Probabilistic representation of parabolic stochastic variational inequality with Dirichlet-Neumann boundary and variational generalized backward doubly stochastic differential equations

TL;DR

This work develops a rigorous probabilistic framework for parabolic variational SPDEs with nonlinear Dirichlet-Neumann boundary by studying variational generalized backward doubly stochastic differential equations with subdifferential operators under non-Lipschitz growth. Existence and uniqueness of the VGBDSDE are established through a Picard scheme together with Yosida approximations, complemented by a comparison principle in the variational setting. A stochastic viscosity solution notion for the variational SPDE is formulated, and a Doss-Sussmann transformation is employed to relate the stochastic problem to PDEs with random coefficients, enabling a robust ω-wise analysis. The main contribution is a probabilistic representation, u(t,x) = Y^{t,x}_t, of the stochastic viscosity solution to the variational SPDE with nonlinear Neumann-Dirichlet boundary, broadening the applicability to convex constraints and non-Lipschitz nonlinearities. These results lay the groundwork for further exploration of uniqueness and extensions to broader boundary and coefficient structures.

Abstract

We derive the existence and uniqueness of the generalized backward doubly stochastic differential equation with sub-differential of a lower semi-continuous convex function under a non Lipschitz condition. This study allows us give a probabilistic representation (in stochastic viscosity sense) to the parabolic variational stochastic partial differential equations with Dirichlet-Neumann conditions.

Theorems & Definitions (32)

• Remark 2.1
• Remark 2.2
• Definition 2.3: see Br and the references therein
• Proposition 2.4
• Definition 2.5
• Proposition 2.6
• Theorem 2.7
• Remark 2.8
• proof
• Remark 2.9