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Second order in time stochastic evolution equations and Wiener chaos approach

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Second order in time stochastic evolution equations and Wiener chaos approach

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Content SECOND ORDER IN TIME STOCHASTIC EVOLUTION EQUATIONS AND WIENER CHAOS APPROACH by Jie Zhong 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) May 2013 Copyright 2013 Jie Zhong Dedication I dedicate this work to my parents and to my wife Lu. ii Acknowledgements First of all, I would like to give my sincere thanks to my advisor Prof. Sergey Lototsky for introducing me to the fascinating eld of stochastic partial dierential equations. Without his constantly encouragement and support, this thesis could not be nished. I am so grateful to Prof. Boris Rozovsii for his invitation of my visit to ICERM at Brown. My heart was very much touched by his great personality as well as his unbelievable mathematical intuition. I also deeply appreciate the interesting discussions and comments from my committee members: Prof. Remigijus Mikulevicius and Prof. Roger Ghanem. iii Table of Contents Dedication ii Acknowledgements iii Abstract v Chapter 1 Introduction 1 Chapter 2 Stochastic parabolicity and evolution equation 7 2.1 Stochastic heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 First order in time evolution equation with constant coecients . . . . . . 10 2.3 Well-posedness of abstract parabolic equation . . . . . . . . . . . . . . . . 13 Chapter 3 Second order in time equation with constant coefficients 18 3.1 The deterministic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 The stochastic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Chapter 4 Malliavin Calculus and Wiener chaos decomposition 32 4.1 The Skorohod integral with respect to Gaussian white noise . . . . . . . . 32 4.2 Connections with Wick product . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 The weighted Wiener chaos space . . . . . . . . . . . . . . . . . . . . . . . 39 Chapter 5 Second order in time equation with unbounded damping op- erators 48 5.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2 An overview of the deterministic equation . . . . . . . . . . . . . . . . . . 54 5.3 Wiener chaos solution and its propagator . . . . . . . . . . . . . . . . . . 58 5.4 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.5 Optimality of the weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 References List 75 iv Abstract This thesis aims to study the well-posedness of second order in time stochastic evolution equations. Motivated by the well known stochastic parabolicity condition, a class of second order in time stochastic partial dierential equations with constant coecients is studied, and well-posedness in Sobolev spaces on R d is established by investigating the eigenvalues of a matrix with entries in the form of symbols of the dierential operators. There are also many important equations, especially with multiplicative noises, can hardly admit a square integrable solution. A new method to study second order in time stochastic evolution equations is developed. The idea is to separate the deterministic and stochastic components of the equation by projecting the equation on a suitable basis in the probability space. The coecients in the expansion, called the propagator of the stochastic equation, are characterized by a system of deterministic equations. A well- posedness result is obtained by constructing a weighted solution space. v Chapter 1 Introduction In this thesis, we focus on the second order in time stochastic evolution equations. In particular, we investigate the well-posedness, namely, existence, uniqueness and continu- ous dependence on inputs, of the solution. This is the rst step to study an equation and provides fundamental guidelines for other interesting problems such as large deviations, martingale problems, regularity of the probabilistic law of the solution and also stochastic computations. The second order in time equation plays an important role in applications, and usually describes wave motion and mechanical vibration, the most commonly observed physical phenomena in the real world. Not like parabolic equations (for example the heat equa- tion), the second order in time equations do not improve the regularity of the inputs. This makes the mathematical problem more challenging and interesting. Two dierent methods are developed to solve our equations. One is the classical Fourier transform to study the equation with constant coecients in the framework of square integrability. The other is a new analytical method, called the Wiener chaos approach, using the tools from innite-dimensional analysis. 1 We rst investigate well-posedness in Sobolev spaces onR d of the initial value problem for a second order in time stochastic evolution equation with constant coecients and multiplicative time-only Gaussian white noise. It is well known (for example, see [37]) that the stochastic parabolic It^ o equation u t =au xx +u x _ w(t); t> 0;x2R (1.0.1) is well-posed in the sense of square integrability in Sobolev spaceH r (R) if and only if the stochastic parabolicity condition 2a 2 holds. On the other hand, the stochastic wave equation u tt +au x +c 2 u xx = 1 u x _ w 1 (t) + 2 u t _ w 2 (t) is well-posed in every Sobolev space for all real a and 1 ; 2 as long as c 2 > 0. However, the ultra sonic wave equation with Kelvin-Voigt damping (see Meyers and Chawla [28]) and random perturbed viscosity u tt =Eu xx + ( + _ w(t))u txx (1.0.2) whereE 1 is the Young's modulus (or elasticity) and is the viscosity of the material, does not have a square integrable solution if is not zero. These interesting facts motivate us to investigate how the coecients in the deterministic and stochastic parts of a second order in time equation contribute to the well-posedness. 1 In this thesis, we use the standard notation E for the Young's modulus; and this E for expectation in probability. 2 The diculty to study this problem is that by Fourier method the derived second order (stochastic) dierential equation with variable coecients does not, in general, have a closed-form solution, compared to parabolic equations or rst order equations. To overcome this obstacle, we follow the idea from physicists, who study the oscillators with random frequency, see for example, Gitterman [9], that we rewrite the equation as a rst order system and derive a single third order (fourth order in general) dierential equation for the second moment. This technique was in fact systematically and rigorously explored in mathematics in Khasminskii [13], while studying the stability of stochastic ordinary dierential equations. However, in our situation, the symbols of dierential operators are possibly complex valued, and thus we introduce four quantities Ej^ u(t)j 2 , Ej^ u t (t)j 2 , E^ u t (t)^ u (t) and E^ u(t)^ u t (t), where ^ u is the Fourier transform of u, and the symbol means the complex conjugate. We build up a system of dierential equations for these four quantities and conclude that a second order in time stochastic evolution equation is well-posed in the Sobolev spaces if and only if there exists a real number c such that all eigenvalues of a 4 4 matrix with entries in the form of symbols of the dierential operators are in the regionfz2C :<z <cg. If all symbols are real, we can reduce the matrix to 3 3 and get the same result. Even in this case, there is no simple relation as the stochastic parabolicity condition since the cubic formula is much more complicated to solve third order polynomial equations. Possibly repeated solutions make the problem even harder. Instead, we are trying to verify the general theory in particular examples. By a suitable generalization of the Routh-Hurwitz criterion (see for example, the real case, 3 Mekin [27, Theorem 4.6]; the complex case, Morris [34]), we obtain an interesting result that the equation u tt + A 1" u t +cA 1+" u =Au _ w(t); where A is a pseudo-dierential operator with the symbol A(y) = y 2 , and "2 (0; 1), is well-posed if > 0 and 2 c> 2 . In the second part of this thesis, we develop the Wiener chaos method to solve a class of damped systems. Wiener chaos expansion is a stochastic analogue of Fourier series, which separates the deterministic and stochastic components of the equation. The coecients satisfy a system of deterministic partial dierential equations, often referred to as the propagator. The idea of the propagator originated from the work by Mikulevicius and Rozovsky in [29]. It can also be derived for certain nonlinear equations: in particular, stochastic Navier-Stokes equation in [30{33], and stochastic Burger's equation in [12]. As a powerful computation tool, the Wiener chaos approach is successfully applied to nonlinear ltering problem [20,21] and engineering simulations, for example [39]. The simplest example of a second order in time equation is the wave equation u tt =u xx ; (1.0.3) which is conservative in some energy form. However, in reality there is no such system and there are always dissipative mechanisms causing the energy to decay during any positive time interval. Therefore, a