)dy Lp(0;) C 0 (ts) =2 kf(s;)k Lp(0;) (6.3) by Lemma 48. Therefore, by the commutativity of , P D ?f(t;) H 2+ p (0;) C 0 Z t 0 (ts) (2+ )=2 kf(s;)k H p (0;) ds C 0 kfk L1(0;T ;H p (0;)) Z t 0 (ts) (2+ )=2 ds C 1 kfk L1(0;T ;H p (0;)) T ( )=2 ; completing the proof. Denote the Neumann heat kernel on (0;) by P N (t;x;y) = 1 X k=0 e k 2 t ~ m k (x) ~ m k (y); ~ m 0 (x) = 1 p ; ~ m k (x) = r 2 cos(kx); k 1: Corollary 50. Under the same assumptions of Theorem 49, the convolution map f7!P N ?f is continuous from L 1 0;T ;H p (0;) to L 1 0;T ;H 2+ p (0;) . In other words, P N ?f L1(0;T ;H 2+ p (0;)) CT ( )=2 kfk L1(0;T ;H p (0;)) (6.4) for some constant C > 0 depending only on and . 59 6.2 The Mild Solution and its Regularity LetfW (x)g x2[0;] be a standard Brownian motion on a probability space ( ;F;P). Here,F denotes the ltration generated by W . It is known that Brownian motion is inC (0;) for any 0< < 1=2. Let W " be smooth approximations of W such that kW " Wk C (0;) ! 0 as "! 0: Consider the approximated equations of (5.1) for "> 0: @u " (t;x) @t = @ 2 u " (t;x) @x 2 +u " (t;x) @ @x W " (x); t> 0; 0<x<; u " (t; 0) =u " (t;) = 0; u " (0;x) =u 0 (x): (6.5) Since W " is smooth, the equation (6.5) has the classical solution u " . Denote P 0 (t;x) = 8 > > < > > : u 0 (x) if t = 0 Z 0 P D (t;x;y)u 0 (y)dy if t> 0: Note that as t! 0 + , we see that P 0 (t;x)! u 0 (x) for each x2 [0;]. Then, the mild formulation for the equation (6.5) is given by u " (t;x) = P 0 (t;x) + Z t 0 Z 0 P D (ts;x;y)u " (s;y)@ y W " (y)dyds: (6.6) 60 By integration by parts with respect to y in the last term above, we rewrite (6.6) as u " (t;x) = P 0 (t;x) Z t 0 Z 0 P D y (ts;x;y)u " (s;y)W " (y)dyds Z t 0 Z 0 P D (ts;x;y)u " y (s;y)W " (y)dyds: (6.7) Denition 51. We say that u is a mild solution of (5.1) if for every 0<t<T and 0<x<, u is continuous in t, continuously dierentiable in x, and it satises the equation u(t;x) = P 0 (t;x) Z t 0 Z 0 P D y (ts;x;y)u(s;y)W (y)dyds Z t 0 Z 0 P D (ts;x;y)u y (s;y)W (y)dyds (6.8) for every 0tT , u(t; 0) =u(t;) = 0; for every 0x, lim t!0 +u(t;x) =u 0 (x). For the existence and the uniqueness of mild solution, we use a contraction mapping (or xed point argument) on L 1 0;T ;H 1+ p (0;) with 0< < 1 and p 1. Dene a map M :L 1 0;T ;H 1+ p (0;) !L 1 0;T ;H 1+ p (0;) by (Mu) (t;x) = P 0 (t;x) Z t 0 Z 0 P D y (ts;x;y)u(s;y)W (y)dyds Z t 0 Z 0 P D (ts;x;y)u y (s;y)W (y)dyds: (6.9) 61 We rst prove the well-posedness of (6.9). Lemma 52. Let 0< < < 1. If u 0 2H p (0;) with p 1, kP 0 k L1(0;T ;H 1+ p (0;)) <1: Proof. It is clear from Lemma 48. Theorem 53. Let 0 < < < 1=2. If u 0 2 H p (0;) with p 1, the mappingM on L 1 0;T ;H 1+ p (0;) is well-dened. Also, there exists a xed point ofM. That is, the xed point is the unique mild solution of (5.1). Proof. Since P D y (t;x;y) =P N x (t;x;y) for each t> 0 and x;y2 [0;], we have Z t 0 Z 0 P D y (ts;x;y)u(s;y)W (y)dyds = Z t 0 Z 0 P N x (ts;x;y)u(s;y)W (y)dyds: Then, we rewrite (6.9) by (Mu) (t;x) = P 0 (t;x) +P N x ? (uW )(t;x)P D ? (u x W )(t;x): We show the well-denedness term by term. By Lemma 52, P 0 2L 1 (0;T ;H 1+ p (0;)): 62 Fix t2 (0;T ). We have P N x ?uW (t;) H 1+ p (0;) = @ x P N ?uW (t;) H 1+ p (0;) P N ? (uW )(t;) H 2+ p (0;) : Then, by Corollary 50 and [16, Lemma 5.2], P N x ?uW L1(0;T ;H 1+ p (0;)) P N ?uW L1(0;T ;H 2+ p (0;)) C 1 T ( )=2 kWk C (0;) kuk L1(0;T ;H p (0;)) C 2 T ( )=2 kWk C (0;) kuk L1(0;T ;H 1+ p (0;)) and similarly by (6.3), kP D ?u x Wk L1(0;T ;H 1+ p (0;)) C 3 T 1=2 kWk C (0;) kuk L1(0;T ;H 1+ p (0;)) : Let u;v2L 1 0;T ;H 1+ p (0;) . Then, k (MuMv)k L1(0;T ;H 1+ p (0;)) P N x ? (uv)W L1(0;T ;H 1+ p (0;)) +kP D ? (u x v x )Wk L1(0;T ;H 1+ p (0;)) CT 1=2 kWk C (0;) kuvk L1(0;T ;H 1+ p (0;)) for some C > 0. Choose > 0 such that C 1=2 kWk C (0;) < 1. Then, clearly there exists a xed point ofM up to . Consider a time partition 0 = t 0 < < t n = T such that 63 t i+1 t i for i = 0; ;n 1. We now dene u recursively on (t i ;t i+1 ] with the initial condition u(t i ;x) for i = 0; ;n 1: (Mu)(t;x) = Z 0 P D (t i+1 ;x;y)u(t i ;y)dy Z t i+1 0 Z 0 P D y (t i+1 s;x;y)u(s;y)W (y)dyds Z t i+1 0 Z 0 P D (t i+1 s;x;y)u y (s;y)W (y)dyds: (6.10) The existence of the xed point to (6.10) is guaranteed by the above arguments since t i+1 t i . Since n is nite, we obtain the xed point solution over the whole time interval (0;T ). We note that the xed point solution satises the mild formulation (6.8). Since the xed point solution is unique, the uniqueness of mild solution clearly holds. Remark 54. From Theorem 53, we have that for all t2 [0;T ], u(t;)2H 1+ p (0;); p 1: By the Sobolev embedding theorem, we have H 1+ p (0;)C 1+1=p (0;) (6.11) for any p 1. This shows that the mild solution u is indeed almost H older 3/2 continuous in space. We show the mild solution of (5.1) is the limit of classical solutions u " of (6.5) in L 1 (0;T ;H 1+ p (0;)). 64 Theorem 55. Let 0< < < 1=2. If u 0 2H p (0;), then we have ku " uk L1(0;T ;H 1+ p (0;)) ! 0 as "! 0: Proof. For simplicity, we denotekk C =kk C (0;) and kk =kk L1(0;T ;H 1+ p (0;)) : Then, we can write u " u by P N x ?u " W " P N x ?uWP D ?u " x W " +P D ?u x W: By the triangle inequality, ku " uk P N x ? (u " u)W " + P N x ?u(WW " ) +kP D ? (u " x u x )W " k +kP D ?u x (WW " )k C 0 1 T 1=2 kW " k C ku " uk +C 0 2 T 1=2 kukkWW " k C C 0 1 T 1=2 (kWk C +kWW " k C )ku " uk +C 0 2 T 1=2 kukkWW " k C : Note that the constants C 0 1 ;C 0 2 > 0 are independent of ". Choose > 0 such that C 0 1 1=2 (kWk C +kWW " k C )< 1 65 for small "> 0. Then, sincekWW " k C ! 0; ku " uk! 0 as "! 0: We now consider a time partition 0 = t 0 < < t n = T such that t i+1 t i for i = 0; ;n 1. Finally, we iterate the above argument for u " u recursively on (t i ;t i+1 ] to get ku " (t i ;)u(t i ;)k! 0 as "! 