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Pré-Publication, Document De Travail Année : 2024

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

Résumé

In this paper, we study linear backward parabolic SPDEs 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. To this end, we derive a new It\^{o}'s formula for the $L^p$-norm of the solution, extending the classical result in the $L^2$-setting. This formula is then used to improve further the regularity of the first component of the solution up to $L^\infty$. Additionally, we present applications such as the controllability of backward SPDEs with $L^p$-controls and a local existence result for a semilinear equation without imposing any growth condition on the nonlinear term.
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Dates et versions

hal-04626627 , version 1 (27-06-2024)

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Víctor Hernández-Santamaría, Kévin Le Balc'H, Liliana Peralta. $L^{p}$-estimates, controllability and local-well posedness for backward SPDEs. 2024. ⟨hal-04626627⟩
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