damping term is naturally introduced to the equation. Moreover, to describe as many physical models as possible, we allow the damping operator 4 to be unbounded in the natural state space. To this end, we can look at the following abstract evolution equation u(t) =A 1 u(t) +A 2 _ u(t) +f(t); (1.0.4) where A 1 ;A 2 are linear unbounded operators on some Hilbert space H, and f is the inhomogeneous term. This deterministic equation (1.0.4) in fact has been extensively studied in many literature. For example, Chen and Russel in [4] presented a mathematical model exhibiting the empirically observed damping rates in elastic systems. The model studied is the homogeneous form of equation (1.0.4) and A 2 is related in various ways to the positive square root of A 1 . More general equations were investigated by Banks et al. [1], where the order of the operator A 2 can be relaxed up to the same as the order of A 1 . Their considerations were prompted by the use of bonded or embedded piezoceramic pathes as components in a smart material technology. Our purpose here is to investigate the well-posedness of the stochastic version of (1.0.4), that is, u(t) =A 1 u(t) +A 2 ^ u(t) +f(t) +(B 1 (t)u(t) +B 2 (t) _ u(t)); (1.0.5) with initial conditions u(0) = u 0 ; _ u(0) = u 1 , where B 1 (t);B 2 (t) are linear unbounded operators on H for all t2 [0;T ]; is the Skorohod integral. One of the major advantages of using Skorohod integral, instead of It^ o's calculus, is that we can cover all types of Gaussian random perturbations: time only, space-time 5 or even space only. For example, given a separable Hilbert spaceU. Let h2U, then (h) = _ W (h). IfU =L 2 (0;T );h = [0;t) , then(h) =W (t), the standard Wiener process; ifU =L 2 (G),G is a domain inR d , then (h) can be referred to as a purely spatial noise; and ifU =L 2 (0;T )L 2 (G), we may say (h) is a space-time noise. The strong connection with Wiener chaos expansion is another important feature of the Skorohod integral. In [23], [24], several simple equations of parabolic type have been shown that they do not possess a square-integrable solution in the probability space and must be solved in special weighted spaces. Thanks to the Cameron-Martin basis, a generalized solution in terms of a formal Wiener chaos series leads to natural weights and a natural replacement of the square integrability condition. It is also worth pointing out an interesting observation of equation (1.0.5) that it is unbiased in that it preserves the mean dynamics due to the property of the Skorohod integral and the linearity of the equation. To be specic, the function u := Eu solves equation (1.0.4) with the inhomogeneous term Ef. Last but not least, as a generalization, the Skorohod integral coincides with the clas- sical It^ o's integral when the integrand is adapted and square integrable. As a result, the generalized Wiener chaos solution is a bona de generalization of the usual It^ o solution. 6 Chapter 2 Stochastic parabolicity and evolution equation 2.1 Stochastic heat equation A stochastic partial dierential equation can be considered as a deterministic partial dif- ferential equation perturbed by some noise. It is always an interesting question that how the noise or stochasticity aects the solvability and properties of the solution. Fedrizzi and Flandoli in [7] present a surprising result about the stochastic linear transport equa- tion with multiplicative noise. It is proved that a certain Sobolev degree of regularity of the solution is maintained, while in the deterministic case smooth initial conditions may develop discontinuities. There are also examples showing that the noise may improve the pathwise uniqueness in innite dimensional spaces; see [3] or [8] for a review. However, in most cases the noise makes the situation worse. We start with the initial value problem for the heat equation u t =au xx ; t> 0;x2R; u(0;x) =(x): (2.1.1) 7 Taking the Fourier transform in the space variable, we nd ^ u t =ay 2 ^ u; or ^ u(t;y) = ^ (y)e ay 2 t ; where ^ f(y) = 1 p 2 Z R f(x)e ixy dx: Therefore, in order to invert the Fourier transform, we need to assume that a is nonneg- ative; otherwise, we have to require that decays very fast. If a is strictly positive, then we get the familiar formula for the solution u(t;x) = 1 p 4at Z R e jxyj 2 =(4at) (y)dy: (2.1.2) Next, we consider the stochastic heat equation du =au xx dt +u x dw(t); t> 0;x2R; u(0;x) =(x): (2.1.3) Then d^ u =ay 2 ^ u +i y^ udw(t) or ^ u(t;y) = ^ (y)e cy 2 t+i w(t) ; where c =a 2 =2. We denote by H r =H r (R);r2R, the Sobolev space onR, i.e., H r (R) = f : Z R j ^ f(y)j 2 (1 +jyj 2 ) r dy<1 : 8 We assume for simplicity that is non-random. Then given c 0, Ekuk 2 H r = Z R Ej^ u(t;y)j 2 (1 +jyj 2 ) r dy = Z R j ^ (y)j 2 e 2cy 2 t (1 +jyj 2 ) r dy<1; for each r2 R, which implies that the equation (2.1.3) is well-posed in every Sobolev space H r if and only if 2a 2 ; (2.1.4) called the stochastic parabolicity condition. Now we give an alternative way to solve the equation (2.1.3) and derive the same con- dition, based on the famous It^ o-Wentzell formula (see for example Kunita [15, Theorem 3.3.1] or Rozovsky [37, Theorem 1.4.9]). Assume u(t;x) is a classical solution of equation (2.1.3), and 2C 1 b (R). Dene v(t;x) =u(t;xw(t)); t 0;x2R: Then by It^ o-Wentzell formula, v(t;x) =(x) + Z t 0 v x (s;x)(dw(s)) + 1 2 Z t 0 v xx (s;x) 2 ds + Z t 0 au xx (s;xw(s))ds + Z t 0 u x (s;xw(s))dw(s) 2 Z t 0 u xx (s;xw(s))ds; or equivalently, dv = (a 2 =2)v xx dt; v(0;x) =(x): 9 Suppose the stochastic parabolicity condition holds, then we obtain thatu(t;x) =v(t;x+ w(t)), where v solve the deterministic heat equation (2.1.1). 2.2 First order in time evolution equation with constant coecients Now we extend the results in the previous section to a more general setting. Consider a deterministic evolution equation in R d that is rst order in time. Using the standard multi-index notations = ( 1 ;:::; d ); i 2f0; 1; 2;:::g; jj = d X i=1 i ; x =x 1 1 x d d ; @x =@x 1 1 @x d d ; (2.2.1) dene partial dierential operator A = X jjm A A @ jj @x ; (2.2.2) with constant coecients A 2R, and the corresponding symbol A(y) = X jjm A i jj A y ; y2R d ; i = p 1: (2.2.3) Consider the evolution equation @u(t;x) @t + Au(t;x) = 0; t> 0; x2R d ; (2.2.4) 10 with initial condition u(0;x) =u 0 (x). Let U(t;y) = 1 (2) d=2 Z R d e ixy u(t;x)dx (2.2.5) be the Fourier transform of the solution of (2.2.4), assuming it exists. Then U t (t;y) +A(y)U(t;y) = 0 (2.2.6) and U(t;y) =U 0 (y)e A(y)t : (2.2.7) The traditional denition (see Evans [6]) of well-posedness of a partial dierential equation is existence, uniqueness and continuous dependence of the solution on the initial conditions and other input. In this section, we will interpret it in a more relaxed sense. Denition 2.2.1 We say that an evolution equation is well-posed in a space H if, for every initial condition from H, the solution exists and is an element of H for all t> 0. Now recall that the Sobolev spaceH onR d is the collection of (generalized) functions f such that kfk 2 = Z R d (1 +jyj 2 ) j ^ f(y)j 2 dy<1; ^ f is the Fourier transform of f. Therefore, with< denoting the real part of a complex number, equality (2.2.7) implies that existence of a number c2R such that <A(y)c for all y2R d (2.2.8) 11 is necessary and sucient for equation (2.2.4) to be well-posed in the Sobolev spaces. Note also that, under condition (2.2.8), we can invert the Fourier transform and construct the solution u = u(t;x) from U(t;y); with this approach, uniqueness of solution of (2.2.6) implies uniqueness of solution of (2.2.4). There is an immediate extension of (2.2.8) to stochastic equations. Let F = ( ;F; (F t ) t0 ;P) be a stochastic basis with the usual conditions, and letw =w(t) be a standard Brownian motion onF. Beside A, consider another operator B = X jjm B B @ jj @x ; with constant coecients B 2 R, and the corresponding stochastic evolution equation in the It^ o form du(t;x) + Au(t;x)dt = Bu(t;x)dw(t); t> 0; x2R d ; (2.2.9) with some initial condition u 0 independent of w. Similar to the deterministic case, we say that equation (2.2.9) is well posed in the Sobolev spaces H if, for every 2 R, Eku 0 k 2 <1 impliesEku(t;)k 2 <1, t> 0. The Fourier transform U =U(t;y) of u satises dU t (t;y) +A(y)U(t;y)dt =B(y)U(t;y)dw(t); (2.2.10) 12 where B(y) is the corresponding symbol of B, so that U(t;y) =U 0 (y) exp A(y) + 1 2 B 2 (y) +B(y)w(t) : (2.2.11) Then EjU(t;y)j 2 =EjU 0 (y)j 2 E exp 2<A(y) +< B 2 (y) + 2< B(y) w(t) =EjU 0 (y)j 2 exp 2<A(y) +< B 2 (y) + 2 <B(y) 2 =EjU 0 (y)j 2 exp 2<A(y) +jB(y)j 2 ; and the stochastic analogue of (2.2.8) becomes a well-known condition 2<A(y)jB(y)j 2 c for all y2R d : (2.2.12) An immediate consequence of (2.2.12) is the order condition on the operators A; B, m A 2m B ; (2.2.13) which is necessary but not sucient for well-posedness of (2.2.9). 