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ASYMPTOTIC PROBLEMS IN STOCHASTIC PARTIAL DIFFERENTIAL
EQUATIONS:
A WIENER CHAOS APPROACH
by
Sivaditya Kaligotla
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(APPLIED MATHEMATICS)
May 2012
Copyright 2012 Sivaditya Kaligotla
Object Description
| Title | Asymptotic problems in stochastic partial differential equations: a Wiener chaos approach |
| Author | Kaligotla, Sivaditya |
| Author email | sivaditya@gmail.com;kaligotl@usc.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Applied Mathematics |
| School | College of Letters, Arts And Sciences |
| Date defended/completed | 2012-03-23 |
| Date submitted | 2012-05-04 |
| Date approved | 2012-05-04 |
| Restricted until | 2012-05-04 |
| Date published | 2012-05-04 |
| Advisor (committee chair) | Lototsky, Sergey V. |
| Advisor (committee member) |
Mikulevicius, Remigijus Ghanem, Roger |
| Abstract | It has been known for a while that certain non-linear as well as bilinear stochastic partial differential equations driven by a singular noise must be interpreted in the re-normalized, or Wick, form. In this dissertation we study two such equations. ❧ In the first part of the dissertation, we study the stochastic Burgers equation with Wick non-linearity. We show that, in the stochastic Burgers equation, Wick nonlinearity forces the solution to be a generalized process no matter how regular the random perturbation is. On the other hand, certain multiplicative random perturbations of the deterministic Burgers equation can only be interpreted in the Wick form and, for such equations, existence and uniqueness of a solution in appropriate generalized spaces is shown. This dissertation is based on the analysis of the coefficients of the chaos expansion of the solution at different stochastic scales. ❧ In the second part of the dissertation, we investigate a homogenization problem for stochastic bilinear elliptic equations. It is known that for such equations, solutions exist in generalized spaces. In this dissertation, homogenization results are proved in appropriate generalized spaces. The main tools of the investigation are analysis of the coefficients of the chaos expansion and the well known concept of two-scale convergence. |
| Keyword | stochastic; pde; spde; wiener chaos; wce;burgers equation; wick; kondratiev; generalized random elements |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m |
| Rights | Kaligotla, Sivaditya |
| Access conditions | The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given. |
| Repository name | University of Southern California Digital Library |
| Repository address | USC Digital Library, University of Southern California, University Park Campus MC 7002, 106 University Village, Los Angeles, California 90089-7002, USA |
| Repository email | cisadmin@usc.edu |
| Archival file | uscthesesreloadpub_Volume4/etd-KaligotlaS-752.pdf |
Description
| Title | Page 1 |
| Full text | ASYMPTOTIC PROBLEMS IN STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS: A WIENER CHAOS APPROACH by Sivaditya Kaligotla A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) May 2012 Copyright 2012 Sivaditya Kaligotla |
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