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

arXiv comment25 pages. arXiv admin note: text overlap with arXiv:2112.00393

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• Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness
• Multidimensional SDE with distributional drift and Lévy noise
• Multiple points of the Brownian sheet in critical dimensions
• Path-by-path uniqueness of multidimensional SDE's on the plane with nondecreasing coefficients
• Stochastic integration with respect to local time of the Brownian sheet and regularising properties of Brownian sheet paths

Cited by in corpus (2)

• Smoothness of solutions of hyperbolic stochastic partial differential equations with $L^{\infty }$-vector fields
• Strong solutions of fractional Brownian sheet driven SDEs with integrable drift