2.3 Well-posedness of abstract parabolic equation In this section, we summarize the well-posedness of stochastic parabolic equation within the framework of Hilbert space theory. The details can be found in the books [5], [37] 13 or [25]; see also [17]. We also characterize the corresponding stochastic parabolicity based on the energy estimates, instead of Fourier transform. For a Hilbert spaceX, (;) X andkk X denote the inner product and the norm inX. Denition 2.3.1 The triple (V;H;V 0 ) of Hilbert spaces is called normal if and only if 1. V ,!H ,!V 0 and both embeddingsV ,!H andH ,!V 0 are dense and continuous; 2. The space V 0 is the dual of V relative to the inner product in H; 3. There exists a constant C > 0 so thatj(h;v) H j Ckvk V khk V 0 for all v2 V and h2H. For example, the Sobolev spaces (H l+ 2 (R d );H l 2 (R d );H l 2 (R d )), > 0, l2 R, form a normal triple. Denote byhv 0 ;vi, v 0 2V 0 , v2V , the duality between V and V 0 relative to the inner product in H. The properties of the normal triple imply thatjhv 0 ;vij Ckvk V kv 0 k V 0, and, if v 0 2H and v2V , thenhv 0 ;vi = (v 0 ;v) H ; Let F = (F;F;fF t g t0 ;P) be a stochastic basis with the usual assumptions. In par- ticular, the sigma-algebrasF andF 0 areP-complete, and the ltrationfF t g t0 is right- continuous; for details, see [19, Denition I.1.1]. We assume that F is rich enough to carry a collection w k =w k (t); k 1; t 0 of independent standard Wiener processes. Given a normal triple (V;H;V 0 ) and a family of linear bounded operatorsA(t) :V ! V 0 ,M k (t) :V !H, t2 [0;T ], consider the following equation: u(t) =u 0 + Z t 0 (Au(s) +f(s))ds + Z t 0 (M k u(s) +g k (s))dw k (s); 0tT; (2.3.1) 14 where T <1 is xed and non-random and the summation convention is in force. Assume that, for all v2V , X k1 kM k (t)vk 2 H <1; t2 [0;T ]: (2.3.2) The input data u 0 ;f, and g k are chosen so that E 0 @ ku 0 k 2 H + Z T 0 kf(t)k 2 V 0dt + X k1 Z T 0 kg k (t)k 2 H dt 1 A <1; (2.3.3) u 0 isF 0 -measurable, and the processes f;g k areF t -adapted, that is, f(t) and each g k (t) areF t -measurable for each t 0. Denition 2.3.2 An F t -adapted process u 2 L 2 (F;L 2 ((0;T );V )) is called a square- integrable solution of equation (2.3.1)) if, for every v2V , the equality (u(t);v) H = (u 0 ;v) H + Z t 0 hAu(s) +f(s);vids + X k1 (M k u(s) +g k (s);v) H dw k (s) (2.3.4) holds P-a.s. for all 0tT , Existence and uniqueness of the square integrable solution for (2.3.1) can be estab- lished when the equation is parabolic. Denition 2.3.3 (stochastic parabolicity condition) Equation (2.3.1) is calledparabolic if there exists a positive number c A and a real number M so that, for all v2 V and t2 [0;T ], 2hA(t)v;vi + X k1 kM(t) k vk 2 H +c A kvk 2 V Mkvk 2 H : (2.3.5) 15 Theorem 2.3.4 If conditions (2.3.3) and (2.3.5) hold, then there exists a unique square integrable solution of (2.3.1). The solution process u is an element of the space L 2 (F;L 2 ((0;T );V )) \ L 2 (F;C((0;T );H)) and satises E sup 0<t<T ku(t)k 2 H + Z T 0 ku(t)k 2 V dt C(C 0 ;;T )E 0 @ ku 0 k 2 H + Z T 0 kf(t)k 2 V 0dt + X k1 Z T 0 kg k (t)k 2 H dt 1 A : (2.3.6) As an application of Theorem 2.3.4, consider the second order (in space) equation du(t;x) = (a ij (t;x)u x i x j (t;x) +b i (t;x)u x i (t;x) +c(t;x)u(t;x) +f(t;x))dt + ( ik (t;x)u x i (t;x) + k (t;x)u(t;x) +g k (t;x))dw k (t) (2.3.7) with 0<tT; x2R d ; and initial condition u(0;x) =u 0 (x). Assume that (S1) The functions a ij are bounded and Lipschitz continuous, the functions b i , c, ik , and are bounded measurable. (S2) There exists a positive number "> 0 so that (2a ij (x) ik (x) jk (x))y i y j "jyj 2 ; x;y2R d ; t2 [0;T ]: (S3) There exists a positive number K so that, for all x2R d , P k1 j k (x)j 2 K: 16 (S4) The initial conditionu 0 2L 2 (F ;L 2 (R d )) isF 0 -measurable, the processesf2L 2 (F [0;T ];H 1 2 (R d )) and g k 2L 2 (F [0;T ];L 2 (R d )) areF t -adapted, and X k1 Z T 0 Ekg k k 2 L 2 (R d ) (t)dt<1 . Theorem 2.3.5 Under assumptions (S1)-(S4), equation (2.3.7) has a unique square in- tegrable solution u2L 2 (F;L 2 ((0;T );H 1 2 (R d ))) \ L 2 (F;C((0;T );L 2 (R d ))); and the solution satises E sup 0<t<T kuk 2 L 2 (R d ) (t) + Z T 0 kuk 2 H 1 2 (R d ) (t)dt C(K;";T )E 0 @ ku 0 k 2 L 2 (R d ) + Z T 0 kfk 2 H 1 2 (R d ) (t)dt + X k1 Z T 0 kg k k 2 L 2 (R d ) (t)dt 1 A : (2.3.8) Remark 2.3.6 Whatever in the form of (2.1.4), (2.2.12) or (2.3.5) , the stochastic parabolicity condition essentially means that the size of the noise or stochasticity should be relatively small, or the deterministic part of the equation dominates the stochastic part, to guarantee the well-posedness. 17 Chapter 3 Second order in time equation with constant coefficients 3.1 The deterministic case As discussed in Section 2.2, the transition from the rst order in time deterministic equations to its stochastic counterpart is relatively straightforward. It is mainly because that for every integrable function f, the equation x 0 (t) =f(t)x(t) has an easy closed-form formula for solution. However, the situation is dierent for higher-order equations, where, in particular, equations with constant coecients x 00 (t) +ax 0 (t) +bx(t) = 0 18 have a rather complicated formula for the solution and equations with variable coecients, such as x 00 (t) +f(t)x(t) = 0; (3.1.1) usually have no closed-form solution at all. If f in (3.1.1) is periodic, then it is the well known Hill's equation (see [26]), which is already quite a sophisticated object to study. As a result, there are signicant complications in transition from the rst-order deter- ministic equations to second-order deterministic ones, and then again from deterministic to stochastic within the second-order setting. We rst focus on the second order in time deterministic equations. Let us consider the following equation u tt (t;x) + Au t (t;x) + Mu(t;x) = 0; t> 0; x2R d ; (3.1.2) with initial conditionsu 0 (x) =u(0;x),v 0 (x) =u t (0;x) and one more constant-coecient partial dierential operator M. Taking the Fourier transform, we get U tt (t;y) +A(y)U t (t;y) +M(y)U(t;y) = 0; (3.1.3) where U is the Fourier transform of u is space variable, and A, M are symbols of A, M, respectively. 19 Now, we have a second-order ordinary dierential equation to solve. The solution of (3.1.3) depends on the roots p(y);q(y) of the characteristic equation r 2 +A(y)r +M(y) = 0: (3.1.4) Dene X 1 (t;y) = 8 > > > > < > > > > : q(y)e p(y)t p(y)e q(y)t q(y)p(y) ; if p(y)6=q(y); (1q(y)t)e q(y)t ; if p(y) =q(y); X 2 (t;y) = 8 > > > > < > > > > : e p(y)t e q(y)t p(y)q(y) ; if p(y)6=q(y); te q(y)t ; if p(y) =q(y): (3.1.5) Then direct computations show that U(t;y) =U 0 (y)X 1 (t;y) +V 0 (y)X 2 (t;y); (3.1.6) V 0 is the Fourier transform of v 0 . Even without explicit expressions for p(y) and q(y) in terms of A(y) and M(y), we see that (3.1.6) is much more complicated than (2.2.7). Still, it is natural to say that equation (3.1.2) is well-posed in the Sobolev spaces if, for every 2R, there exists a =( ) so thatu 0 2H andv 0 2H implyu(t;)2H and v(t;)2H for all t> 0. The example of the wave equation with A = 0, M =, for which U(t;y) =U 0 (y) cos(ty) + V 0 (y) y sin(ty); 20 and therefore = 1 shows that there indeed should be some dependence between and . Formulas (3.1.5) and (3.1.6) suggest that the underlying condition for well-posedness of (3.1.2) is best formulated not explicitly in terms of the symbols A(y) andM(y) of the operators in the equation, but in terms of the roots p(y) and q(y) of the characteristic equation (3.1.4). Note that both p and q are algebraic functions of y andjpj;jqj grow at most polynomially asjyj! +1. Then we conclude that equation (3.1.2) is well-posed in Sobolev spaces if and only if there exists a c2R such that, for all y2R d , <p(y)c; <q(y)c: (3.1.7) Since the analogue of (3.1.4) for the rst-order equation (2.2.4) isr +A(y) = 0, condition (3.1.7) can indeed be considered as an analogue of (2.2.8). Because (3.1.7) also holds for everyc 1 >c, there is no dierence between strict and non-strict inequalities. Verication of (3.1.7) for a particular equation can be carried out in various ways, for example, using a suitable generalization of the Routh-Hurwitz criterion; this is also an example where a strict inequality in (3.1.7) is more convenient. To summarize, while the symbols of the dierential operators determine well-posedness of the equation, the condition for well-posedness is best stated in terms of the correspond- ing characteristic roots rather than directly in terms of the symbols of the dierential operators. 