0 for i = 1; ;n 1. Theorem 56. Let 0< < 1=2. If u 0 2C 1+ (0;), then the mild solution of (5.1) is indeed in C (1+ )=2;1+ t;x ((0;T ) (0;)): Proof. Let 0 < < < 1=2. Recall, for each " > 0, the approximated parabolic Anderson model @u " (t;x) @t = @ 2 u " (t;x) @x 2 +u " (t;x) @ @x W " (x); t> 0; 0<x<; u " (t; 0) =u " (t;) = 0; u " (0;x) =u 0 (x): From the classical parabolic theory, there exists the unique classical solution u " . We note that by Theorem 55, u " converges to the limit u in L 1 (0;T ;H 1+ p (0;)) 66 sinceC 1+ (0;)H p (0;) for any p 1: On the other hand, let v " satisfy the equation @v " (t;x) @t = @ 2 v " (t;x) @x 2 2W " @ @x v " (t;x) + (W " ) 2 v " (t;x); t> 0; 0<x<; v " (t; 0) =v " (t;) = 0; v " (0;x) =u 0 (x)e R x 0 W " (y)dy : We observe that u " (t;x) =v " (t;x)e R x 0 W " (y)dy : By Theorem 40, u " converges to a limit in C (1+ )=2;1+ t;x ((0;T ) (0;)) and the limit of u " is unique in L 2 ((0;T );H 1 0 (0;)) T L 1 ((0;T );L 2 (0;)) by Theorem 45, where H 1 0 (0;) is the closure of the set of smooth functions with compact support in (0;) with respect to the normkk H 1 2 (0;) . Since, for p 2, L 1 (0;T ;H 1+ p (0;))L 2 (0;T );H 1 0 (0;) \ L 1 ((0;T );L 2 (0;)); the mild solution u of (5.1) is indeed in C (1+ )=2;1+ t;x ((0;T ) (0;)): 67 Remark 57. The reason why we set the upper bound of regularity less than 1=2 in Theorem 53, 55 and 56 is due to the regularity of Brownian motion. In fact, the Brownian motion W can be replaced by any pathwisely H older continuous process with 0< < 1 for Theorem 53, 55 and 56. For example, W can be a standard fractional Brownian motion W H with the Hurst index 0<H < 1. Remark 58. Consider the following equation on the whole lineR @u(t;x) @t = @ 2 u(t;x) @x 2 +u(t;x) @ @x W (x); t> 0; x2R; u(0;x) =u 0 (x); x2R: (6.12) Theorem 53, 55 and 56 will also work on the whole lineR if the H older norm of W onR is bounded with 0< < 1; The modied Schauder estimate result in Theorem 49 is sharper in H older spaces if P D is replaced by the Gaussian heat kernelP (t;xy) onR: for anyf2 (0;T ;C (R)), we have kP ?fk L1(0;T ;C 2+ (R)) CT =2 kfk L1(0;T ;C (R)) ; for some C > 0, which is independent of T . Note that a Brownian motion onR do not have a sample trajectory that has a bounded H older norm. Clearly, P y (t;xy) =P x (t;xy) holds for t> 0. 68 Chapter 7 Stochastic PAM in the Wick-It ^ o-Skorokhod Interpretation The objective of this chapter is to establish optimal space-time regularity of the solution of u t (t;x) =u xx (t;x) +u(t;x) _ W (x); t> 0; 0<x<; u(t; 0) =u(t;) = 0; u(0;x) =u 0 (x); (7.1) and to dene and investigate the corresponding fundamental solution. We show that the solution of (7.1) is almost H older 3/4 continuous in time and is almost H older 3/2 continuous in space. In order to construct a solution of the equation (7.1), one may introduce Wick product based on the Wiener chaos expansion. We use the following notations: T n s;t =f(s 1 ;:::;s n )2R n : s<s 1 <s 2 <<s n <tg; 69 0s<t; n = 1; 2;:::; (g;h) 0 = Z 0 g(x)h(x)dx; kgk 0 = p (g;g) 0 ; g k = (g;m k ) 0 ; wherefm k ; k 1g is an orthonormal basis in L 2 (0;); dx n =dx 1 dx 2 dx n : 7.1 The Chaos Spaces Let ( ;F;P) be a probability space. Recall that a Gaussian white noise _ W on L 2 (0;) is a collection of Gaussian random variables _ W (h); h2L 2 (0;); such that E _ W (g) = 0; E _ W (g) _ W (h) = (g;h) 0 : (7.2) For a Banach space X, denote by L p (W ;X), 1p<1, the collection of random elements that are measurable with respect to the sigma-algebra generated by _ W (h); h2 L 2 (0;); and such thatEkk p X <1. In what follows, we x the Fourier sine basisfm k ; k 1g in L 2 (0;): m k (x) = r 2 sin(kx); (7.3) and dene k = _ W (m k ): (7.4) 70 By (7.2), k ; k 1; are iid standard Gaussian random variables, and _ W (h) = X k1 (m k ;h) 0 k : As a result, _ W (x) = X k1 m k (x) k (7.5) becomes an alternative notation for _ W ; of course, the series in (7.5) diverges in the traditional sense. It follows from (7.2) that W (x) = _ W ( [0;x] ) is a standard Brownian motion on [0;], where [0;x] is the indicator function of the interval [0;x]. Dene the collection of random variables =f ; 2Jg by = Y k H k ( k ) p k ! ; where k is from (7.4) and H n (x) = (1) n e x 2 =2 d n dx n e x 2 =2 (7.6) is the Hermite polynomial of order n. By a theorem of Cameron and Martin [1], is an orthonormal basis inL 2 (W ;X) as long asX is a Hilbert space. Accordingly, in what follows, we always assume that X is a Hilbert space. For 2L 2 (W ;X), dene =E 2X. Then = X 2M ; Ekk 2 X = X 2M k k 2 X : 71 We will often need spaces other than L 2 (W ;X): The space D n 2 (W ;X) = n = X 2M 2L 2 (W ;X) : X 2M jj n k k 2 X <1 o ; n> 0; The space L 2;q (W ;X) = n = X 2M 2L 2 (W ;X) : X 2M q jj k k 2 X <1 o ; q> 1; The spaceL 2;q (W ;X), 0<q< 1, which is the closure ofL 2 (W ;X) with respect to the norm kk L 2;q (X) = X 2M q jj k k 2 X ! 1=2 : It follows that L 2;q 1 (W ;X)L 2;q 2 (W ;X); q 1 >q 2 ; and, for every q> 1, L 2;q (W ;X) \ n>0 D n 2 (W ;X): It is also known [23, Section 1.2] that, for n = 1; 2;:::; the space D n 2 (W ;X) is the domain of D n , the n-th power of the Malliavin derivative. Here is another useful property of the spaces L 2;q (W ;X). 72 Proposition 59. If 1<p<1, and q>p 1, then L 2;q (W ;X)L p (W ;X): Proof. Let 2L 2;q (W ;X). The hypercontractivity property of the Ornstein-Uhlenbeck op- erator [23, Theorem 1.4.1] implies 1 0 @ E X jj=n p X 1 A 1=p (p 1) n=2 0 @ X jj=n k k 2 X 1 A 1=2 : It remains to apply the triangle inequality, followed by the Cauchy-Schwarz inequality: Ekk p X 1=p 1 X n=0 (p 1) n=2 0 @ X jj=n k k 2 X 1 A 1=2 1 X n=0 p 1 q n ! 1=2 kk L 2;q (W ;X) : Denition 60. For 2L 2 (W ;X) and 2L 2 (W ;R), the Wick product is dened by = X ; 2M:+ = ! ! ! 1=2 : (7.7) To make sense of , the denition requires at least one of ; to be real-valued. The normalization in (7.7) ensures that, for every n;m;k, H n ( k )H m ( k ) =H n+m ( k ); 1 In fact, a better reference is the un-numbered equation at the bottom of page 62 in [23]. 