Smoothness of solutions of hyperbolic stochastic partial differential equations with $L^{\infty}$-vector fields

TL;DR

This paper proves the existence, uniqueness, and regularity of solutions to a hyperbolic stochastic PDE with merely bounded measurable vector fields, extending techniques from Malliavin calculus and variational methods.

Contribution

It establishes the first strong, Malliavin differentiable solution for such HSPDEs with non-smooth vector fields, improving upon existing SDE results.

Findings

Unique strong solution exists for the HSPDE with bounded measurable vector fields.

Solution is Malliavin differentiable and Sobolev differentiable w.r.t. initial data.

Advances the understanding of hyperbolic SPDEs with singular drift.

Abstract

In this paper we are interested in a quasi-linear hyperbolic stochastic differential equation (HSPDE) when the vector field is merely bounded and measurable. Although the deterministic counterpart of such equation may be ill-posed (in the sense that uniqueness or even existence might not be valid), we show for the first time that the corresponding HSPDE has a unique (Malliavin differentiable) strong solution. Our approach for proving this result rests on: 1) tools from Malliavin calculus and 2) variational techniques introduced in [Davie, Int. Math. Res. Not., Vol. 2007] non trivially extended to the case of SDEs in the plane by using an algorithm for the selection of certain rectangles. As a by product, we also obtain the Sobolev differentiability of the solution with respect to its initial value. The results derived here constitute a significant improvement of those in the current literature on SDEs on the plane and can be regarded as an analogous equivalent of the pioneering works by [Zvonkin, Math. URSS Sbornik, 22:129-149] and [Veretennikov, Theory Probab. Appl., 24:354-366] in the case of one-parameter SDEs with singular drift.