21 3.2 The stochastic case For a stochastic version of equation (3.1.2), the situation is even more complicated and in general requires analysis of eigenvalues and eigenvectors of a 4-by-4 matrix with complex entries. To state the corresponding result, let B k ; k 1 and N k ; k 1, be partial dierential operators with constant coecients and the symbols B k (y) = X jjm B k i jj B ;k y ; N k (y) = X jjm N k i jj N ;k y ; (3.2.1) and let w k = w k (t); t 0; k 1, be independent standard Brownian motions. We assume that X k1 X jjm B k B 2 k; + X k1 X jjm N k N 2 k; <1: (3.2.2) Consider the equation u tt (t;x) + Au t (t;x) + Mu(t;x) = X k1 B k u t (t;x) + N k u(t;x) _ w k (t); (3.2.3) fort> 0,x2R d , and with initial conditionsu 0 (x) =u(0;x),v 0 (x) =u t (0;x) independent of all w k . Of course, (3.2.3) must be interpreted as a system of two It^ o equations 0 B B @ du dv 1 C C A = 0 B B @ 0 1 N A 1 C C A 0 B B @ u v 1 C C A dt + 0 B B @ 0 P k1 B k v + N k u dw k 1 C C A ; but, similar to the deterministic case, we will often work with a more compact form (3.2.3). The reason to consider innitely many Brownian motions is potential future extension of the results to equations with variable coecients. Otherwise, to get arbitrary correlation 22 between the noise terms in the position u and speed u t , two independent Brownian motions are enough. To state precisely the denition of well-posedness for (3.2.3), we need the spaceS(R d ) of (rapidly decreasing) test functions, the dual spaceS 0 (R d ) of tempered distributions (see for example [38], and the duality (;) between the two spaces. Given a partial dierential operator A as in (2.2.2) with (real) constant coecients, we denote by A its adjoint A = X jjm A (1) jj A @ jj @x ; (3.2.4) that is, (A'; ) = ('; A ) for all'; 2S(R d ). By (2.2.2) and (3.2.4), the corresponding symbol A (y) of A is the complex conjugate of A(y): (A )(y) = A(y) : Denition 3.2.1 Equation (3.2.3) is said to be well-posed in the Sobolev spaces if, for every 2R, there exists a = ( )2R such that, for every u 0 2 L 2 (F;H ) and v 0 2 L 2 (F;H ), there exists a unique pair of processes u;v with the following properties: for 23 all t> 0, u(t;)2L 2 (F;H ), v(t;)2L 2 (F;H ), and, for every test function '2S(R d ), the equality 0 B B @ (u;')(t) (v;')(t) 1 C C A = 0 B B @ (u 0 ;')(t) (v 0 ;')(t) 1 C C A + 0 B B @ R t 0 (v;')(s)ds R t 0 (u; M ')(s)ds R t 0 (v; A ')(s)ds 1 C C A + 0 B B @ 0 P k1 R t 0 (v; B k ')(s) + (u; N k ')(s) dw k (s) 1 C C A holds with probability one. For y2R d , dene the functions F B (y) = X k1 jB k (y)j 2 ; F N (y) = X k1 jN k (y)j 2 ; F BN (y) = X k1 B k (y)N k (y): (3.2.5) Here is the main result of the this section. Theorem 3.2.2 Equation (3.2.3) is well posed in the Sobolev spaces if and only if there exists a real number c such that, for every y2R d , all eigenvalues of the matrix M = 0 B B B B B B B B B B @ 0 1 1 0 M(y) A(y) 0 1 M (y) 0 A (y) 1 F N (y) M (y) +F BN (y) M(y) +F BN (y) 2<A(y) +F B (y): 1 C C C C C C C C C C A (3.2.6) are in the regionfz2C :<z<cg. 24 Proof: For simplicity, we only prove the theorem for one Brownian motion. The general case can be showed by independence and changing notations. Suppose U(t;y) and V (t;y) are the Fourier transforms of u(t;x) and v(t;x), respec- tively. It follows from equation (3.2.3) that 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : dU =Vdt dU =V dt dV = (MUAV )dt + (BV +NU)dw dV = (M U A V )dt + (B V +N U )dw: Then by It^ o's formula, we have djUj 2 = (UV +U V )dt; d(U V ) =U (MUAV )dt +jVj 2 dt +U (BV +NU)dw; d(UV ) =U(M U A V )dt +jVj 2 dt +U(B V +N U )dw; and djVj 2 = (jBj 2 jVj 2 +BN U V +NB UV +jNj 2 jUj 2 M U V 2<AjVj 2 MUV )dt + (2<BjVj 2 +NUV +N U V )dw: 25 Set z 1 = EjUj 2 ;z 2 = E(U V );z 3 = E(UV ) and z 4 = EjVj 2 , we obtain a system of equations 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : z 0 1 =z 2 +z 3 ; z 0 2 =Mz 1 Az 2 +z 4 ; z 0 3 =M z 1 A z 3 +z 4 ; z 0 4 =F N z 1 + (F BN M )z 2 + (F BN M)z 3 + (F B 2<A)z 4 ; or equivalently, ~ z 0 =M~ z; where~ z = (z 1 ;z 2 ;z 3 ;z 4 ) T . Since the trace of the matrix (3.2.6) is4<A(y) +F B (y) and F B (y) 0, we get the following necessary conditions for well-posedness: <A(y) ~ c; m A 2 max k m B k ; in particular, the orders of B k cannot be arbitrarily large. When all operators have real symbols, then Theorem 3.2.2 simplies as follows. 26 Theorem 3.2.3 If the symbols of all the operators are real, then Equation (3.2.3) is well posed in the Sobolev spaces if and only if there exists a real number c such that, for every y2R d , all eigenvalues of the matrix ~ M = 0 B B B B B B @ 0 2 0 M(y) A(y) 1 F N (y) 2M(y) + 2F BN (y) 2A(y) +F B (y) 1 C C C C C C A (3.2.7) are in the regionfz2C :<z<cg. Proof: If A(y) =A (y); M(y) =M (y); F BN (y) =F BN (y); (3.2.8) then the matrix M has a simple eigenvalue A(y) and the corresponding eigenvec- tor (0 1 1 0) T , and there is a one-to-one correspondence between the remaining eigenvalues/eigenvectors ofM and the eigenvalues/eigenvectors of ~ M as follows. Let v = (v 1 ;v 2 ;v 3 ;v 4 ) T be a column vector such thatMv =v and 6=A(y). Then (3.2.8) implies v 2 =v 3 , and then the vector ~ v = (v 1 ;v 2 ;v 4 ) > satises ~ M~ v =~ v. Conversely, if ~ v = (v 1 ;v 2 ;v 3 ) > satises ~ M~ v = ~ v, then, under (3.2.8), the vector v = (v 1 ;v 2 ;v 2 ;v 3 ) > satisesMv =v. 3.3 Examples Now we consider the following equation with "2 [1; 1] u tt + A 1" u t +cA 1+" u =Au _ w(t); (3.3.1) 27 for t> 0;x2R, and with initial conditions u 0 (x) =u(0;x);v 0 (x) =u t (0;x) independent of w. We assume that the symbol A of A is y 2 and all the coecients are real. Suppose U is the Fourier transform of u, then it satises the ordinary equation U(t) + y 2(1") _ U(t) +cy 2(1+") U(t) =y 2 U(t) _ w(t); (3.3.2) for xed y. It is easy to observe that equation (3.3.1) in fact contains some interesting concrete examples. In particular, we have u tt + u txxxx +cu =u xx _ w(t); " =1; u tt u txx cu xx =u xx _ w(t); " = 0; u tt + u t +cu xxxx =u xx _ w(t); " = 1: Since the symbols involving coecients are real, we can apply Theorem 3.2.3 to equa- tion (3.3.1). Note that in this case we have A = y 2(1") ; M =cy 2(1+") ; B = 0; N =y 2 ; 28 and F N = N 2 ;F B = F BN = 0 in equation (3.3.3). Therefore, we conclude that the equation (3.3.1) is well-posed in the Sobolev spaces if and only if there exists a real number b such that, for every y2R d , all eigenvalues of the matrix ~ M = 0 B B B B B B @ 0 2 0 cy 2(1+") y 2(1") 1 2 y 4 2cy 2(1+") 2 y 2(1") 1 C C C C C C A (3.3.3) are in the regionfz2 C :<z < bg. Equivalently, equation (3.3.1) is well-posed if and only if<z(y)<b uniformly in y, where z is the root of the characteristic equation p(z) :=p(z(y);y) := det(zI ~ M(y)) = 0: After some easy computation, we have p(z) =z 3 +a 1 z 2 +a 2 z +a 3 ; (3.3.4) where a 1 = 3 y 2(1") ; a 2 = 2 2 y 4(1") + 4cy 2(1+") a 3 = 2(2c 2 )y 4 : It is observed that the condition<z < b is equal to<(zb) < 0, so we may apply the Routh-Hurwitz criterion, which provides a sucient and necessary condition for all of the 29 roots of a polynomial with real coecients to lie in the left half of the complex plane. Given the polynomial f(z) =z n +a 1 z n1 + +a n1 z +a n ; the Routh-Hurwitz criterion for n = 2; 3; 4 are summarized as follows: n = 2 :a 1 > 0 and a 2 > 0. n = 3 :a 1 > 0;a 2 > 0;a 3 > 0; and a 1 a 2 >a 3 . n = 4 :a i > 0;i = 1; 2; 3; 4; and a 1 a 2 a 3 >a 2 3 +a 2 1 a 4 . For the general result, see for example [27, Theorem 4.6]. First, let us consider the case when "2 (0; 1) in equation (3.3.1). We have to keep in mind that it is enough for our purpose to derive all the conditions based on the large value of y. Now back to equation (3.3.4). It is easy to see that a 1 ;a 2 and a 3 are all strictly positive if > 0;c> 0 and 2c > 2 . Then we for free have a 1 a 2 = 6 3 y 6(1") + 12c y 4 > 4c y 4 a 3 : Therefore, equation (3.3.1) is well posed if > 0;c > 0 and 2c > 2 . In fact, the statement is still true if we replace the last inequality by equality, or 2c = 2 . Under this condition, a 3 = 0 and equation (3.3.4) reduces to p(z) =z(z 2 +a 1 z +a 2 ) = 0: 30 Thus, one of the roots of p is zero and the real parts of rest of two are bounded above as long as > 0;c> 0. Conversely, we assume that there exists b2 R such that<z(y) < b uniformly in y. Without loss of generality, we may assume b> 0. Then ~ z =zb satises<~ z< 0 and q(~ z) :=p(~ z +b) =p(z) = 0: Thus, we have q(~ z) = ~ z 3 + ~ a 1 ~ z 2 + ~ a 2 ~ z + ~ a 3 ; (3.3.5) where ~ a 1 =a 1 + 3b; ~ a 2 =a 2 + 2b 2 + 2a 1 b; ~ a 3 =a 3 +b 3 +a 1 b 2 +a 2 b: By the Routh-Hurwiz criteria, we must have that > 0;c> 0 and 2c 2 . For other values of "2 [1; 1], we can repeat the argument similarly. To conclude, we summarize the well-posedness of equation (3.3.1) as follows "2 [1; 0]: equation (3.3.1) is well-posed if and only if > 0; "2 (0; 1): equation (3.3.1) is well-posed if and only if > 0; 2c 2 ; " = 1: equation (3.3.1) is well-posed if and only if c> 0. 31 Chapter 4 Malliavin Calculus and Wiener chaos decomposition 4.1 The Skorohod integral with respect to Gaussian white noise Let F = ( ;F;P) be a complete probability space. Suppose thatU is a real separable Hilbert space with inner product (;) U . On F, consider an isonormal Gaussian process, i.e., a centered Gaussian family _ W = n _ W (h); h2U o such that E _ W (h 1 ) _ W (h 2 ) = (h 1 ;h 2 ) U : We x an orthonormal basis U =fu k ;k 1g inU, and dene a collectionf k ;k 1g of independent standard Gaussian random variables by k = _ W (u k ). For our purpose, it suces to assume thatF is the -algebra generated byf k ;k 1g. Given a real 32 separable Hilbert space X, we denote by L 2 (F;X) the Hilbert space of square-integrable F-measurable X-valued random elements f. In particular, (f;g) 2 L 2 (F;X) :=E(f;g) 2 X : When X =R, we write L 2 (F) instead of L 2 (F;R). Denition 4.1.1 A Gaussian white noise _ W onU is a formal series _ W = X k1 u k k : (4.1.1) Instead of the process _ W , we can also start with a system of independent standard Gaussian random variablesf k ;k 1g and dene an isonormal Gaussian process onU by _ W (h) = X k1 (h;u k ) U k : Example 4.1.2 IfU =L 2 ((0;T )), then _ W (h) = R T 0 h(s)dW (s), whereW (t) = _ W ( [0;t) ) is the standard Wiener process. To proceed, we review some denitions and conventions related to multi-indices. Let J be the collection of multi-indices with = ( 1 ; 2 ; ) so that each k is a non- negative integer andjj := P k1 k <1. For ;2J , we dene + = ( 1 + 1 ; 2 + 2 ; ); ! = Y k1 k !: 33 By denition, > 0 ifjj> 0 and if k k for all k 1. If , then = ( 1 1 ; 2 2 ; ): Remark 4.1.3 The setJ is not closed under substraction and the expression " k is undened if k = 0. In the sequel, we make the following convention: if there is k such that k < k , then = 0. By (0) we denote the multi-index with all zeroes. By " i we denote the multi-index with i = 1 and j = 0 for j6= i. With this notation, n" i is the multi-index with i =n and j = 0 for j6=i. Now we give a description of a multi-index withjj = n > 0 by its characteristic vector K = (i 1 ; ;i n ); where i 1 i 2 i n . The rst entry in K is the position index of the rst nonzero element of . The second entry is the same as the rst if the rst nonzero element of is greater than one; otherwise, it is the position index of the second nonzero element of and so on. As a result, if k > 0, then there exist exactly k copies of k in K . For example, if = (1; 0; 2; 0; 4; 0; 0; ) with nonzero elements 1 = 1; 3 = 2; 5 = 4, then jj = 7 and K = (1; 3; 3; 5; 5; 5; 5). It is observed that for every sequence (b k ;k 1) of positive numbers, Y k1 b k k =b i 1 b i 2 b in = Y i2K b i ; (4.1.2) which can serve as an equivalent denition of K . 