Backward Ornstein-Uhlenbeck transition operators and mild solutions of non-autonomous Hamilton-Jacobi equations in Banach spaces

TL;DR

This paper extends the existence of mild solutions for non-autonomous Hamilton-Jacobi equations from Hilbert to Banach spaces using Ornstein-Uhlenbeck operators, broadening the applicability in infinite-dimensional stochastic PDEs.

Contribution

It demonstrates that the well-known mild solution existence results can be generalized to Banach spaces, utilizing the regularizing properties of Ornstein-Uhlenbeck operators.

Findings

Extension of mild solution existence to Banach spaces

Use of Ornstein-Uhlenbeck operators for regularization

Applicable to non-autonomous stochastic PDEs in Banach spaces

Abstract

In this paper we revisit the mild-solution approach to second-order semi-linear PDEs of Hamilton-Jacobi type in infinite-dimensional spaces. We show that a well-known result on existence of mild solutions in Hilbert spaces can be easily extended to non-autonomous Hamilton-Jacobi equations in Banach spaces. The main tool is the regularizing property of Ornstein-Uhlenbeck transition evolution operators for stochastic Cauchy problems in Banach spaces with time-dependent coefficients