Huaizhong Zhao

Stationary solutions of SPDEs and infinite horizon BDSDEs

In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between Lρ2(Rd;R1)⊗Lρ2(Rd;Rd) valued solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. Moreover, we prove the existence and uniqueness of the solutions of BDSDEs on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary solutions (independent of any initial value) of SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.

Approximate Travelling Waves for Generalized KPP Equations and Classical Mechanics

We consider the existence of approximate travelling waves of generalized KPP equations in which the initial distribution can depend on a small parameter μ which in the limit μ → 0 is the sum of some δ-functions or a step function. Using the method of Elworthy & Truman (1982) we construct a classical path which is the backward flow of a classical newtonian mechanics with given initial position and velocity before the time at which the caustic appears. By the Feynman–Kac formula and the Maruyama–Girsanov–Cameron–Martin transformation we obtain an identity from which, with a late caustic assumption, we see the propagation of the global wave front and the shape of the trough. Our theory shows clearly how the initial distribution contributes to the propagation of the travelling wave. Finally, we prove a Huygens principle for KPP equations on complete riemannian manifolds without cut locus, with some bounds on their volume element, in particular Cartan–Hadamard manifolds.

Random periodic solutions of random dynamical systems

Abstract In this paper, we give the definition of the random periodic solutions of random dynamical systems. We prove the existence of such periodic solutions for a C 1 perfect cocycle on a cylinder using a random invariant set, the Lyapunov exponents and the pullback of the cocycle.

Two-parameter p, q-variation paths and integrations of local times

In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter

Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding

p, q

On stochastic diffusion equations and stochastic Burgers’ equations

-variation path integrals. Our condition of locally bounded

Pathwise random periodic solutions of stochastic differential equations

p,q

Generalized Ito formulae and space-time Lebesgue-Stieltjes integrals of local times

-variation is more natural and easy to verify than those of Young. This result can be easily generalized to multi-parameter case. The second result is to define the integral of local time

The propagation of travelling waves for stochastic generalized KPP equations

\int_{-\infty}^\infty\int_0^t g(s,x)d_{s,x}L_s(x)

Two properties of stochastic KPP equations: ergodicity and pathwise property

pathwise and then give generalized It