Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space

TL;DR

This paper studies the Kolmogorov equation linked to a stochastic reflection problem in a Hilbert space, proving existence and uniqueness of smooth solutions with Neumann boundary conditions.

Contribution

It establishes the existence and uniqueness of smooth solutions to an elliptic Kolmogorov equation with reflection in an infinite-dimensional setting.

Findings

Proved existence of smooth solutions for the Kolmogorov equation.

Established uniqueness of solutions under reflection boundary conditions.

Extended results to infinite-dimensional Hilbert spaces with convex sets.

Abstract

We consider the stochastic reflection problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $K$ with nonempty interior and regular boundary $\Sigma$ in a Hilbert space $H$. We prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite-dimensional Kolmogorov equation with Neumann boundary condition on $\Sigma$.