Malliavin differentiability of solutions of hyperbolic stochastic partial differential equations with irregular drifts

TL;DR

This paper establishes path-by-path uniqueness and Malliavin differentiability of solutions to hyperbolic stochastic PDEs with irregular drifts, extending previous results and applying the Yamada-Watanabe principle for Brownian sheet-driven equations.

Contribution

It proves path-by-path uniqueness for hyperbolic SPDEs with monotone Borel measurable drifts and demonstrates Malliavin differentiability of solutions under bounded drift conditions.

Findings

Proved path-by-path uniqueness for hyperbolic SPDEs with irregular drifts.

Established Malliavin differentiability of solutions with bounded drifts.

Extended previous results to equations with spatial linear growth drifts.

Abstract

We prove path-by-path uniqueness of solution to hyperbolic stochastic partial differential equations when the drift coefficient is the difference of two componentwise monotone Borel measurable functions of spatial linear growth. The Yamada-Watanabe principle for SDE driven by Brownian sheet then allows to derive strong uniqueness for such equation and thus extending the results in [Bogso, Dieye and Menoukeu Pamen, Elect. J. Probab., 27:1-26, 2022] and [Nualart and Tindel, Potential Anal., 7(3):661--680, 1997]. Assuming that the drift is globally bounded, we show that the unique strong solution is Malliavin differentiable. The case of spatial linear growth drift coefficient is also studied.