34 Dene the collection of random variables =f ;2Jg as follows: = 1 p ! Y k H k ( k ); (4.1.3) where H n (x) = (1) n e x 2 =2 d n dx n e x 2 =2 (4.1.4) is the Hermite polynomial of order n. Theorem 4.1.4 The collection =f ; 2Jg is an orthonormal basis in L 2 (F;X): if 2L 2 (F;X) and =E( ), then = P 2J and Ekk 2 X = P 2J k k 2 X . This is another version of the classical result of Cameron and Martin [2], and thus we refer to as the Cameron-Martin basis in L 2 (F;X). The Malliavin derivative D (see [35]) is a continuous linear operator from D 1;2 (F;X) := ( 2L 2 (F;X) : X 2J jjk k 2 X <1 ) (4.1.5) to L 2 (F;X U). It is known that the domain D 1;2 (F;X) of the operator D is a dense subspace of L 2 (F;X); see [35]. In particular, D( " k ) = D( k ) = D( _ W (u k )) =u k ; and by the relation H 0 n (x) =nH n1 (x) for n 1, we have D( ) = X k1 p k " k u k : (4.1.6) 35 The adjoint operator , also known as the Skorohod integral, of the Malliavin derivative is a closed unbounded linear operator fromL 2 (F;X U) toL 2 (F;X) such that E('(f)) =E(D(');f) U (4.1.7) for all '2D 1;2 (F;R) and f2D 1;2 (F;X U). Similar to (4.1.6), we need to derive the expressions for the operator in the basis . Proposition 4.1.5 For 2 , h2X, and u k 2 U, ( h u k ) =h p k + 1 +" k : (4.1.8) Proof: It suces to show that E( ( h u k )) =E( h p k + 1 +" k ) for all 2J . In fact, it follows from (4.1.6) and (4.1.7) that E( ( h u k )) =E(D( ); h u k ) U = p k hE( " k ) = 8 > > > > < > > > > : p k h; if =" k ; 0; if 6=" k = 8 > > > > < > > > > : p k+1 h; if =" k ; 0; if 6=" k =E( h p k+1 +" k ): 36 As a generalization of the Laplace operator to an innite-dimensional setting, the composite operatorL := D, also known as the Ornstein Uhlenbeck operator, is linear and unbounded on L 2 (F). It follows from (4.1.6) and (4.1.8) that the random variables are eigenfunctions of this operator: L( ) =jj : (4.1.9) 4.2 Connections with Wick product Like the classical integration, we next give an alternative characterization of the Skoro- hod integral operator as an innite sum of products. However, instead of the usual multiplication, we introduce a new operation on the elements of . Denition 4.2.1 For , from , dene the Wick product := s ( +)! !! + : (4.2.1) In particular, choosing = k" i and = n" i in (4.1.3), it follows directly from the denition that H k ( i )H n ( i ) =H k+n ( i ): (4.2.2) An immediate consequence of Proposition 4.1.5 and Denition 4.2.1 is the following iden- tity: ( h u k ) =h k ; h2X: (4.2.3) 37 More generally, we dene the operation on formal series: X 2J f ! X 2J g ! = X 2J 0 @ X ; 2J:+ = s ! ! ! f g 1 A for f 2X, g 2R. Theorem 4.2.2 Assume f = X k1 f k u k ; f k = X 2J f k; 2L 2 (F;X); and f is in the domain of , then (f) = X k1 f k k ; (4.2.4) and ((f)) = X k1 p k f k;" k : (4.2.5) Proof: In fact, by linearity and (4.2.3), (f) = X k1 X 2J ( f k; u k ) = X k1 X 2J f k; k = X k1 f k k ; which is (4.2.4). On the other hand, we have by (4.1.8), (f) = X k1 X 2J f k; p k + 1 +" k = X k1 X 2J f k;" k p k ; and (4.2.5) follows. 38 4.3 The weighted Wiener chaos space Before dening the weighted space, we give a characterization of the space L 2 (F;X) by the multiple Skorohod integral expansion. With the help of the characteristic vector, we have the following analog of the well known result of It^ o, connecting multiple Wiener integrals and Hermite polynomials; see [11]. Proposition 4.3.1 Let 2J be a multi-index withjj = n 1 and characteristic vector K = (i 1 ;i 2 ; ;i n ). Then = i 1 i 2 in p ! : (4.3.1) Proof: By the denition of the Wick product, we have " i 1 " i 2 = p (" i 1 +" i 2 )! " i 1 +" i 2 : Then by induction, we obtain " i 1 " i 2 " in = p (" i 1 +" i 2 + +" in )! " i 1 +" i 2 ++" in : But note the fact that " i 1 +" i 2 + +" in =, and thus " i 1 " i 2 " in = p ! ; which completes the proof. 39 Now by induction, we dene the multiple Skorohod integral ^ n , n> 1 on the space L 2 (F;X U ^ n ), whereU ^ n is the symmetric tensor power of cU. Then relation (4.2.3) leads to an alternative form of (4.3.1): = 1 jj! p ! ^ jj (E ); (4.3.2) where E = X 2P u i (1) u i (n) ; (4.3.3) with summation taken over all permutationsP on the setf1; ;ng. Consequently, we have the following generalization of the multiple Wiener integral expansion for the elements of L 2 (F;X). Theorem 4.3.2 If 2 L 2 (F;X), then there is a unique collection of the deterministic elements n , n 0, with 0 =E2X and n 2X U ^ n , n 1, such that = 0 + X n1 1 n! ^ n ( n ); (4.3.4) with the property that Ekk 2 X =k 0 k 2 X + X n1 1 n! kk n k U ^ n k 2 X : (4.3.5) 40 Proof: The multiple Skorohod integral representation is obtained by (4.3.2) and Theorem 4.1.4, i.e., = 0 + 1 X n=1 X jj=n 1 jj! p ! ^ jj (E ) = 0 + 1 X n=1 1 n! X jj=n ^ n 1 p ! E ; and we get the equality in (4.3.4) with n = X jj=n 1 p ! E : Note that forn 1,fE ;jj =ng is an orthogonal set inU ^ n , and by direct computation we have kE k 2 U ^ n =n!!: (4.3.6) Thus, k n k U ^ n k 2 X = X jj=n 1 p ! E U ^ n 2 X = X jj=n 1 ! k k 2 X kE k 2 U ^ n =n! X jj=n k k 2 X : Therefore, equality (4.3.5) is a result of the fact that Ekk 2 X =k 0 k 2 X + X k1 X jj=k k k 2 X : 41 Denition 4.3.3 A sequence Q =fq k ; k 1g is called a weight sequence if q k 1 for all k 1. For 2J and a real number r we write q r = Y k1 q r k k : Given a weight sequence Q, denote by Q the following self-adjoint operator onU: Q u k =q k u k ; k 1: Then, for every r2 R, the operator r Q is dened. The domain of every r Q contains nite linear combinations of u k and is therefore dense inU. For f2D( r Q ), the domain of r Q , dene the norm kfk Q;r =k r Q fk U : (4.3.7) The operator r Q extends to every tensor productX U n ; we will keep the same notation for this extension and, in the case X =R, for the corresponding norm (4.3.7). Denition 4.3.4 The space (L) Q;r (F;X) is the closure of the set of random elements of the form = 0 + N X n=1 1 n! ^ n ( n ); N 1; 42 with respect to the norm kk 2 Q;r;X =k 0 k 2 X + N X k=n 1 (n!) 2 kk n k Q;r k 2 X ; (4.3.8) where 0 2X, n 2D( r Q ) T X U ^ n . Denote by (L) Q;r (F) the dual space of (L) Q;r (F) relative to the inner product inL 2 (F). If 2 (L) Q;r (F;X) and 2 (L) Q;r (F), then the duality hh;ii is dened and is an element of X. As a stochastic analogue of Theorem V. 13-14 in Reed and Simon [36], we now give the characterizations of (L) Q;r (F;X) and (L) Q;r (F) by formal series. Theorem 4.3.5 1. A formal series = P 2J is an element of (L) Q;r (F;X) if and only if X 2J 1 jj! q 2r k k 2 X <1: (4.3.9) The left-hand side of (4.3.9), if nite, is equal tokk 2 Q;r;X . 2. A formal series = P 2J is an element of (L) Q;r (F) if and only if X 2J jj!q 2r j j 2 <1: (4.3.10) In this case, kk 2 (L) Q;r (F) = X 2J jj!q 2r j j 2 ; 43 and hh;ii = X 2J : (4.3.11) Proof: For the rst part, it suces to consider =f ;f2X by the orthonormality of . It follows from Theorem 4.3.2 and Denition 4.3.4 and that kk 2 Q;r;X = 1 (jj!) 2 ! kE k 2 Q;r kfk 2 X : SincekE k 2 Q;0 =kE k 2 L 2 (0;T) =jj!!, and by the denition of the operator Q we have kE k 2 Q;r =q 2r jj!!; and thus kk 2 Q;r;X = 1 jj! q 2r kfk 2 X : The second part is in fact a corollary of the rst one. If 2 (L) Q;r (F;X) and 2 (L) Q;r (F), then X 2J 2 X = X 2J 1 p jj! q r p jj!q r 2 X X 2J 1 jj! q 2r k k 2 X ! X 2J jj!q 2r j j 2 ! =kk 2 Q;r;X kk 2 (L) Q;r (F) : 44 Fix h2U, i.e., h = P k h k u k with h k = (h;u k ) U . Dene the stochastic exponential E(h) = exp _ W (h) 1 2 khk 2 U ; (4.3.12) or equivalently, E(h) = exp X k h k k 1 2 X k h 2 k ! : Now we claim that = 1 p ! @ h E(h)j h=0 ; where @ h = Q k @ k =(@h k ) k . In fact, E(h) = exp 1 2 X k h 2 k + X k h k k ! = Y k exp 1 2 h 2 k +h k k = Y k F (h k ; k ); where F (t;x) = exp(t 2 =2 +tx) = P 1 n=0 t n H n (x)=n! and F (0;x) = 1. Thus @ h E(h)j h=0 = Y k @ k (@h k ) k E(h)j h=0 = Y k Y l6=k F (h l ; l ) @ k (@h k ) k F (h k ; k )j h=0 = Y k H k ( k ) = p ! ; which completes the proof of the claim. Therefore, we can expandE(h) as E(h) = X 2J h p ! ; h = Y k1 h k k : (4.3.13) 45 Proposition 4.3.6 E(h)2 (L) Q;r (F) if and only ifkhk Q;r < 1. Proof: By (4.3.13) and the multinomial expansion, we have X 2J jj!q 2r j(E T (h)) j 2 = X 2J jj! ! h 2 q 2r = X n0 X jj=n n! ! Y k1 h 2 k q 2r k k = X n0 0 @ X k1 h 2 k q 2r k 1 A n = X n0 khk 2 Q;r n : Thus, we complete the proof by the characterization of (L) Q;r (F) in Theorem 4.3.5. Remark 4.3.7 1. L 2 (F;X)6= (L) Q;0 (F;X). In general, for r 1 r 2 0, (L) Q;r 1 (F;X) (L) Q;r 2 (F;X) L 2 (F;X) (L) Q;0 (F;X) (L) Q;r 2 (F;X) (L) Q;r 1 (F;X): 2. For r 0, (L) Q;r (F) (L) Q;r (F)L 2 (F) (L) Q;r (F) (L) Q;r (F): Remark 4.3.8 In applications to stochastic evolution equations, for example ultrasonic wave equation with random viscosity, the number r in the solution space (L) Q;r (F;X) is more likely to be negative, as we want to help the series in (4.3.9) to converge. As a result, the following denition is appropriate. 