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

Víctor Hernández-Santamaría, Kévin Le Balc'h, Liliana Peralta

TL;DR

This paper studies linear backward parabolic BSPDEs in bounded domains and establishes $L^p$- and $L^ fty$-estimates for weak solutions under minimal regularity of coefficients and data. It introduces a constructive Itô formula for the $L^p$-norm of the backward solution, enabling sharp a priori estimates and improved regularity for the first component. The results yield local well-posedness for semilinear BSPDEs without growth restrictions and provide new local controllability results for both linear and nonlinear backward SPDEs. The methods extend to higher dimensions and avoid strong differentiability assumptions, contributing quantitative tools for backward BSPDE analysis and stochastic control.

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 Itô'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.

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

TL;DR

This paper studies linear backward parabolic BSPDEs in bounded domains and establishes - and -estimates for weak solutions under minimal regularity of coefficients and data. It introduces a constructive Itô formula for the -norm of the backward solution, enabling sharp a priori estimates and improved regularity for the first component. The results yield local well-posedness for semilinear BSPDEs without growth restrictions and provide new local controllability results for both linear and nonlinear backward SPDEs. The methods extend to higher dimensions and avoid strong differentiability assumptions, contributing quantitative tools for backward BSPDE analysis and stochastic control.

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 -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 Itô's formula for the -norm of the backward solution, tailored to this setting and extending the classical result in the -framework. This formula is then used to improve further the regularity of the first component of the solution up to . 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.

Paper Structure

This paper contains 15 sections, 19 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Notation and main results
  3. Applications
  4. Outline of the paper
  5. Itô's formula for the $L^{p}$-norm
  6. Technical lemmas
  7. Proof of the Ito's formula for the $L^p$-norm
  8. Extra regularity for the process $Y$ and proof of \ref{['thm:ito']}
  9. Uniform $L^p$- and $L^\infty$-estimates
  10. Local well-posedness of semilinear equations
  11. Applications to controllability
  12. The linear case
  13. The nonlinear control problem
  14. On the optimality of the regularity
  15. Auxiliary results

Key Result

Theorem 1.2

Let $\alpha,\beta$ be given coefficients satisfying Hy. If $y_T\in L^2_{\mathcal{F}_T}(\Omega; L^2(\mathcal{O}))$ and $F\in L^2_{\mathbb{F}}(0,T;L^2(\mathcal{O}))$, then eq:intro admits a unique weak solution in the sense of def:weak. Moreover, the following energy estimate holds for a positive constant $C>0$ independent of $y_T$ and $F$.

Theorems & Definitions (50)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 1.8
  • Proposition 1.9
  • Definition 1.10: Local controllability
  • ...and 40 more