73 where k is one of the standard Gaussian random variables (7.4) and H n is the Hermite polynomial (7.6). Remark 61. If 2L 2 W ;L 2 (0;) and is adapted, that is, for every x2 [0;], the ran- dom variable(x) is measurable with respect to the sigma-algebra generated by _ W ( [0;y] ); 0 yx, then, by [10, Proposition 2.5.4 and Theorem 2.5.9], Z x 0 (x) _ W (x)dx = Z x 0 (x)dW (x); where the right-hand side is the It^ o integral with respect to the standard Brownian motion W (x) = _ W ( [0;x] ). This connection with the It^ o integral does not help when it comes to equation (7.1): the structure of the heat kernel implies that, for every x2 (0;), the solution u =u(t;x) of (7.1) depends on all of the trajectory of W (x); x2 (0;); and therefore is not adapted as a function of x. Given a xed 2M, the sum in (7.7) contains nitely many terms, but, in general, P 2M 2 X =1 so that is not square-integrable. Here is a sucient condition for the Wick product to be square-integrable. Proposition 62. If = X k b k k ; b k 2R; (7.8) and P k b 2 k <1, then 7! is a bounded linear operator fromD 1 2 (W ;X) toL 2 (W ;X). Proof. By (7.7), Ekk 2 X = X 2M X k p k b k (k) 2 X : 74 By the Cauchy-Schwarz inequality, X k p k b k (k) 2 X jj X k b 2 k k (k) k 2 X : After summing over all and shifting the summation index, Ekk 2 X X k b 2 k X 2M jj + 1 k k 2 X ; concluding the proof. Note that, while _ W (x) is of the form (7.8) (cf. (7.5)), Proposition 62 does not apply: for a typical value of x2 [0;], P k jm k (x)j 2 = +1. Thus, without either adaptedness of or square-integrability of _ W , an investigation of the Wick product _ W (x) requires additional constructions. One approach (cf. [19]) is to note that if (7.8) is a linear combination of k , then, by (7.7), the number () = X k p k b k (k) is well-dened for every2M regardless of whether the series P k b 2 k converges or diverges. This observation allows an extension of the operation to spaces much bigger thanL 2 (W ;X) and L 2 (W ;R); see [19, Proposition 2.7]. In particular, both _ W and _ W , with _ W = X k p k m k (k) ; (7.9) become generalized random elements with values in L 2 (0;). 75 An alternative approach, which we will pursue in this paper, is to consider _ W and _ W as usual (square integrable) random elements with values in a space of generalized functions. For 2R, dene the operator = I @ 2 @x 2 =2 (7.10) on L 2 (0;) by f (x) = 1 X k=1 1 + (k 1) 2 =2 f k m k (x); (7.11) where, for a smooth f with compact support in (0;), f k = Z 0 f(x)m k (x)dx; recall thatfm k ; k 1g is the Fourier cosine basis (7.3) in L 2 (0;) so that 2 m k (x) =m k (x) +m 00 k (x) = 1 + (k 1) 2 m k (x): If > 1=2, then, by (7.11), f (x) = Z 0 R (x;y)f(y)dy; (7.12) where R (x;y) = X k1 1 + (k 1) 2 =2 m k (x)m k (y): (7.13) 76 Denition 63. The Sobolev space H 2 (0;) is L 2 (0;) . The normkfk in the space is dened by kfk =k fk 0 : The next result is a variation on the theme of Proposition 62. Theorem 64. If > 1=2; then7! _ W is a bounded linear operator fromD 1 2 W ;L 2 (0;) to L 2 W ;H 2 (0;) . Proof. By (7.9), if = X 2M with 2L 2 (0;), then _ W (x) = X 2M X k p k m k (x) (k) (x) ! ; so that Ek _ Wk 2 = X 2M X k p k m k (k) 2 0 : By the Cauchy-Schwarz inequality, X k p k m k (k) 2 0 jj X k Z 0 m k (k) 2 (x)dx: After summing over all and shifting the summation index, Ek _ Wk 2 X 2M jj + 1 X k Z 0 m k 2 (x)dx: 77 By (7.12) and Parsevals's equality, X k Z 0 m k 2 (x)dx = Z 0 Z 0 R 2 (x;y) 2 (y)dydx; and then (7.13) implies Z 0 R 2 (x;y)dx = X k1 1 + (k 1) 2 m 2 k (y) 2 X k0 1 (1 +k 2 ) ; that is, Z 0 Z 0 R 2 (x;y) 2 (y)dydxC k k 2 0 ; C = 2 X k0 1 (1 +k 2 ) : As a result, Ek _ Wk 2 C X 2M jj + 1 k k 2 0 ; concluding the proof of Theorem 64. 7.2 The Chaos Solution Let (V;H;V 0 ) be a normal triple of Hilbert spaces, that is VHV 0 and the embeddings VH and HV 0 are dense and continuous; The space V 0 is dual to V relative to the inner product in H; There exists a constant C H > 0 such thatj(u;v) H jC H kuk V kvk V 0 for allu2V and v2H: 78 An abstract homogeneous Wick-It^ o-Skorohod evolution equation in (V;H;V 0 ), driven by the collectionf k ; k 1g of iid standard Gaussian random variables, is _ u(t) = Au(t) + X k M k u(t) k ; t> 0; (7.14) where A and M k are bounded linear operators from V to V 0 . Except for Section 7.6, every- where else in the paper, the initial condition u(0)2H is non-random. Denition 65. The chaos solution of (7.14) is the collection of functionsfu =u (t); t> 0; 2Mg satisfying the propagator _ u (0) (t) = Au (0) ; u (0) (0) =u(0); _ u = Au + X k p k M k u (k) ; u (0) = 0; jj> 0: It is known [19, Theorem 3.10] that if the deterministic equation _ v = Av is well-posed in (V;H;V 0 ), then (7.14) has a unique chaos solution u (t) = 1 p ! X 2Pn Z t 0 Z sn 0 ::: Z s 2 0 tsn M k (n) s 2 s 1 M k (1) s 1 u 0 ds 1 :::ds n ; (7.15) where P n is the permutation group of the set (1;:::;n); K =fk 1 ;:::;k n g is the characteristic set of; t is the semigroup generated by A: u (0) (t) = t u 0 . 79 Once constructed, the chaos solution does not depend on the particular choice of the basis in L 2 (W ;H) [19, Theorem 3.5]. In general, though, X 2M ku (t)k 2 H =1; that is, the chaos solution belongs to a space that is bigger than L 2 (W ;H); cf. [19, Remark 3.14]. On the one hand, equation (7.1) is a particular case of (7.14): Af(x) =f 00 (x) with zero boundary conditions, M k f(x) = m k (x)f(x), H = L 2 (0;); V = H 1 (0;), V 0 = H 1 (0;). The corresponding propagator becomes @u (0) (t;x) @t = @ 2 u (0) (t;x) @x 2 ; u (0) (0;x) =u 0 (x); @u (t;x) @t = @ 2 u (t;x) @x 2 + X k p k m k (x)u (k) (t;x); u (0;x) = 0;jj> 0: (7.16) Then existence and uniqueness of the chaos solution of (7.1) are immediate: Proposition 66. If u 0 2 L 2 (0;), then equation (7.1), considered in the normal triple H 1 (0;);L 2 (0;);H 1 (0;) ; has a unique chaos solution. Proof. This follows from [19, Theorems 3.10]. On the other hand, equation (7.1) has two important features that are, in general, not present in (7.14): The semigroup t has a kernel P D (t;x;y): t f(x) = Z 0 P D (t;x;y)f(y)dy; t> 0; (7.17) 80 where P D (t;x;y) = X k1 e k 2 t m k (x)m k (y) = 2 1 X k=1 e k 2 t sin(kx) sin(ky): (7.18) By Parseval's equality, X k Z 0 f(x)m k (x)dx 2 = Z 0 f 2 (x)dx: (7.19) In fact, the properties of the chaos solution of (7.1) are closely connected with the prop- erties of the functionP D (t;x;y) from (7.18). Below are some of the properties we will need. Proposition 67. For t> 0 and x;y2 [0;], 0P D (t;x;y) 1 p t ; (7.20) jP D x (t;x;y)j 4 t ; jP D xx (t;x;y)j 27 t 3=2 ; jP D t (t;x;y)j 27 t 3=2 : Proof. The maximum principle implies 0 P D (t;x;y). To derive other inequalities, note that, by integral comparison, X k1 e k 2 t Z 1 0 e x 2 t dx = p 2 p t ; t> 0; and more generally, for t> 0; r 1, X k1 k r e k 2 t r 2t (r+1)=2 + Z 1 0 x r e x 2 t dx (r + 1) (r+1) t (r+1)=2 : (7.21) 81 To complete the proof, we use jP D (t;x;y)j 2 X k1 e k 2 t ;jP D x (t;x;y)j X k1 ke k 2 t ; jP D xx (t;x;y)j X k1 k 2 e k 2 t ;jP D t (t;x;y)j X k1 k 2 e k 2 t : The main consequence of (7.17) and (7.19) is Proposition 68. (1) Forjj = 0, ku (0) (t;)k 0 ku 0 k 0 ; t> 0; (7.22) and ju (0) (s;y)jC(p;s)ku 0 k Lp(0;) ; 0<st; 0y; (7.23) with C(p;s) = 8 > > > > > > > > < > > > > > > > > : s 1=2 ; if p = 1; 1=p 0 s 1=2 ; if 1<p< +1; p 0 = p p1 ; 1; if p = +1: In particular, C(p;s)s 1=2 (7.24) for all 0<st and 1p +1. 