Pao-Liu Chow

Stationary solutions of nonlinear stochastic evolution equations

General theorems concerning the existence and uniqueness of invariant measures are proved for a certain class of regular diffusion processes in Separable Banach spaces under some weak compactness and other conditions. Then, based on these theorems, some verifiable sufficient conditions are obtained to ensure the existence and uniqueness of an invariant distribution for the strong solution to some nonlinear evolution equations in a Hilbert space. The results are applied to certain monotone parabolic Ito equations as well as to the 2-D Navier-Stokes equations under random perturbations

Exponential estimates in exit probability for some diffusion processes in hilbert spaces

The paper is concerned with a large deviation type of estimates for some diffusion processes in Hilbert spaces, including the Brownian motion, the ltd integral and the solution of a stochastic evolution equation. Exponential upper bounds in exit probability are obtained and proved for such processes to leave a ball of radius r before a finite time t. An application to parabolic Ito equations is given as an example

Infinite-dimensional Hamilton-Jacobi-Bellman equations in Gauss-Sobolev spaces

We consider the strong solution of a semi linear HJB equation associated with a stochastic optimal control in a Hilbert space H. By strong solution we mean a solution in a L2(μ,H)-Sobolev space setting. Within this framework, the present problem can be treated in a similar fashion to that of a finite-dimensional case. Of independent interest, a related linear problem with unbounded coefficient is studied and an application to the stochastic control of a reaction-diffusion equation will be given.

Infinite-dimensional Kolmogorov equations in Gauss-Sobolev spaces

We consider the Kolmogorov equation in infinite dimensions associated with a nonlinear stochastic evolution equation in some Hilbert space. By introducing a proper class of Gauss-Sobolev spaces, a L2 -theory of the Kolmogorov equation is developed. Under some suitable conditions, the existence and uniqueness of solutions of the Cauchy problem and the related elliptic problem are proved in such a Gauss-Sobolev space setting

Method of Lyapunov functions for analysis of absorption and explosion in Markov chains

The main goal of this paper is to derive some sufficient conditions for testing the absorption, explosion, and nonexplosion of time-inhomogeneous Markov chains with a countable state space. The method of Lyapunov functions is used for this purpose. Several theorems concerned with such sufficient conditions are proved for a general class of Markov chains. Then they are applied to some problems in time-inhomogeneous birth-death processes and branching processes.

Additive control of stochastic linear systems

This paper gives an overview of some results on control of diffusion processes when the control acts additively on the state of the system. These results are mainly based on the paper [14].

Method of regularization for second-order stochastic ∗

The paper is concerned with the strong solution of secondorder stochastic evolution equations in a Hilbert space. We introduce the method of regularization to prove the existence, uniqueness of strong solution for such equations without the usual coercivity assumption. The result is applied to stochastic wave and plate equations to yield the existence of a unique strong solution for each of such problems arising from physical application.

On-line estimation of smooth signals with partial observation

The paper concerns the estimation of a smooth signal S(t) and its derivatives in the presence of a noise depending on a small parameter ε based on a partial observation. A nonlinear Kalman-type filter is proposed to perform on-line estimation. For the signal S in a given class of smooth functions, the convergence rate for the estimation risks, as ε → 0, is obtained. It is proved that such rates are optimal in a minimax sense. In contrast to the complete observation case, the rates are reduced, due to incomplete information.

Almost sure convergence of stochastic integrals in Hilbert Spaces

It is proved that, under the usual measurability and the square-integrability conditions, the stochastic integral of an operator-valued integrand with respect to a continuous local martingale in Hilbert spaces can be defined for almost every sample path via a sequence of simple progressively measurable approximations. The result generalizes McKeanls construction of the It6 integral with respect to a one-dimensional Brownian motion

Applications of functional integrals to wave propagation problems

A functional integral approach to solution of a class of evolution equations is described. This is done by introducing the Feynman-Kac formula. Then selected applications to quantum mechanical and classical wave propagation problems are presented. A connection with the nonlinear filtering theory is also discussed.