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