82 (2) Forjj =n 1; X jj=n ju (t;x)j 2 n! Z (0;) n Z T n 0;t P D (ts n ;x;y n )P D (s 2 s 1 ;y 2 ;y 1 )u (0) (s 1 ;y 1 )ds n ! 2 dy n : (7.25) Proof. (1) Forjj = 0, u (0) (s;y) = Z 0 P D (s;y;z)u 0 (z)dz; with P D from (7.18). Then ku (0) (s;)k 2 0 = X k1 e 2k 2 s u 2 0;k ; from which (7.22) follows. To derive (7.23) when p<1, we use the H older inequality and (7.20); if p = +1, then we use R 0 P D (s;y;z)dz = 1 instead of the upper bound in (7.20). (2) It follows from (7.15) that, forjj 1, u (t;x) = 1 p ! X 2Pn Z (0;) n Z T n 0;t P D (ts n ;x;y n )m k (n) (y n ) P D (s 2 s 1 ;y 2 ;y 1 )m k (1) (y 1 )u (0) (s 1 ;y 1 )ds n dy n : (7.26) 83 Using (7.17) and notations e (y 1 ;:::;y n ) = 1 p n!! X 2Pn m k (n) (y n )m k (1) (y 1 ); F n (t;x;y 1 ;:::;y n ) = Z T n 0;t P D (ts n ;x;y n )P D (s 2 s 1 ;y 2 ;y 1 )u (0) (s 1 ;y 1 )ds n ; (7.27) we re-write (7.26) as u (t;x) = p n! Z (0;) n F n (t;x;y 1 ;:::;y n )e (y 1 ;:::;y n )dy n : (7.28) The collectionfe ; jj = ng is an orthonormal basis in the symmetric part of the space L 2 (0;) n , so thatu becomes the corresponding Fourier coecient of the functionF n , and (7.25) becomes Bessel's inequality. Remark 69. It follows from (7.28) that X jj=n ju (t;x)j 2 =n! Z (0;) n e F 2 n (t;x;y 1 ;:::;y n )dy n ; where e F n (t;x;y 1 ;:::;y n ) = 1 n! X 2Pn F n (t;x;y (1) ;:::;y (n) ) is the symmertrization of F n from (7.27). By the Cauchy-Schwarz inequality, k e F n k L 2 ((0;) n ) kF n k L 2 ((0;) n ) ; 84 and a separate analysis is necessary to establish a more precise connection betweenk e F n k L 2 ((0;) n ) andkF n k L 2 ((0;) n ) . The upper bound (7.25) is enough for the purposes of this paper. 7.3 Basic Regularity of the Chaos Solution The objective of this section is to show that, for each t> 0, the chaos solution of (7.1) is a regular, as opposed to generalized, random variable, and to introduce the main techniques necessary to establish better regularity of the solution. Theorem 70. If u 0 2L 2 (0;), then, for every t> 0, the solution of (7.1) satises u(t;)2 \ q>1 L 2;q W ;L 2 (0;) : (7.29) Proof. It follows from (7.25) that X jj=n ju (t;x)j 2 n! Z (0;) n Z T n 0;t Z T n 0;t P D (ts n ;x;y n )P D (s 2 s 1 ;y 2 ;y 1 )u (0) (s 1 ;y 1 ) P D (tr n ;x;y n )P D (r 2 r 1 ;y 2 ;y 1 )u (0) (r 1 ;y 1 ) ds n dr n dy n : (7.30) We now integrate both sides of (7.30) with respect to x and use the semigroup property Z 0 P D (t;x;y)P D (s;y;z)dy =P D (t +s;x;z) (7.31) 85 together with (7.20) to evaluate the integrals over (0;) on the right-hand side, starting from the outer-most integral. We also use (7.22). The result is X jj=n ku (t;)k 2 0 n!ku 0 k 2 0 Z T n 0;t Z T n 0;t (2ts n r n ) 1=2 (s n +r n s n1 r n1 ) 1=2 (s 2 +r 2 s 1 r 1 ) 1=2 ds n dr n : (7.32) Next, we use the inequality 4pq (p +q) 2 , p;q> 0, to nd (p +q) 1=2 p 1=4 q 1=4 ; (7.33) so that Z T n 0;t Z T n 0;t (2ts n r n ) 1=2 (s n +r n s n1 r n1 ) 1=2 (s 2 +r 2 s 1 r 1 ) 1=2 ds n dr n 0 B @ Z T n 0;t (ts n ) 1=4 (s n s n1 ) 1=4 (s 2 s 1 ) 1=4 ds n 1 C A 2 = (3=4) n ((3=4)n + 1) ! 2 t 3n=2 ; (7.34) where is the Gamma function (y) = Z 1 0 t y1 e t dt: The last equality in (7.34) follows by induction using Z t 0 s p (ts) q ds =t p+q+1 (1 +p)(1 +q) (2 +p +q) ; p;q>1: (7.35) 86 Combining (7.30), (7.32), and (7.34), X jj=n ku (t;)k 2 0 n! (3=4) n ((3=4)n + 1) ! 2 t 3n=2 ku 0 k 2 0 : As a consequence of the Stirling formula, (1 +p) p 2pp p e p and n! 2 p n n e n ; meaning that X jj=n ku (t;)k 2 0 C n (t)n n=2 ku 0 k 2 0 ; t> 0; (7.36) with C(t) = 4=3 3=2 e 1=2 2 (3=4)t 3=2 : Since Eku(t;)k 2 L 2;q (L 2 (0;)) = 1 X n=0 q n X 2M:jj=n ku (t;)k 2 0 ; and the series X n1 C n n n=2 = X n1 C p n n converges for every C > 1, we get (7.29) and conclude the proof of Theorem 70. Corollary 71. Ifu 0 2L 2 (0;), then the chaos solution is anL 2 (0;)-valued random process and, for all t 0, Eku(t;)k p 0 <1; 1p<1: 87 Proof. This follows from (7.29) and Proposition 59. We will need a slightly more general family of integrals than the one appearing on the right-hand side of (7.34): I 1 (t;;) = Z t 0 (ts) s ds; I n (t;;) = Z T n 0;t (ts n ) n Y k=2 (s k s k1 ) 1=4 s 1 ds n ; n = 2; 3;:::; (7.37) for 2 (0; 1); 2 [0; 1). Note that I 1 (t;;) = Z t 0 (ts) s ds = (1)(1) (2) t 1 and I n (t;;) = Z t 0 (ts n ) I n1 (s n ; 1=4;)ds n ; n 1: By induction and (7.35), I n (t;;) = (3=4) n1 (1)(1) (3n + 5 4 4)=4 t (3n+144)=4 ; and then n!I 2 n (t;;)C n (;;t)n n=2 ; (7.38) cf. (7.36). Next, we show that the chaos solution of (7.1) is, in fact, a random eld solution, that is, u(t;x) is well-dened as a random variable for every t> 0, x2 [0;]. 88 Theorem 72. If u 0 2L p (0;) for some 1p1, then, for every t> 0 and x2 [0;], u(t;x)2 \ q>1 L 2;q (W ;R): (7.39) Proof. By Proposition 68, inequality (7.30) becomes X jj=n ju (t;x)j 2 n! 2 ku 0 k 2 Lp(0;) Z (0;) n ZZ T n 0;t T n 0;t P D (ts n ;x;y n )P D (s 2 s 1 ;y 2 ;y 1 )s 1=2 1 P D (tr n ;x;y n )P D (r 2 r 1 ;y 2 ;y 1 )r 1=2 1 ds n dr n dy n : (7.40) We now use the semigroup property (7.31) together with (7.20) to evaluate the integrals over (0;) on the right-hand side of (7.40) starting from the inner-most integral with respect to y 1 . The result is X jj=n ju (t;x)j 2 n! 2 ku 0 k 2 Lp(0;) ZZ T n 0;t T n 0;t (2ts n r n ) 1=2 (s n +r n s n1 r n1 ) 1=2 (s 2 +r 2 s 1 r 1 ) 1=2 s 1=2 1 r 1=2 1 ds n dr n : (7.41) 89 Next, similar to (7.34), we use (7.33) and (7.37) to compute ZZ T n 0;t T n 0;t (2ts n r n ) 1=2 (s n +r n s n1 r n1 ) 1=2 (s 2 +r 2 s 1 r 1 ) 1=2 s 1=2 1 r 1=2 1 ds n dr n 0 B @ Z T n 0;t (ts n ) 1=4 (s n s n1 ) 1=4 (s 2 s 1 ) 1=4 s 1=2 1 ds n 1 C A 2 =I 2 n (t; 1=4; 1=2): (7.42) Combining (7.41) with (7.42) and (7.38), X jj=n ju (t;x)j 2 C n (t)n n=2 ku 0 k 2 Lp(0;) ; (7.43) for a suitable C(t). Then (7.43) leads to (7.39) in the same way as (7.36) lead to (7.29), completing the proof of Theorem 72. Corollary 73. For every t> 0, x2 [0;], and 1p<1, Eju(t;x)j p <1: Proof. This follows from (7.39) and Proposition 59. Finally, we establish a version of the maximum principle for the chaos solution. 