$L^{p}$-estimates, local well-posedness and controllability for linear and semilinear backward SPDEs

TL;DR

This paper develops new $L^p$-estimates and regularity results for linear backward SPDEs, enabling applications to local existence and controllability of semilinear backward SPDEs with minimal assumptions.

Contribution

It introduces a new Itô's formula for the $L^p$-norm of backward solutions and applies it to establish regularity, local existence, and controllability results for backward SPDEs.

Findings

Established $L^p$-estimates with minimal regularity assumptions.

Developed a new Itô's formula for the $L^p$-norm of solutions.

Proved local existence and controllability for semilinear backward SPDEs.

Abstract

In this paper, we study linear backward parabolic SPDEs in bounded domains and present new a priori estimates for their weak solutions. Inspired by the seminal work of Y. Hu, J. Ma and J. Yong from 2002 on strong solutions, we establish $L^p$-estimates requiring minimal assumptions on the regularity of the coefficients, the terminal data, and the external force. Our approach relies on direct, constructive, and quantitative arguments, adapted from known methods in the theory of SPDEs to this setting. In particular, we develop a new It\^{o}'s formula for the $L^p$-norm of the backward solution, tailored to this setting and extending the classical result in the $L^2$-framework. This formula is then used to improve further the regularity of the first component of the solution up to $L^\infty$. We also present two applications: a local existence result for a semilinear equation without imposing any growth condition on the nonlinear term, and a novel local controllability result for semilinear backward SPDEs that partially resolves an open problem in the field.