46 Denition 4.3.9 We say the element h2U is suciently small relative to the sequence Q if there exists a positive number r such that khk r;Q < 1: 47 Chapter 5 Second order in time equation with unbounded damping operators 5.1 An example As a motivation, we give an example based on the ultrasonic wave equation with no square integrable solution. Let us consider the ultrasonic wave, which may be visualized as an innite number of oscillating particles connected by means of elastic springs. The motion of particles may follow Newtonian mechanics. For simplicity, we restrict ourselves to an ideal material in which the particles only move in one direction. If u(t;x) is the particle displacement from equilibrium at position x and time t, then the force on a volume element is given by Newton's second law as follows dF =dxdydzu tt ; (5.1.1) 48 where is the density of the material. On the other hand, since the stress is dened as the force per unit area, we have dF = ((x +dx)(x))dydz = x dxdydz; (5.1.2) which leads to the continuum equation u tt = x : (5.1.3) Since the displacementu depends on the position, the distortion occurs, which is measured by a dimensionless quantity - strain. The initial length dx of the element is increased by u(x +dx)u(x), and so the strain is given by (x) = u(x +dx)u(x) dx =u x : (5.1.4) It is now clear that the form of the equation (5.1.3) depends on the constitutive rela- tion between the stress and strain . In the Kelvin-Voigt model, this relation can be simplied as a linear dierential equation (in t) =E + _ ; (5.1.5) 49 where E is the Young's modulus (or elasticity) and is the viscosity of the material. Then it follows from (5.1.3)-(5.1.5) that u tt =Eu xx +u xxt : (5.1.6) Without loss of generality, we may assume = 1 in (5.1.6). We consider the ultrasonic wave equation with the random perturbation of the viscosity as follows: u tt =Eu xx + ( + _ W (t))u txx ; (5.1.7) where2R and _ W is the Gaussian white noise in time. To study the square integrability, we consider the Fourier transform of (5.1.7) as in [23] for the heat equation. However, dierent from the rst order dierential equation, the second order equation with variable coecients in general does not have an explicit solution. Instead, we will apply It^ o's formula to obtain a third order dierential equation for the second moment and use asymptotic analysis to construct an example with no square integrable solution. We now assume there is a solution of (5.1.7), then its Fourier transform in space ^ u(t;y) = 1 p 2 Z R e ixy u(t;x)dx satises ^ u(t) + ( + _ W (t))y 2 _ ^ u(t) +Ey 2 ^ u(t) = 0: (5.1.8) 50 We rewrite (5.1.8) as a system of stochastic dierential equations in the sense of It^ o as follows: 8 > > > > < > > > > : d^ u = ^ vdt; d^ v =Ey 2 ^ udty 2 ^ vdty 2 ^ vdW: (5.1.9) Then by It^ o's formula, d(^ u 2 ) = 2^ ud^ u = 2^ u^ vdt; d(^ v 2 ) = 2^ vd^ v + 2 y 4 ^ v 2 dt =2Ey 2 ^ u^ vdt 2y 2 ^ v 2 dt 2^ v 2 dW (t) + 2 y 4 ^ v 2 dt; and d(^ u^ v) = ^ ud^ v + ^ vd^ u = ^ v 2 dtEy 2 ^ u 2 dty 2 ^ u^ vdty 2 ^ u^ vdW: Set z 1 =E(^ u 2 );z 2 =E(^ v 2 );z 3 =E(^ u^ v), we obtain _ z 1 = 2z 3 ; (5.1.10) _ z 2 = 2 y 4 2y 2 z 2 2Ey 2 z 3 ; (5.1.11) _ z 3 =Ey 2 z 1 +z 2 y 2 z 3 : (5.1.12) It follows from (5.1.10) that z 3 = 1 2 _ z 1 ; (5.1.13) 51 which gives us z 2 = 1 2 z 1 + 1 2 y 2 _ z 1 +Ey 2 z 1 : (5.1.14) Plugging (5.1.13) and (5.1.14) into (5.1.11), we obtain a third order dierential equation for z 1 , ... z 1 + (3y 2 2 y 4 ) z 1 + (4E + 2 2 y 2 y 4 )y 2 _ z 1 + 2E(2y 2 2 y 4 )y 2 z 1 = 0; (5.1.15) and the characteristic polynomial is given by f(;y;;E;) := 3 +a 1 2 +a 2 +a 3 ; (5.1.16) where a 1 = 3y 2 2 y 4 ;a 2 = (4E + 2 2 y 2 y 4 )y 2 ;a 3 = 2E(2y 2 2 y 4 )y 2 . We denote by H r (R);r2R, the Sobolev space onR, i.e., H r (R) = f : Z R j ^ f(y)j 2 (1 +jyj 2 ) r dy<1 : Then Eku(t)k 2 H r (R) = Z R Ej^ u(t;y)j 2 (1 +jyj 2 ) r dy: Therefore, if we assume thatEku(0)k 2 H r (R) <1;Eku t (0)k 2 H (R) <1 for somer;2R, then the sucient and necessary condition forEku(t)k 2 H r (R) <1;t> 0 is that there exists a constant c2R such that <(y)c; (5.1.17) 52 for all y2R, where< is the real part of a complex number. Now let us analyze the solution of the characteristic equation (5.1.16). Set = =y 4 ;" = 1=y 2 , then dividing y 6 in equation (5.1.16), we get 3 + (3" 2 ) 2 + (4E" 3 + 2 2 " 2 2 ") + 4E" 4 2E 2 " 3 = 0: (5.1.18) Suppose = 0 + P 1 k=1 k " k , then = 0 when " = 0. Consequently, it follows from (5.1.18) that 3 0 2 2 0 = 0: (5.1.19) Thus, 0 = 0 or 0 = 2 . This implies that there is a asymptotically (in y) equivalent to 2 y 4 , violating the condition (5.1.19). Hence, we claim that as long as the noise exists, i.e., 6= 0, the equation (5.1.7) does not have a square integrable solution. Remark 5.1.1 We here have to emphasize the dierence between stability in time and asymptotics in space. The former is a common question in stochastic dierential equa- tion(s); and in particular, with constant coecients the n-th order equation is stable if and only if the Routh-Hurwitz condition is satised; see [13]. However, in stochastic par- tial dierential equations, by the Fourier transform the asymptotical behavior in space is quite related to the regularity; and it may need more careful analysis. Remark 5.1.2 We also point out an interesting observation. Instead of perturbing the viscosity, we can consider the equation with random Young's modulus, u tt = (E + _ W (t))u xx +u txx : (5.1.20) 53 By the similar technique applied to the random viscosity model, we conclude that equation (5.1.20) has a square integrable solution as long as > 0, no matter how large the noise is. 5.2 An overview of the deterministic equation Our approach to establishing the existence and uniqueness of solution for the stochastic equation is to derive the Wiener chaos expansion and study the chaos coecients, which are determined by a lower-triangular system of second order in time equations. There- fore, we rst need to understand the well-posedness of the corresponding deterministic equations. Let (V 1 ;H;V 0 1 ) be a normal triple of Hilbert spaces. That is, V 1 is continuously and densely embedded in H, and we identify the dual space H 0 of H with itself. The space V 0 1 is the dual of V 1 relative to the inner product, denoted by ( ; ), in H. The duality pairingh ; i 1 :=h ; i V 0 1 ;V 1 is the extension by continuity of the inner product in H from V 1 H toV 0 1 H. We assume the operatorA 1 to be symmetric, continuous and elliptic. To wit, A 1 satises (H1) For all u;v2V 1 ,hA 1 u;vi 1 =hu;A 1 vi 1 . (H2) There exists C 1 > 0 such that for all u;v2V 1 jhA 1 u;vi 1 jC 1 kuk V 1 kvk V 1 : 54 (H3) There exists 1 > 0 such that for all u2V 1 hA 1 u;ui 1 + 1 kuk 2 V 1 0: To make assumptions on A 2 , we similarly introduce another normal triple (V 2 ;H;V 0 2 ). Since we often interpretA 2 as a damping operator and we would like to describe as many models in physics as possible, we may allow V 2 to be the same asV 1 or the same asH or somewhere between V 1 and H. To be precise, we write V 1 ,!V 2 ,!H =H 0 ,!V 0 2 ,!V 0 1 ; where the notation \,! "means being continuously and densely embedded. We also denote byh ; i 2 the duality between V 2 and V 0 2 . Then we assume A 2 satises (H4) There exists C 2 > 0 such that for all u;v2V 2 jhA 2 u;vi 2 jC 2 kuk V 2 kvk V 2 : (H5) There exist 2 > 0 and C2R such that for all u2V 2 hA 2 u;ui 2 + 2 kuk 2 V 2 Ckuk H : Later, we also use these notations:V i (T ) =L 2 (0;T ;V i ) andV 0 i (T ) =L 2 (0;T ;V 0 i ),i = 1; 2. 