90 Theorem 74. If u 0 (x) 0 for all x2 [0;]; and u = u(t;x) is a random eld solution of (7.1) such that u2L 2 [0;T ];L p (0;) ; then, with probability one, u(t;x) 0 for all t2 [0;T ] and x2 [0;]. Proof. Let h =h(x) be a smooth function with compact support in (0;) and dene V (t;x;h) =E u(t;x) exp _ W (h) 1 2 khk 2 L 2 (0;) : Writing h(x) = P 1 k=1 h k m k (x) and h = Q k h k k ; we nd V (t;x;h) = X 2M h u (t;x) p ! : By (7.16), the function V =V (t;x;h) satises @V (t;x;h) @t = @ 2 V (t;x;h) @x 2 +h(x)V (t;x;h); 0<tT; x2 (0;); with V (0;x;h) = u 0 (x) and V x (t; 0;h) = V x (t;;h) = 0, and then the maximum principle impliesV (t;x;h) 0 for allt2 [0;T ]; x2 [0;]. The conclusion of the theorem now follows, because the collection of the random variables exp _ W (h) 1 2 khk 2 L 2 (0;) ; h smooth with compact support in (0;) is dense in L 2 (W ;R); cf. [25, Lemma 4.3.2]. 91 Remark 75. If u =u(t;x) is continuous in (t;x), then there exists a single probability-one subset 0 of such that u =u(t;x;!)> 0 for all t2 [0;T ], x2 [0;], and !2 0 . 7.4 Time Regularity of the Chaos Solution The objective of this section is to show that the chaos solution of (7.1) has a modication that is almost H older 3=4 continuous in time. To simplify the presentation, we will not distinguish dierent modications of the solution. Theorem 76. If u 0 2C 3=2 (0;), then the chaos solution of (7.1) satises u(;x)2C 3=4 (0;T ) for every T > 0 and x2 [0;]. Proof. We need to show that, for every x2 [0;], h2 (0; 1), "2 (0; 3=4), t2 (0;T ), and p2 (1; +1), Eju(t +h;x)u(t;x)j p 1=p C(p;T;")h 3=4" : Then the statement of the theorem will follow Theorem 28. Recall that u (0) (t;x) is the solution of @u (0) (t;x) @t = @ 2 u (0) (t;x) @x 2 ; u (0) (0;x) =u 0 (x); with boundary conditions @u (0) (t; 0) @x = @u (0) (t;) @x = 0; 92 that is, @u (0) (t;x) @t = (1 2 )u (0) (t;x); the operator is dened in (7.10). Applying [17, Theorem 5.3] to the equation, we conclude that, for each x2 [0;], u (0) (;x)2C 3=4 (0;T ): (7.44) For n 1 and h2 (0; 1), similar to (7.25), X jj=n ju (t +h;x)u (t;x)j 2 n! Z (0;) n Z T n 0;t+h P D (t +hs n ;x;y n )P D (s 2 s 1 ;y 2 ;y 1 )u (0) (s 1 ;y 1 )ds n Z T n 0;t P D (ts n ;x;y n )P D (s 2 s 1 ;y 2 ;y 1 )u (0) (s 1 ;y 1 )ds n ! 2 dy n : (7.45) We add and subtract Z T n 0;t P D (t +hs n ;x;y n )P D (s 2 s 1 ;y 2 ;y 1 )u (0) (s 1 ;y 1 )ds n 93 inside the square on the right-hand side of (7.45), and then use (p +q) 2 2p 2 + 2q 2 to re-write (7.45) as X jj=n ju (t +h;x)u (t;x)j 2 2n! Z (0;) n Z t+h t Z T n1 0;sn P D (t +hs n ;x;y n )P D (s 2 s 1 ;y 2 ;y 1 )u (0) (s 1 ;y 1 )ds n ! 2 dy n +2n! Z (0;) n Z T n 0;t h P D (t +hs n ;x;y n )P D (ts n ;x;y n ) i P D (s n s n1 ;y n ;y n1 ) P D (s 2 s 1 ;y 2 ;y 1 )u (0) (s 1 ;y 1 )ds n ! 2 dy n : (7.46) To estimate the rst term on the right-hand side of (7.46), we follow computations similar to (7.34) and (7.41), and use P D (t;x;y) 0; ku (0) (s;)k L1(0;) ku 0 k L1(0;) ; s 0; (7.47) as well as (7.38): 2n! Z (0;) n t+h Z t Z T n1 0;sn P D (t +hs n ;x;y n )P D (s 2 s 1 ;y 2 ;y 1 )u (0) (s 1 ;y 1 )ds n ! 2 dy n 2n!ku 0 k 2 L1(0;) t+h Z t Z T n1 0;sn (t +hs n ) 1=4 (s n s n1 ) 1=4 (s 2 s 1 ) 1=4 ds n ! 2 dy n 2n!ku 0 k 2 L1(0;) t+h Z t (t +hs n ) 1=4 I n1 (s n ; 1=4; 1=4)ds n ! 2 ku 0 k 2 L1(0;) C n (t)n n=2 h 3=2 ; (7.48) 94 with a suitable C(t). To estimate the second term on the right-hand side of (7.46), dene I(t;h;s;r;x) = Z 0 P D (t +hs;x;y)P D (ts;x;y) P D (t +hr;x;y)P D (tr;x;y) dy: By (7.18), I(t;h;s;r;x) = 4 2 X k1 (e k 2 h 1) 2 e k 2 (ts)k 2 (tr) sin 2 (kx): Using (3.13) and taking 0< < 3=4, we conclude that I(t;h;s;r;x)h 2 X k1 k 4 e k 2 (2tsr) : Then (7.21) implies I(t;h;s;r;x) (4 + 1) 4 +1 h 2 (2tsr) (1=2+2 ) : Note that 1=2 + 2 < 2. 95 We now carry out computations similar to (7.41), and use (7.38) and (7.47): 2n! Z (0;) n Z T n 0;t h P D (t +hs n ;x;y n )P D (ts n ;x;y n ) i P D (s 2 s 1 ;y 2 ;y 1 )P D (s 2 s 1 ;y 2 ;y 1 )u (0) (s 1 ;y 1 )ds n ! 2 dy n 2n!ku 0 k 2 L1(0;) ZZ T n 0;t T n 0;t I(t;h;s n ;r n ;x) n2 Y k=1 (s k+1 +r k+1 s k r k ) 1=2 (s 1 +r 1 2s) 1=2 ds n dr n h 2 ku 0 k 2 L1(0;) 4(4 + 1) 4 +1 n!I 2 n t; 1=4 + ; 1=4 ku 0 k 2 L1(0;) C n (t)n n=2 h 2 ; (7.49) with a suitable C(t). Combining (7.48) and (7.49), X jj=n ju (t +h;x)u (t;x)j 2 h 3=22" ku 0 k 2 L1(0;) C n (t;")n n=2 ; (7.50) "2 (0; 3=4); n 1; and then, by (7.44) and Proposition 59, Eju(t +h;x)u(t;x)j p 1=p 1 X n=0 (p 1) n=2 0 @ X jj=n ju (t +h;x)u (t;x)j 2 1 A 1=2 C(p;T;")ku 0 k L1(0;) h 3=4" ; for all 1 < p < +1; t2 (0;T ); "2 (0; 3=4); 0 < h < 1, completing the proof of Theorem 76. 96 7.5 Space Regularity of the Chaos Solution The objective of this section is to show that, for every t> 0, the chaos solution u(t;x) = X 2J u (t;x) of (7.1) has a modication that is inC 3=2 (0;). As in the previous section, we will not distinguish between dierent modications of u. To streamline the presentation, we will break the argument in two parts: existence of u x as a random eld, followed by H older 1=2 continuous regularity of u x in space. Dene v(t;x) = X 2J v (t;x) ; where v (t;x) = @u (t;x) @x = 1 p ! Z (0;) n X 2Pn Z t 0 Z sn 0 ::: Z s 2 0 P D x (ts n ;x;y n )m k (n) (y n ) P D (s n s n1 ;y n ;y n1 )m k (n1) P D (s 2 s 1 ;y 2 ;y 1 )m k (1) (y 1 )u 0 (s 1 ;y 1 )ds n dy n : Theorem 77. Assume that u 0 2 L p (0;) for some 1 p1. Then, for every t> 0 and x2 (0;), u x (t;x)2 \ q>1 L 2;q (W ;R): 97 Proof. By construction, v =u x as generalized processes. It remains to show that v(t;x)2 \ q>1 L 2;q (W ;R): (7.51) Similar to (7.25), X jj=n jv (t;x)j 2 n! Z (0;) n Z T n 0;t P D x (ts n ;x;y n )P D (s n s n1 ;y n ;y n1 ) P D (s 2 s 1 ;y 2 ;y 1 )u (0) (s 1 ;y 1 )ds n ! 2 dy n : (7.52) Using (7.24), X jj=n jv (t;x)j 2 n!ku 0 k 2 Lp(0;) 2 Z (0;) n Z T n 0;t P D x (ts n ;x;y n )P D (s n s n1 ;y n ;y n1 ) P D (s 2 s 1 ;y 2 ;y 1 )s 1=2 1 ds n ! 2 dy n : By (7.18), Z 0 P D x (t;x;y)P D x (s;x;y)dy = 4 2 1 X k=1 k 2 e k 2 (t+s) cos 2 (kx) 27 (t +s) 3=2 ; 98 and then X jj=n jv (t;x)j 2 27n! 2 ku 0 k 2 Lp(0;) Z T n 0;t (ts n1 ) 3=4 (s n1 s n2 ) 1=4 (s 2 s 1 ) 1=4 s 1=2 1 ds n ! 2 ds = 27 2 ku 0 k 2 Lp(0;) n!I 2 n t; 3=4; 1=2 ku 0 k 2 Lp(0;) C n (t)n n=2 with a suitable C(t); cf. (7.38). Then (7.51) follows in the same way as (7.29) followed from (7.36). Remark 78. Similar to the proof of Theorem 76, an interested reader can conrm that u x (;x)2C 1=4 ([;T ]) for every x2 [0;] and T >> 0. Theorem 79. If u 0 2L p (0;) for some 1p1, then, for every t> 0, u x (t;)2C 1=2 (0;): Proof. We continue to use the notationv =u x . Then the objective is to show that, for every suciently small h> 0 and every x2 (0;), t> 0, p> 1, and 2 (0; 1=2); Ejv(t;x +h)v(t;x)j p 1=p C(t;p; )h ; (7.53) then the conclusion of the theorem will follow from the Kolmogorov continuity criterion. 