Pathwise mild solutions for quasilinear stochastic partial differential equations

Mild Solutions of Neutral Stochastic Partial Functional Differential Equations

Pathwise Stationary Solutions of Stochastic Partial Differential Equations and Backward Doubly Stochastic Differential Equations on Infinite Horizon

The main purpose of this paper is to study the existence of stationary solution for stochastic partial differential equations. We establish a new connection between backward doubly stochastic differential equations on infinite time horizon and the stationary solution of the SPDEs. For this we study the existence of the solution of the associated BDSDEs on infinite time horizon and prove it is a...

Pathwise Random Periodic Solutions of Stochastic Differential Equations

In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify them as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general cases. We then use the argument of the relative compactness of Wiener-Sobolev spaces in C([0, T ], L(Ω)) and generalized Schauder’s fixed point theorem to prove...

Quasilinear Parabolic Stochastic Partial Differential Equations: Existence, Uniqueness

In this paper, we provide a direct approach to the existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone.

Degenerate Parabolic Stochastic Partial Differential Equations: Quasilinear case

In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an L1-contraction property. In comparison to the previous works of the authors concerning stochastic...