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

Antoine-Marie Bogso, Olivier Menoukeu Pamen

TL;DR

This work establishes strong well-posedness and Malliavin regularity for a two-parameter hyperbolic SPDE driven by a Brownian sheet with drift $b$ that is the difference of two componentwise monotone functions and exhibits spatial linear growth. The authors implement a Yamada–Watanabe strategy for Brownian sheets by proving path-by-path uniqueness via a Davie-type averaging estimate on the plane and a sheet-based Grönwall argument, leading to weak existence and hence a unique strong solution. They further prove Malliavin differentiability of the unique strong solution when $b$ is uniformly bounded, and obtain Malliavin regularity for small times under linear growth, leveraging Gaussian white noise techniques, local time-space integration formulas, and a robust compactness framework on the plane. The results extend existing work on one-parameter SDEs to hyperbolic SPDEs with irregular drifts, providing tools for density analysis and regularity in this broader stochastic PDE setting.

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.

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

TL;DR

This work establishes strong well-posedness and Malliavin regularity for a two-parameter hyperbolic SPDE driven by a Brownian sheet with drift that is the difference of two componentwise monotone functions and exhibits spatial linear growth. The authors implement a Yamada–Watanabe strategy for Brownian sheets by proving path-by-path uniqueness via a Davie-type averaging estimate on the plane and a sheet-based Grönwall argument, leading to weak existence and hence a unique strong solution. They further prove Malliavin differentiability of the unique strong solution when is uniformly bounded, and obtain Malliavin regularity for small times under linear growth, leveraging Gaussian white noise techniques, local time-space integration formulas, and a robust compactness framework on the plane. The results extend existing work on one-parameter SDEs to hyperbolic SPDEs with irregular drifts, providing tools for density analysis and regularity in this broader stochastic PDE setting.

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.
Paper Structure (9 sections, 26 theorems, 129 equations)

This paper contains 9 sections, 26 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Existence and uniqueness results
  3. Preliminary results
  4. Main results and proofs
  5. Proof of Theorem \ref{['maintheuniq2']}
  6. Malliavin regularity
  7. Basic facts on Malliavin calculus and compactness criterion on the plane
  8. Malliavin differentiability for bounded drifts
  9. Malliavin regularity under linear growth condition

Key Result

Proposition 2.1

Let $W:=\left(W^{(1)}_{s,t},\ldots,W^{(d)}_{s,t};(s,t)\in[0,1]^2\right)$ be a $\mathbb{R}^d$-valued Brownian sheet defined on a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$, where $\mathbb{F}=(\mathcal{F}_{s,t};s,t\in[0,1])$. Let $b\in\mathcal{C}\left([0,1]^2,\mathcal{C}^1 Here $\nabla_yb$ denotes the gradient of $b$ with respect to the third variable, $|\cdot|$ is the u

Theorems & Definitions (44)

  • Definition 1.1
  • Proposition 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Theorem 2.8
  • Theorem 2.9
  • ...and 34 more