99 Similar to (7.25), X jj=n jv (t;x +h)v (t;x)j 2 n! Z (0;) n Z T n 0;t P D x (ts n ;x +h;y n )P D x (ts n ;x;y n ) P D (s n s n1 ;y n ;y n1 )P D (s 2 s 1 ;y 2 ;y 1 )u (0) (s 1 ;y 1 )ds n ! 2 dy n ; and then X jj=n jv (t;x +h)v (t;x)j 2 n! 2 ku 0 k 2 Lp(0;) Z (0;) n Z T n 0;t P D x (ts n1 ;x +h;y n )P D x (ts n1 ;x;y n ) P D (s n1 s n2 ;y n ;y n1 )P D (s 1 s;y 2 ;y 1 )s 1=2 1 ds n ! 2 dy n ; (7.54) cf. (7.52). Next, dene J(t;s;r;x;y;h) = Z 0 P D x (ts;x +h;y)P D x (ts;x;y) P D x (tr;x +h;y)P D x (tr;x;y) dy: From (7.18), J(t;s;r;x;y;h) = 2 X k1 k 2 e k 2 (2tsr) cos(k(x +h)) cos(kx) 2 : 100 Using cos' cos =2 sin((' )=2) sin((' + )=2) and (3.14), and taking 2 (0; 1=2), J(t;s;r;x;y;h) 2h 2 (2tsr) 3=2 : Note that 3=2 + < 2: After expanding the square and using the semigroup property, (7.54) becomes X jj=n jv (t;x +h)v (t;x)j 2 2h 2 n! 2 ku 0 k 2 Lp(0;) ZZ T n 0;t T n 0;t (2ts n1 r n1 ) 3=2 n2 Y k=1 (s k+1 +r k+1 s k r k ) 1=2 s 1=2 1 r 1=2 1 ds n dr n 2h 2 2 ku 0 k 2 Lp(0;) n!I 2 n t; 3=4 + ( =2); 1=2 C n (t; )n n=2 ; (7.55) cf. (7.32) and (7.38). Then Proposition 59 implies (7.53), completing the proof of Theorem 79. 101 7.6 The Fundamental Chaos Solution Denition 80. The fundamental chaos solution of (7.1) is the collection of functions fP (t;x;y); t> 0; x;y2 [0;]; 2Mg dened by P (0) (t;x;y) =P D (t;x;y); P (t;x;y) = 1 p ! X 2Pn Z (0;) n Z T n 0;t P D (ts n ;x;y n )m k (n) (y n ) P D (s 2 s 1 ;y 2 ;y 1 )m k (1) (y 1 )P D (s 1 ;y 1 ;y)ds n dy n : (7.56) The intuition behind this denition is that (7.56) is the chaos solution of (7.1) with initial condition u 0 (x) =(xy). More precisely, it follows from (7.26) that if P(t;x;y) = X 2M P (t;x;y) ; (7.57) then u(t;x) = Z 0 P(t;x;y)u 0 (y)dy (7.58) is the chaos solution of (7.1) with non-random initial condition u(0;x) = u 0 (x). Before developing these ideas any further, let us apply the results of Sections 7.2{7.5 to the random function P. 102 Theorem 81. The function P dened by (7.57) has the following properties: P(t;x;y)2 \ q>1 L 2;q (W ;R); t> 0; uniformly in x;y2 [0;]; (7.59) P(t;x;y) 0; P(t;x;y) =P(t;y;x); t> 0; x;y2 [0;]; (7.60) P(;x;y)2C 3=4 (;T ); 0<<T; x;y2 [0;]; (7.61) P(t;;y)2C 3=2 (0;); t> 0; y2 [0;]: (7.62) Proof. Using (7.20), (7.30), (7.34), and (7.38), X jj=n jP (t;x;y)j 2 n! Z (0;) n Z T n 0;t Z T n 0;t P D (ts n ;x;y n )P D (s 2 s 1 ;y 2 ;y 1 )P D (s 1 ;y 1 ;y) P D (tr n ;x;y n )P D (r 2 r 1 ;y 2 ;y 1 )P D (r 1 ;y 1 ;y) ds n dr n dy n n!I 2 n t; 1=4; 1=2 C n (t) n n=2 ; from which (7.59) follows. To establish (7.60), note that (7.58) and Theorem 74 imply P 0, whereas, by (7.25), using P D (t;x;y) =P D (t;y;x) and a suitable change of the time variables in the integrals, X jj=n jP (t;x;y)P (t;y;x)j 2 = 0; n 1; which implies P(t;x;y) =P(t;y;x). 103 To establish (7.61) and (7.62), we compute, for n 1, X jj=n jP (t +h;x;y)P (t;x;y)j 2 h 2 C n (t; )n n=2 ; 2 (0; 3=4); cf: (7.50); and X jj=n jP ;x (t;x +h;y)P ;x (t;x;y)j 2 h 2 C n (t; )n n=2 ; 2 (0; 1=2); cf: (7.55): Note that P (0) (t;x;y) =P D (t;x;y) is innitely dierentiable in t and x for t> 0 but is unbounded as t& 0; cf. (7.20). Now we can give full justication of the reason why P is natural to call the fundamental chaos solution of equation (7.1). Theorem 82. If u 0 2 L 2;q W ;L 2 (0;) for some q > 1, then the chaos solution of (7.1) with initial condition u(0;x) =u 0 (x) is u(t;x) = Z 0 P(t;x;y)u 0 (y)dy; (7.63) and u(t;x)2L 2;p (W ;R) (7.64) for every p<q, t> 0, and x2 [0;]. 104 Proof. Let u 0 (x) = X 2M u 0; (x) be the chaos expansion of the initial condition. By denition, the chaos solution of (7.1) is u(t;x) = X 2M u (t;x) ; where @u (0) (t;x) @t = @ 2 u (0) (t;x) @x 2 ; u (0) (0;x) =u 0;(0) (x); @u (t;x) @t = @ 2 u (t;x) @x 2 + X k p k m k (x)u (k) (t;x); u (0;x) =u 0; (x);jj> 0: By [18, Theorem 9.8], if u(0;x) =f(x) for some f2L 2 (0;) and2M, then u(t;x) = Z 0 P(t;x;y) f(y)dy: Then (7.63) follows by linearity. Next, given 1 p < q, take p 0 = qp=(qp), so that p 01 +q 1 = p 1 . Then, by (7.59) and [20, Theorem 4.3(a)], P(t;x;)u 0 2L 2;p W ;L 2 (0;) ; which implies (7.64). 105 Chapter 8 Concluding Remarks 8.1 Conclusion The results of Stratonovich interpretation imply several important remarks as follows: Space-only (or time-homogeneous) noise _ W (x) aects only spatial regularity. In other words, for t> 0, u is innitely dierentiable in time; The paper [12] gives the spatial (H older) regularity only less than 1=2. However, we showed that the optimal spatial (H older) regularity of the solution u is 3=2" for any "> 0 as long as u 0 2C 3=2 (0;); We achieved the spatial regularity higher than 1=2; Since the standard Brownian mo- tion W is H older 1=2" continuous almost surely, it is possible to apply Young's integral: for each s<t and x, Z 0 P D (ts;x;y)u(s;y) @ @y W (y)dy := Z 0 P D (ts;x;y)u(s;y)dW (y) 106 appearing in the classical mild formulation of (5.1); The regularity 3=4" in time and 3=2" in space for " > 0 are indeed in the line with the standard parabolic partial dierential equation theory. In addition, there are two main contributions in the stochastic PAM with Wick product: The paper [28] gave the regularity results: u is almost H older 1/2 continuous in time and space, But, we established the optimal space-time regulairty: u is almost H older 3/4 continuous in time and almost H older 3/2 continuous in space; We found the fundamental solution, not just from results in deterministic PDE theory, but directly from the stochastic model. 8.2 Further Directions 8.2.1 Wick Product vs. Usual Product It is known that for any initial function u 0 2C 3=2 (0;), there exists the unique Wick-It^ o- Skorokhod solution u satisfying the equation u (t;x) = P 0 (t;x) + Z t 0 Z 0 P D (ts;x;y)u (s;y) _ W (y)dyds (8.1) almost surely inC 3=4";3=2" t;x ((0;T ) (0;)) for any "> 0. The natural further question is to nd a meaningful relation between the usual solution of (5.1) and the Wick-It^ o-Skorokhod solution of (7.1) with Dirichlet boundary condition in the mild formulation. 107 There are relations between the usual product and the Wick product (e.g., [21] and [22]). For any u;v2 L 2 ( ), Cameron-Martin theorem [1] gives the fact: for each t2 [0;T ] and x2 [0;], u and v can be written as u(t;x) = X 2M u (t;x) and v(t;x) = X 2M v (t;x) : Also, we have the identity [21, Theorem 2.3] uv =uv + X 2M 0 @ X 6=(0) X (0) p !