BSDE and generalized Dirichlet forms: the infinite dimensional case

TL;DR

This paper develops a probabilistic framework using generalized Dirichlet forms and stochastic calculus to solve infinite-dimensional backward PDE systems with degenerate operators, extending martingale representation and solution concepts.

Contribution

It introduces a novel approach to solving infinite-dimensional backward PDEs via generalized Dirichlet forms and stochastic calculus, including a generalized martingale representation theorem.

Findings

Probabilistic representation of solutions via backward SDEs

Extension of martingale representation theorem to infinite dimensions

Solution framework for degenerate second order operators with measurable coefficients

Abstract

We consider the following quasi-linear parabolic system of backward partial differential equations on a Banach space $E$: $(\partial_t+L)u+f(\cdot,\cdot,u, A^{1/2}\nabla u)=0$ on $[0,T]\times E,\qquad u_T=\phi$, where $L$ is a possibly degenerate second order differential operator with merely measurable coefficients. We solve this system in the framework of generalized Dirichlet forms and employ the stochastic calculus associated to the Markov process with generator $L$ to obtain a probabilistic representation of the solution $u$ by solving the corresponding backward stochastic differential equation. The solution satisfies the corresponding mild equation which is equivalent to being a generalized solution of the PDE. A further main result is the generalization of the martingale representation theorem in infinite dimension using the stochastic calculus associated to the generalized Dirichlet form given by $L$. The nonlinear term $f$ satisfies a monotonicity condition with respect to $u$ and a Lipschitz condition with respect to $\nabla u$.