55 Denition 5.2.1 We say a functionu2V 1 (T ) with _ u2V 2 (T ) is a (variational) solution of equation (1.0.4) if the equality in (1.0.4) holds in V 0 1 for a.e. 0 t T , and u(0) = 0; _ u(0) =u 1 . Now we state the fundamental well-posedness result of equation (1.0.4); see [1] for the detailed discussions. Theorem 5.2.2 SupposeA 1 andA 2 satisfy (H1)-(H5), and that u 0 2V 1 , u 1 2H. Then there exists a unique solution u2 C(0;T ;V 1 ) of (1.0.4) with _ u2 C(0;T ;H) T V 2 (T ), u2V 0 1 (T ) and kuk 2 C(0;T;V 1 ) +k _ uk 2 C(0;T;H) +k _ uk 2 V 2 (T) C(T ) ku 0 k 2 V 1 +ku 1 k 2 H +kfk 2 V 0 2 (T) : (5.2.1) Moreover, let A = 0 B B @ 0 I A 1 A 2 1 C C A ;~ u = 0 B B @ u _ u 1 C C A ;~ u 0 = 0 B B @ u 0 u 1 1 C C A ; ~ f = 0 B B @ 0 f 1 C C A : (5.2.2) ThenA generates a C 0 -semigroup S(t) such that ~ u(t) =S(t)~ u 0 + Z t 0 S(ts) ~ f(s)ds: (5.2.3) 56 We omit the complete proof of this theorem, but still we would like to give a simple example to illustrate the basic idea of the energy estimate method. Let us consider the following equation u(t) =u xx (t)u(t) + _ u xx (t) +f(t); (5.2.4) with initial conditions u(0) =u 0 ; _ u(0) =u 1 . We use the abbreviations H 1 ;L 2 ;H 1 for the spaces H 1 (R);L 2 (R);H 1 (R), respec- tively. We also denote by ( ; ) the inner product in L 2 . To get the a priori estimates, we here assume all the functions are smooth. We rst multiply the equation (5.2.4) by _ u and integrate it onR, so we obtain ( u; _ u) = (u xx ; _ u) (u; _ u) + ( _ u xx ; _ u) + (f; _ u): (5.2.5) But since d dt ( u; _ u) = 1 2 d dt j _ uj 2 L 2 ; d dt (u xx ; _ u) = d dt ju x j 2 L 2 ; d dt (u; _ u) = 1 2 d dt juj 2 L 2 ; ( _ u xx ; _ u) =j _ u x j 2 L 2 ; juj 2 H 1 =juj 2 L 2 +ju x j 2 L 2 ; then the equality (5.2.5) becomes d dt j _ uj 2 L 2 +juj 2 H 1 + 2j _ u x j 2 L 2 = 2(f; _ u): (5.2.6) 57 Upon integrating the equality (5.2.6), we have j _ uj 2 L 2 (t) +juj 2 H 1 (t) + 2 Z t 0 j _ u x j 2 L 2 (s)ds =ju 1 j 2 L 2 +ju 0 j 2 H 1 + 2 Z t 0 (f(s); _ u(s))ds: (5.2.7) Adding 2 R t 0 j _ uj 2 L 2 (s)ds to both sides of (5.2.7) and using 2j(f; _ u)jjfj 2 H 1 +j _ uj 2 H 1 , we nd j _ uj 2 L 2 (t) +juj 2 H 1 (t) + Z t 0 j _ uj 2 H 1 (s)dsju 1 j 2 L 2 +ju 0 j 2 H 1 + 2 Z t 0 j _ uj 2 L 2 (s)ds + 2 Z t 0 jf(s)j 2 H 1 ds: (5.2.8) Finally, by Gronwall's inequality sup 0tT j _ uj 2 L 2 (t) +juj 2 H 1 (t) + Z T 0 j _ uj 2 H 1 (t)dtC ju 1 j 2 L 2 +ju 0 j 2 H 1 + Z T 0 jf(t)j 2 H 1 dt : (5.2.9) Therefore, we can conclude by our formal argument that if u 0 2 H 1 ;u 1 2 L 2 and f2 L 2 (0;T ;H 1 ), there is a unique solutionu2C(0;T ;H 1 ) with _ u2C(0;T ;L 2 ) T L 2 (0;T ;H 1 ), which is consistent with the previous well-posedness result stated in Theorem (5.2.2). 5.3 Wiener chaos solution and its propagator In this section, we show the well-posedness of the following damped second order in time equation u(t) =A 1 u(t) +A 2 _ u(t) +f(t) +(B 1 (t)u(t) +B 2 (t) _ u(t)); (5.3.1) with initial conditions u(0) =u 0 ; _ u(0) =u 1 . 58 To motivate the denition of the solution of (5.3.1), we for now assume u h (t) =E(u(t)E(h)); f h (t) =E(f(t)E(h)) are well dened as well as the quantity E((B 1 (t)u(t) +B 2 (t) _ u(t))E(h)). Then the inte- gration by parts formula (4.1.7) and the simple fact that D(E(h)) =hE(h) lead to E((B 1 (t)u(t) +B 2 (t) _ u(t))E(h)) = (B 1 (t)u h (t) +B 2 (t) _ u(t);h) U : Therefore, we obtain a family of functions u h (t;x) satisfying u h (t) =A 1 u h (t) +A 2 _ u h (t) +f h (t) + (B 1 (t)u h (t) +B 2 (t) _ u h (t);h) U : (5.3.2) Remark 5.3.1 We point out that E(g 1 g 2 ) is nothing buthhg 1 ;g 2 ii as long as g 1 ;g 2 2 L 2 (F). Formally, (5.3.2) is a result of an application of the dualityhh u(t);E h ii to both sides of (5.3.1). This suggests the following denition of the \variational solution ", which is quite typical for partial dierential equations; see [16], [18], [37], etc. Denition 5.3.2 Givenu 0 2 S r (L) Q;r (F;V 1 );u 1 2 S r (L) Q;r (F;H);f2 S r (L) Q;r (F;V 0 2 (T )). We say u is a solution of (5.3.1) if there exist a weight sequence Q 0 and a real number r 0 such that u2 (L) Q 0 ;r 0(F;V 1 (T ) and the function u h (t) =hhu(t);E h ii 59 satises u h (0) =hhu 0 ;E h ii; _ u h (0) =hhu 1 ;E h ii and (5.3.2) holds in V 0 1 for every h2U that is suciently small relative to the sequence Q 0 , To state the solvability of the stochastic equation (5.3.1), we also need to make as- sumptions for the operatorsB 1 andB 2 . Suppose thatU is a real separable Hilbert space, and B 1 (t) : V 1 ! V 0 2 U and B 2 (t) : V 2 ! V 0 2 U are continuous operators for every t2 [0;T ]. Then, for every v2V 1 , there exists a collection v k 2V 0 2 ;k 1, such that B 1 v = X k1 v k u k : We therefore dene the operators B 1;k :V 1 !V 0 2 by setting B 1;k v =v k and write B 1 v = X k1 (B 1;k v) u k : Similarly, we can dene the operators B 2;k :V 2 !V 0 2 as well. Therefore, according to Theorem 4.2.2 we can rewrite the random driving force as follows (B 1 (t)u(t) +B 2 (t) _ u(t)) = X k1 (B 1;k (t)u(t) +B 2;k (t) _ u(t)) k ; k = _ W (u k ); (5.3.3) and equation (5.3.1) becomes u(t) =A 1 u(t) +A 2 _ u(t) +f(s) + X k1 (B 1;k (t)u(t) +B 2;k (t) _ u(t)) k : (5.3.4) 60 If the noise _ W itself depends on time, this dependence is encoded in the operatorsB 1 ;B 2 . As a result, (5.3.1) includes many popular second equations we are familiar with. Suppose thatV (t;u 0 ;u 1 ;f) is the unique solution of equation (1.0.4) with initialu 0 ;u 1 and inhomogeneous term f. Then we assume (H6) For every v 1 2V 1 (T ) and v 2 2V 2 (T ), and k 1, there exists C k independent of v 1 ;v 2 such that kV (t; 0; 0;B 1;k v 1 +B 2;k v 2 )k 2 V 1 (T) +k _ V (t; 0; 0;B 1;k v 1 +B 2;k v 2 )k 2 V 2 (T) C 2 k kv 1 k 2 V 1 (T) +kv 2 k 2 V 2 (T) ; (5.3.5) To make the notations more concise, we introduce for k 1 M k = 0 B B @ 0 0 B 1;k B 2;k 1 C C A ; (5.3.6) Then we have an equivalent condition to (H6): (H6 0 ) For every~ v2V 1 (T )V 2 (T ), andk 1, there existsC k independent of~ v such that Z T 0 Z t 0 S(ts)M k (s)~ v(s)ds 2 V 1 V 2 dtC 2 k k~ vk 2 V 1 (T)V 2 (T) : (5.3.7) In virtue of Theorem 4.3.5, the solutionu can be written as a formal series P 2J u with u 2V 1 (T ). The following result provides an alternative characterization of the solution in terms of the Wiener chaos expansion. 61 Theorem 5.3.3 Let u = P 2J u be an element of (L) Q 0 ;r 0(F;V 1 (T )). The process u is a solution of equation (5.3.1) in the sense of Denition 5.3.2 if and only if the functions u 2V 1 (T ) and the system of equations u (t) =A 1 u (t) +A 2 _ u (t) +f (t) + X k1 p k (B 1;k (t)u(t) +B 2;k (t) _ u(t)) k (5.3.8) with u (0) =u 0; ; _ u (0) =u 1; holds in V 0 1 for all 2J . Proof: If u is a solution of (5.3.1), then by (4.3.11) and (4.3.13) we have u h = X 2J u h p ! ; h = Y k h k k ;h k = (h;u k ) U ; and by a general result from functional analysis [14], this power series in h converges in some (innite dimensional) neighborhood of zero. Therefore, the mapping h7! u h is analytic at zero, and thus u = 1 p ! @ h u h j h=0 ; whence by the linearity and continuity, u (t) =A 1 u (t) +A 2 _ u (t) +f (t) + X k1 1 p ! @ h ((B 1;k (t)u h (t) +B 2;k (t) _ u h (t))h k )j h=0 : But for , we have @ h h = ! ()! h ; 62 which leads to X k1 1 p ! @ h ((B 1;k (t)u h (t) +B 2;k (t) _ u h (t))h k )j h=0 = X k1 1 p ! ! p (" k )! (B 1;k (t)u(t) +B 2;k (t) _ u(t)) " k = X k1 p k (B 1;k (t)u(t) +B 2;k (t) _ u(t)) " k ; which gives us (5.3.8). Conversely, if u satises (5.3.8), and set u h = P 2J uh p ! , then by similar compu- tation as above, u h satises (5.3.2) in V 0 1 . This simple but very helpful result establishes the equivalence of the \physical "(5.3.4) and the stochastic analogue of Fourier (5.3.8) forms of equation (5.3.1). The system of equations (5.3.8) is often referred in the literature as the propagator of the equation (5.3.4). To prove the existence and uniqueness of solution of (5.3.1), we assume all the inputs are deterministic, and may easily extend the result to random inputs by linearity and triangle inequalities. Then, the corresponding propagator system is _ ~ u (0) (t) =A~ u (0) (t) + ~ f(t); ~ u (0) (0) =~ u 0;(0) ; (5.3.9) _ ~ u k (t) =A~ u k (t) +M k (t)~ u (0) (t); ~ u k (0) = 0; (5.3.10) _ ~ u (t) =A~ u (t) + X k1 p k M k (t)~ u k (t); ~ u (0) = 0; jj> 1: (5.3.11) 63 Note that the propagator system is lower triangular and can be solved by induction on jj. Assume that ~ V (t;~ u 0 ; ~ f) is the unique (variational) solution of the deterministic equa- tion (by abuse of notation, inputs here may be very generic) _ ~ u(t) =A~ u(t) + ~ f(t); ~ u(0) =~ u 0 ; (5.3.12) then we observe some linear properties, which will be used later: ~ V (t;~ u 0 ; ~ f +~ g) = ~ V (t;~ u 0 ; ~ f) + ~ V (t; 0;~ g). ~ V (t;~ u 0 ;a ~ f +b~ g) =a ~ V (t;~ u 0 ; ~ f) +b ~ V (t; 0;~ g); a;b2R. In particular, ~ V (t; 0; 0) = 0. Theorem 5.3.4 We assume that (5.3.12) has a unique (variational) solution ~ u(t) = ~ V (t;~ u 0 ; ~ f) for allf and~ u 0 . Forjj = 0, let~ u (0) (t) = ~ V (t;~ u 0;(0) ; ~ f) be the unique solution of (5.3.9). For 2J withjj =n 1 and the corresponding characteristic vector K , dene functions ~ F n = ~ F n (t;) by induction as follows: 8 > > > > < > > > > : ~ F 1 (t; k ) = ~ V (t; 0;M k ~ u (0) ); if K = (k); ~ F n (t;) = P n j=1 ~ V (t; 0;M i j ~ F n1 (; i j )); if K = (i 1 ;i 2 ; ;i n ): (5.3.13) Then ~ u (t) = 1 p ! ~ F n (t;): (5.3.14) 64 Proof: Ifjj = 1, say = k , then we have by (5.3.10) ~ u k (t) = ~ V (t; 0;M k ~ u (0) ) = ~ F 1 (t; k ): Ifjj =n> 1, in view of the linear property, equation (5.3.11) becomes ~ u (t) = X k1 p k ~ V (t; 0;M k ~ u k ) = 1 p ! X k1 p ! p k ~ V (t; 0;M k ~ u k ) = 1 p ! X k1 k ~ V (t; 0;M k p ( k )!~ u k ): Now by the denition of the characteristic vector, we have X k1 k ~ V (t; 0;M k p ( k )!~ u k ) = n X j=1 ~ V (t; 0;M i j q ( i j )!~ u i j ); and thus u (t) = 1 p ! n X j=1 ~ V (t; 0;M i j p ( k )!