A Feynman-Kac formula for stochastic Dirichlet problems
M\'at\'e Gerencs\'er, Istv\'an Gy\"ongy
TL;DR
This paper introduces a Feynman-Kac type formula for solving stochastic PDEs with Dirichlet boundary conditions, extending the classical approach to cases where backward characteristics are not well-defined.
Contribution
It provides a new representation formula for stochastic PDE solutions that applies even when backward characteristics are undefined in the Itô sense.
Findings
Established a Feynman-Kac formula for stochastic Dirichlet problems.
Extended the formula to cases lacking well-defined backward characteristics.
Broad applicability to complex stochastic PDE boundary conditions.
Abstract
A representation formula for solutions of stochastic partial differential equations with Dirichlet boundary conditions is proved. The scope of our setting is wide enough to cover the general situation when the backward characteristics that appear in the usual formulation are not even defined in the It\^o sense.
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