( + )!( + )! ! !()! u + v + 1 A : From the fact that standard Brownian motion on [0;] has an explicit formula W (x) = 1 X k=1 Z x 0 m k (y)dy (k) ; we have a formal expression of Gaussian white noise on [0;] given by _ W (x) = 1 X k=1 m k (x) (k) ; where m k 's are dened as before. Let us consider smooth approximations _ W " of _ W (x) by a convolution with " as before: for each x2 [0;], _ W " (x) = 1 X k=1 m " k (x) (k) 2L 2 ( ); where m " k (x) =m k " (x): 108 It is also known by Theorem 72 that for any u 0 2 L p (0;); 1 p <1, the Wick-It^ o- Skorokhod solution u (t;x)2L p ( ); t> 0; x2 [0;] satisfying (8.1). This implies that the approximated Wick-It^ o-Skorokhod solutions (u " ) (t;x)2L p ( ); t> 0; x2 [0;] satisfying (u " ) (t;x) = P 0 (t;x) + Z t 0 Z 0 P D (ts;x;y) (u " ) (s;y) _ W " (y)dyds: (8.2) Then, we get the following relation (u " ) (t;x) _ W " (x) = (u " ) (t;x) _ W " (x) + X 2M X k1 p k + 1 (u " ) +(k) (t;x)m " k (x) : (8.3) Dene the residual by " (t;x) = X 2M " (t;x) := X 2M X k1 p k + 1 (u " ) +(k) (t;x)m " k (x) ; where " (t;x) := X k1 p k + 1 (u " ) +(k) (t;x)m " k (x): 109 Beyond the basic relations, let us nd a further connection between usual solution and Wick-It^ o-Skorokhod solution. Consider the approximated mild solutions of (6.5) u " (t;x) = P 0 (t;x) + Z t 0 Z 0 P D (ts;x;y)u " (s;y) _ W " (y)dyds: (8.4) After we dene Z " (t;x) by Z " (t;x) =u " (t;x) (u " ) (t;x) and using the relation (8.3), we have the following equation Z " (t;x) = Z t 0 Z 0 P D (ts;x;y)Z " (s;y) _ W " (y)dyds Z t 0 Z 0 P D (ts;x;y) " (s;y)dyds: (8.5) Equivalently, the equation (8.5) is the mild formulation of @Z " (t;x) @t = @ 2 Z " (t;x) @x 2 +Z " (t;x) _ W " (x) " (t;x); 0<t<T; 0<x<; Z " (0;x) = 0; Z " (t; 0) =Z " (t;) = 0: (8.6) Set Z(t;x) =u(t;x)u (t;x); whereu is the usual mild solution of (5.1) andu is the Wick-It^ o-Skorokhod solution of (7.1) on [0;] with Dirichlet boundary condition. Then, we see that for any 0 < 1 < 3=4; 0 < 2 < 3=2, Z " !Z inC 1 ; 2 t;x ((0;T ) (0;)) as "! 0: 110 We naturally expect that Z(t;x) satises the equation @Z(t;x) @t = @ 2 Z(t;x) @x 2 +Z(t;x) _ W (x)(t;x); 0<t<T; 0<x<; Z(0;x) = 0; Z(t; 0) =Z(t;) = 0; (8.7) where (t;x) = X 2M X k1 p k + 1u +(k) (t;x)m k (x) : Actually, we can show, for each t > 0, " (t;) converges to (t;) in L 2 ;H r 2 (0;) with r > 1=2. Here, H r 2 is the dual space of the Sobolev space H r 2 . Therefore, we can view the Wick-It^ o-Skorokhod solution of (7.1) on [0;] with Dirichlet boundary condition as an approximation of usual mild solution of (5.1), and moreover, further investigation of the residual equation (8.6) is reasonable to give a rigorous relation between usual solution and Wick-It^ o-Skorokhod solution. 8.2.2 The Stochastic Anderson Model The Anderson model, named after the American physicist Philip Warren Anderson (b. 1923, Nobel Prize 1977) and rst introduced around 1960 to study certain quantum phenomena in solid state physics, is still a subject of active research for both mathematicians and physi- cists. The model is also nding new applications, from understanding interface formation in nanomaterials to improving quality of computer memory. Rigorous mathematical study of the model started around 1990 and is a relatively new area of stochastic analysis. The centerpiece of the Anderson model is a particular Hamiltonian operator, usually represented by the standard Laplacian plus random potential. The operator can then be used 111 in a heat equation or a wave equation, leading, respectively, to the parabolic and hyperbolic Anderson models. The great variety of the particular problems comes from \mixing and matching" from the following list: (1) Discrete or continuous time evolution; (2) Discrete or continuous space structure; (3) Dierent types of random potential; (4) Dierent types of questions about the solution of the resulting equation. 8.2.3 The Stochastic Anderson Operator The energy levels of random particles are corresponding to the eigenvalues of stochastic Anderson operator @ 2 @x 2 + _ W (x) u(x) =u(x); x2 [0;] (8.8) with suitable boundary conditions. The model (8.8) depending on the multiplication operator can be separated into two cases: 1) Stochastic eigenvalue problem with usual path-wise product 2) Stochastic eigenvalue problem with Wick product. 112 1) Using the energy form, the paper [3] proved that the stochastic Anderson operator @ 2 @x 2 + _ W (x) with usual product and Dirichlet boundary condition has the discrete pure point spectrum of nite multiplicity with no accumulation point except for1: namely, 1 2 k : The paper also investigated the asymptotic property of eigenvalues and found the exact for- mula for spectral distribution function. Using the modied Pr ufer transformation, the same results for separated boundary conditions, which are more general than Dirichlet boundary condition, can be shown. Besides the energy form method, dierent methodologies originated from functional anal- ysis can be applied to show the self-adjointness and the discrete pure point spectrum of (8.8) by Introducing quasi-derivative, Approximations by smooth potentials, or Multiplier methods for general boundary conditions. 113 One contribution in the stochastic eigenvalue problem with usual product is the existence of a connection with the stochastic PAM. Indeed, the discrete pure point spectrum suggests the fundamental solutionP(t;x;y) = 1 X k=1 e k t m k (x)m k (y) and a Fourier solution of (5.1): u(t;x) = Z 0 P(t;x;y)u 0 (y)dy; (8.9) where k 's are eigenvalues of (8.8), m k 's are eigenfunctions corresponding to k , and u 0 is the initial condition of (5.1). 2) The Wiener-chaos expansion and Fredholm alternative help nd a recursion formula for the eigenvalues of @ 2 @x 2 + _ W (x) u(x) =u(x); x2 [0;] with Dirichlet and Neumann boundary conditions and prove the existence of the correspond- ing eigenfunctions, similarly to [9]. We have two immediate next steps: To obtain the asymptotic behaviors of the eigenvalues. To investigate a connection between the eigenvalues of the stochastic eigenvalue prob- lem with Wick product and the solution of the stochastic PAM with Wick product u t (t;x) =u xx (t;x) +u(t;x) _ W (x); x2 [0;] motivated by the case of usual product. 