~ u i j ): SinceK i j of i j is obtained from K by removing one copy of i j , by induction we may have ~ u (t) = 1 p ! n X j=1 ~ V (t; 0;M i j ~ F n1 (; i j ) = 1 p ! F n (t;): 65 Corollary 5.3.5 Under the same conditions as in Theorem 5.3.4, the representation (5.3.14) becomes ~ u (t) = 1 p ! X 2P n Z t 0 Z n 0 Z 2 0 S(t n )M i (n) ( n )S( 2 2 )M i (1) ( 1 )~ u (0) ( 1 )d 1 d n ; (5.3.15) where the summation is over all permutations off1; 2; ;ng. Proof: We will prove (5.3.15) by induction onjj. For n = 1, say = k , it follows from (5.3.14) and the semigroup representation that ~ u k (t) = ~ F 1 (t; k ) = ~ V (t; 0;M k ~ u (0) ) = Z t 0 S(ts)M k (s)~ u (0) (s)ds: 66 Now assume (5.3.15) holds forjj = n. Letjj = n + 1 with K = (i 1 ;i 2 ; ;i n+1 ). If P n+1 j is the permutation group off1; ;n + 1gnfjg, then by (5.3.13) we have F n+1 (t;) = n+1 X j=1 ~ V (t; 0;M i j ~ F n (; i j )) = n+1 X j=1 Z t 0 S(t)M i j () ~ F n (s; i j )ds = n+1 X j=1 Z t 0 S(t)M i j () X 2P n+1 j Z 0 Z n 0 Z j1 0 Z j+1 0 Z 2 0 S( n )M i (n) ( n )S( j+1 j1 )M i (j1) ( j1 )S( 2 1 ) M i (1) ~ u (0) ( 1 )d 1 d n = X 2P n+1 Z t 0 Z n+1 0 Z 2 0 S(t n+1 )M i (n+1) ( n+1 )S( 2 2 )M i (1) ( 1 )~ u (0) ( 1 )d 1 d n+1 : Therefore, (5.3.15) is true forjj =n + 1, and it is valid for all n 1 by induction. 5.4 Main result Finally, we are ready to prove our main result: Theorem 5.4.1 Assume (H1)-(H6), and that u 0 2 V 1 ;u 1 2 H, and f2V 2 (T ). Then there exist a weight sequence Q and a negative real number r such that (5.3.1) has a unique solution u2 (L) Q;r (F;V 1 (T )) with _ u2 (L) Q;r (F;V 2 (T )) and kuk 2 Q;r;V 1 (T) +k _ uk 2 Q;r;V 2 (T) C ku 0 k 2 V 1 +k _ uk 2 H +kfk 2 V 0 2 (T) ; (5.4.1) 67 for some constant C > 0 only dependent on T and the operators A 1 ;A 2 ;B 1 ;B 2 . Proof: Since u 0 ;u 1 and f are all deterministic, we have u 0;h = u 0 ;u 1;h = u 1 and f h =f. By assumptions (H1)-(H6), the equation _ ~ u h (t) =A~ u h (t) + ~ f + (M(t)~ u h (t);h) U with u h (0) = u 0 ; _ u h (0) = u 1 has a unique solution as long askhk U is suciently small, and the dependence of u h on h is analytic by linearity. Now we write u(t) = X 2J u (t) ; 68 and then the chaos coecientsu ,jj 1 can be represented by (5.3.15). It follows from (5.2.1) and (5.3.7) that k~ u k 2 V 1 (T)V 2 (T) = 1 ! Z T 0 X 2P n Z t 0 Z n 0 Z 2 0 S(t n )M i (n) ( n ) S( 2 2 )M i (1) ( 1 )~ u (0) ( 1 )d 1 d n 2 V 1 V 2 dt n! ! X 2P n Z T 0 Z t 0 Z n 0 Z 2 0 S(t n )M i (n) ( n ) S( 2 2 )M i (1) ( 1 )~ u (0) ( 1 )d 1 d n 2 V 1 V 2 dt n! ! X 2P n C 2 i (n) Z T 0 Z n 0 Z n1 0 Z 2 0 S( n n1 )M i (n1) ( n1 ) S( 2 2 )M i (1) ( 1 )~ u (0) ( 1 )d 1 d n1 2 V 1 V 2 d n n! ! X 2P n n Y k=1 C 2 i(k) k~ u (0) k 2 V 1 (T)V 2 (T) = (n!) 2 ! Y k1 C 2 k k k~ u (0) k 2 V 1 (T)V 2 (T) : But we also have k~ u (0) k 2 V 1 (T)V 2 (T) C(T;A;M) ku 0 k 2 V 1 +ku 1 k 2 H +kfk 2 V 0 2 (T) by (5.2.1). Thus, for withjj 1, k~ u k 2 V 1 (T)V 2 (T) C(T;A;M) (n!) 2 ! Y k1 C 2 k k ku 0 k 2 V 1 +ku 1 k 2 H +kfk 2 V 0 2 (T) : 69 It is known (see, for example, [10, Proposition 2.3.3 and page 35]) that X 2J (2N) r 1 if and only if r> 1; (5.4.2) andjj!!(2N) 2 . Then we dene the weight sequence Q =fq k ;k 1g by q k = 2k(1 +C k ); (5.4.3) and obtain kuk 2 Q;r;V 1 (T) +k _ uk 2 Q;r;V 2 (T) = X 2J 1 jj! q 2r (k~ u k 2 V 1 (T)V 2 (T) ) C(T;A;M) ku 0 k 2 V 1 +ku 1 k 2 H +kfk 2 V 0 2 (T) 0 @ 1 + X :jj1 jj! ! q 2r Y k1 C 2 k k 1 A C(T;A;M) ku 0 k 2 V 1 +ku 1 k 2 H +kfk 2 V 0 2 (T) 0 @ 1 + X :jj1 (2N) 2(r+1) C 2 (1 +C) 2r 1 A ; whereC 2 = Q k1 C 2 k k and (1+C) 2r = Q k1 (1+C k ) 2r k . Now we chooser<3=2 and get kuk 2 Q;r;V 1 (T) +k _ uk 2 Q;r;V 2 (T) C(T;A;M;Q;r) ku 0 k 2 V 1 +ku 1 k 2 H +kfk 2 V 0 2 (T) ; which completes the proof. 70 5.5 Optimality of the weight In the procedure of bounding the statistical moment, we constantly apply dierent types of inequalities. In each step of the estimate, there is no way to guarantee the sharpness of the bounds, let alone the nal weight we have. However, as long as we can nd an equation within the framework of the abstract equation (5.3.1) such that the solution space has the exact factorial weight, we are satised and in this sense we say the weight optimal. Let us consider the following equation 2u tt ="u xx +u txx : (5.5.1) Set (y) = 8 > > > > < > > > > : y 2 p y 4 8"y 2 4 ; y 2 8" y 2 i p y 4 +8"y 2 4 ; y 2 < 8" so that it is a solution of the characteristic equation 2 2 +y 2 +"y 2 = 0: We also setu 1 (x) = _ u(0;x) = exp(x 2 =2) and chooseu 0 (x) such that ^ u 0 (y) = 1 (y)^ u 1 (y). Therefore, the solution of (5.5.1) will satisfy _ ^ u(t;y) = ^ u 1 (y)e (y)t =e y 2 2 +(y)t : (5.5.2) 71 Now consider the stochastic version of (5.5.1) 2u tt ="u xx +u txx + 2u txx _ w(t) (5.5.3) with the same initial condition as we impose on the deterministic equation. It is known [22] that X jj=n k~ u (t)k 2 L 2 (R)L 2 (R) = t n n! kM 2n ~ v(t)k 2 L 2 (R)L 2 (R) ; (5.5.4) where~ v = (v; _ v) T is the unique solution of (5.5.1). Recall that M = 0 B B @ 0 0 0 @ 2 x 1 C C A ; and thus I " n :=kM 2n ~ v(t)k 2 L 2 (R)L 2 (R) =k@ 2n x _ v(t)k 2 L 2 (R) = Z R jyj 4n e y 2 +2(y)t dy: But I n := Z R jyj 4n e y 2 (t+1) dy = 1 2 p 1 +t (2n + 1 2 ) (1 +t) 2n Using the Stirling's formula, we have t n n! I n = 2 p t 1 +t 2n C(n)n!; (5.5.5) where the number C(n) are uniformly bounded from above and below. 72 Claim 5.5.1 I " n !I n as "! 0. Proof: First of all, we have jI " n I n j Z R y 4n e y 2 je 2t e y 2 t jdy = Z y 2 <8" y 4n e y 2 (t+1) e y 2 2 t 1 dy + Z y 2 8" y 4n e y 2 e (y 2 =2 p y 4 8"y 2 =2)t 1 dy: But Z y 2 <8" y 4n e y 2 (t+1) e y 2 2 t 1 dy = Z y 2 <8" y 4n e y 2 (t=2+1) 1e y 2 2 t dy Z y 2 <8" y 4n e y 2 (t=2+1) dy! 0 as "! 0 by the dominant convergence theorem. We also have Z y 2 8" y 4n e y 2 e (y 2 =2 p y 4 8"y 2 =2)t 1 dy = Z y 2 8" y 4n e y 2 (t=2+1) e p y 4 8"y 2 =2)t e y 2 2 t dy! 0 as"! 0 by the dominant convergence theorem. Combining the above two estimates, we complete the proof of the claim. Therefore, we haveI " n =I n +(") with so that(")! 0 as"! 0. Then it follows from equation (5.5.5) that t n n! I " n = 2 p t 1 +t 2n C(n)n! + t n n! (") = 2 p t 1 +t 2n C(n;")n!; 73 where C(n;") = 2 p t 1 +t 2n n! C(n) + t n n! (") 1 +t 2 p t 2n 1 n! ! : Note that for any t2 (0;T ] t n n! 1 +t 2 p t 2n 1 n! = 1 (n!) 2 (1 +t) 2n 2 2n ; so the quantity t n n! (") 1 +t 2 p t 2n 1 n! is bounded if " is suciently small (for example, " is so small thatjj < 1). Thus the numberC(n;") is uniformly bounded from above and below. Hence, for suciently small "> 0 we have X n1 1 n! X jj=n ku (t)k 2 L 2 (R) +k _ u (t)k 2 L 2 (R) = X 2J 1 jj! ku (t)k 2 L 2 (R) +k _ u (t)k 2 L 2 (R) <1: 74 Reference List [1] H. T. Banks, K. Ito and Y. Wang, Well-posedness for damped second order systems with unbounded input operators, CRSC-TR93-10, June 1993, N.C. State University, Dierential and Integral Equations, Vol. 8, 1995, pp. 587-606. [2] R. H. Cameron and W. T. Martin, The orthogonal development of nonlinear func- tions in a series of Fourier-Hermite function. Annals of Mathematics 48(2), 385-39 (1947) [3] S. Cerrai and G. Da Prato, A basic identity for Kolmogorov operators in the space of continuous functions related to RDEs with multiplicative noise, submitted. [4] G. Chen and D.L. Russell, A mathematical model for linear elastic systems with structural damping, Quarterly of Applied Mathematics, Vol. 39, 1982, 433-454. [5] G. Da Prato, and J. Zabczyk, Stochastic equations in innite dimensions. Cambridge University, Press 1992. [6] L. C. Evans, Partial Dierential Equations, Graudate Studies In Mathematics, Vol- ume 19, American Mathematical Society, Providence, Rhode Island, 1998. [7] E. Fedrizzi and F. Flandoli, Noise prevents singularities in linear transport equations, Journal of Functional Analysis, Volume 264, Issume 6, page 1329-1354, 2013. [8] F. 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Abstract (if available)

Abstract This thesis aims to study the well-posedness of second order in time stochastic evolution equations. ❧ Motivated by the well known stochastic parabolicity condition, a class of second order in time stochastic partial differential equations with constant coefficients is studied, and well-posedness in Sobolev spaces on Rᵈ is established by investigating the eigenvalues of a matrix with entries in the form of symbols of the differential operators. ❧ There are also many important equations, especially with multiplicative noises, can hardly admit a square integrable solution. A new method to study second order in time stochastic evolution equations is developed. The idea is to separate the deterministic and stochastic components of the equation by projecting the equation on a suitable basis in the probability space. The coefficients in the expansion, called the propagator of the stochastic equation, are characterized by a system of deterministic equations. A well-posedness result is obtained by constructing a weighted solution space.

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Creator Zhong, Jie (author)

Core Title Second order in time stochastic evolution equations and Wiener chaos approach

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 04/26/2013

Defense Date 03/13/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,second order in time,stochastic evolution equations,Wiener chaos

Language English

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Advisor Lototsky, Sergey V. (committee chair), Ghanem, Roger G. (committee member), Mikulevičius, Remigijus (committee member)

Creator Email jiezhong@usc.edu,jiezhongmath@gmail.com

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