12 research outputs found
An Asymptotic Comparison of Two Time-homogeneous PAM Models
Both Wick-Ito-Skorokhod and Stratonovich interpretations of the parabolic
Anderson model (PAM) lead to solutions that are real analytic as functions of
the noise intensity e, and, in the limit e->0, the difference between the two
solutions is of order e^2 and is non-random.Comment: 12 page
A Note on Generalized Malliavin Calculus
The Malliavin derivative, divergence operator, and the Ornstein-Uhlenbeck
operator are extended from the traditional Gaussian setting to generalized
processes from the higher-order chaos spaces
Chaos expansion solutions of stochastic magnetic Schr\"odinger equations on curved spaces
We treat some classes of stochastic partial differential equations of
Schr\"odinger type within the framework of white noise analysis, combined with
Wiener-It\^o chaos expansions and pseudodifferential operator methods. The
initial data and potential term of the Schr\"odinger operator are assumed to be
generalized stochastic processes that have both temporal and spatial
dependence. We prove that the equations under consideration have unique
solutions in the appropriate (intersections of weighted)
Sobolev-Kato-Kondratiev spaces.Comment: 18 pages. This is the first version. It is expected that some
revisions will be performed in the futur
Stochastic homogenization of elliptic equation and optimal control
In this thesis, we mainly study the numerical methods for stochastic homogenization of elliptic optimal control problem, where there is random variable involved in the constraint. We start with a simple one-dimensional optimal control problem and derive the effective equations of the original optimal control problem and the theory results regarding the convergence of solutions have been studied in \cite{kesavan}. Then an elliptic optimal control problem with coefficient having ergodicity is studied and convergence theorem is also given regarding the effective equations. Finally, another elliptic optimal control problem, where the normal product is replaced by the wick product, is discussed. An algorithm used to search the optimal solution is obtained. Numerical examples are given in each chapter
On generalized Malliavin calculus
AbstractThe Malliavin derivative, the divergence operator (Skorokhod integral), and the Ornstein–Uhlenbeck operator are extended from the traditional Gaussian setting to nonlinear generalized functionals of white noise. These extensions are related to the new developments in the theory of stochastic PDEs, in particular elliptic PDEs driven by spatial white noise and quantized nonlinear equations