114 Index Adapted process, 9 Additive noise, 26 Burkholder-Davis-Gundy inequality, 14 Cameron-Martin theorem, 19 Chaos expansion, 17 Chaos solution, 79 Classical geometric rough path solution, 43 Classical Schauder's estimate, 56 Classical solution, 42 Dirichlet heat kernel, 56 Elementary process, 9 Fractional Sobolev space, 57 Fundamental chaos solution, 102 Fundamental solution, 52 Generalized geometric rough path solution, 50 Generalized solution, 49 Geometric rough path, 40 H older space, 27 Hermite function, 22 Hermite polynomials, 16 Hida distribution space, 21 Hida test function space, 21 Hypercontractivity property, 73 It^ o integral, 10 It^ o-Stratonovich correction, 15 Kolmogorov's continuity criterion, 28 Mild solution, 61 Modied Schauder estimate, 58 Multi-index, 18 Neumann heat kernel, 59 Normal triple of Hilbert spaces, 78 Ornstein-Uhlenbeck operator, 73 Parseval's equality, 78 Propagator, 79 Random eld solution, 88 Sobolev space, 77 Stochastic Anderson operator, 112 Stochastic chain rule, 15 Stochastic exponential function, 19 Stratonovich integral, 14 White noise, 8 Wick product, 20, 73 Wiener integral, 7 Wiener-It^ o chaos expansion theorem, 20 115 Bibliography [1] R. H. Cameron and W. T. Martin, The orthogonal development of nonlinear functionals in a series of Fourier-Hermite functions, Ann. Math. 48 (1947), no. 2, 385{392. [2] T. Coulhon and X.T. Duong, Maximal regularity and kernel bounds: observations on a theorem by Hieber and Pr us, Adv. Di. Eq., 5 (2000), no. 1-3, 343{368. [3] M. Fukushima and S. Nakao, On spectra of the Schr odinger operator with a white Gaus- sian noise potential, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37 (1977), no. 3, 267{274. MR 0438481. [4] M. Gubinelli, P. Imkeller, and N. Perkowski, Paracontrolled distributions and singular PDEs, preprint (2012). [5] M. Hairer, A theory of regularity structures, Invent. Math. 198, no. 2, (2014), 269{504. [6] M. Hairer and C. Labb e, A simple construction of the continuum parabolic Anderson model onR 2 , Electron. Commun. Probab. 20 (2015), no. 43, 1{11. [7] M. Hairer and E. Pardoux. A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1551{1604. [8] M. Hairer and C. Labb e, Multiplicative stochastic heat equations on the whole space, preprint (2015). [9] H, Manouzi and T.-M. Laleg-Kirati, Solving stochastic eigenvalue problem of Wick type, 1 (2014), no. 5, 102. [10] H. Holden, B. ksendal, J. Ube, and T. Zhang, Stochastic partial dierential equations, second edition, Universitext, Springer, 2010. [11] Y. Hu, Chaos expansion of heat equations with white noise potentials, Potential Anal. 16 (2002), no. 1, 45{66. [12] Y. Hu, J. Huang, D. Nualart, and S. Tindel, Stochastic heat equations with general mul- tiplicative Gaussian noises: H older continuity and intermittency, Electron. J. Probab. 20 (2015), no. 55, 50pp. 116 [13] O. Kallenberg, Foundations of modern probability, Probability and its Applications (New York), 2002. [14] I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Springer, New York, 1991. [15] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in H older Spaces, AMS (1996). [16] N. V. Krylov, An analytic approach to SPDEs, Stochastic Partial Dierential Equa- tions, Six Perspectives, Mathematical Surveys and Monographs (B. L. Rozovskii and R. Carmona, eds.), AMS (1999), 185{242. [17] O. A. Lady zenskaja, V. A. Solonnikov, and N. N. Ural 0 ceva, Linear and quasilinear equations of parabolic type, AMS, Providence, R.I., 23 (1968). [18] S. V. Lototsky and B. L. Rozovskii, Stochastic dierential equations: a Wiener chaos approach, From stochastic calculus to mathematical nance: the Shiryaev festschrift (Yu. Kabanov, R. Liptser, and J. Stoyanov, eds.), Springer, 2006, pp. 433{507. [19] S. V. Lototsky and B. L. Rozovskii, Stochastic partial dierential equations driven by purely spatial noise, SIAM J. Math. Anal. 41 (2009), no. 4, 1295{1322. [20] S. V. Lototsky, B. L. Rozovskii, and D. Sele si, On generalized Malliavin calculus, Stochastic Process. Appl. 122 (2012), no. 3, 808{843. [21] W. Luo, Wiener chaos expansion and numerical solutions of stochastic partial dieren- tial equations (2006) Ph.D. Thesis California Institute of Technology. MR 3078549. [22] R. Mikulevicius and B. L. Rozovskii, On unbiased stochastic Navier-Stokes equations, Probab. Theory Related Fields 154 (2012), no. 3-4, 787{834. MR 3000562. [23] D. Nualart, Malliavin calculus and related topics, 2nd ed., Springer, New York, 2006. [24] D. Nualart, Malliavin calculus and its applications, American Mathematical Society, Providence, RI (2009). [25] B. K. ksendal, Stochastic dierential equations: an introduction with applications, 6th ed., Springer, 2003. [26] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer, Berlin, (2004). [27] A. M. Savchuk and A. A. Shkalikov, Sturm-liouville operators with singular potentials, Math. Notes 66 (1999), no. 6, 741{753. [28] H. Uemura, Construction of the solution of 1-dimensional heat equation with white noise potential and its asymptotic behavior, Stochastic Anal. Appl. 14 (1996), no. 4, 487{506. 117 [29] H. Wendland, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics (2005). [30] E. Wongand M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist. 36 (1965), 1560{1564. 118

Abstract (if available)

Abstract The linear stochastic heat equation is often the starting point in the analysis of various physical phenomena involving randomness. The noise in the equation can enter additively or multiplicatively. In the basic physical setting, additive noise corresponds to random heat source

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Creator Kim, Hyun-Jung (author)

Core Title Time-homogeneous parabolic Anderson model

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 07/12/2018

Defense Date 04/30/2018

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Tag chaos solution,fundamental solution,geometric rough path solution,mild solution,OAI-PMH Harvest,parabolic Anderson model,stochastic integral,Stratonovich interpretation,time-homogeneous Gaussian white noise,Wick-Itô-Skorokhod interpretation

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Advisor Lototsky, Sergey (committee chair), Ghanem, Roger (committee member), Mikulevicius, Remigijus (committee member)

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