Semilinear Feynman-Kac Formulae for $B$-Continuous Viscosity Solutions

TL;DR

This work extends the Feynman--Kac framework to semilinear PDEs in infinite-dimensional spaces by employing $B$-continuous viscosity solutions and a scalar BSDE driven by a cylindrical Wiener process. The authors prove existence of a $B$-continuous viscosity solution $u(t,x)=Y^{t,x}_t$ to the infinite-dimensional PDE and establish a stochastic representation via the backward equation, $u(t,x)=Y^{t,x}_t$, with a uniqueness result under a strengthened comparison principle. The methodology provides a nonlinear Feynman--Kac formula without relying on stochastic control, and it introduces a robust framework for analyzing fully nonlinear extensions in infinite dimensions. The results have potential implications for numerical schemes and high-dimensional SPDE control problems, enabling probabilistic representations where traditional mild-solution approaches are challenging.

Abstract

We prove the existence of a $B$-continuous viscosity solution for a class of infinite dimensional semilinear partial differential equations (PDEs) using probabilistic methods. Our approach also yields a stochastic representation formula for the solution in terms of a scalar-valued backward stochastic differential equation. The uniqueness is proved under additional assumptions using a comparison theorem for viscosity solutions. Our results constitute the first nonlinear Feynman-Kac formula using the notion of $B$-continuous viscosity solutions and thus introduces a framework allowing for generalizations to the case of fully nonlinear PDEs.

Theorems & Definitions (25)

• Remark 1.1
• Remark 2.3
• Remark 2.5
• Proposition 3.1
• proof
• Definition 3.2
• Definition 3.3
• Remark 3.4
• Proposition 3.5
• proof