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

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 (International Mathematics Research Notices, vol. 2007, Duke University Press, Durham, 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 of the pioneering works by Zvonkin (Math. URSS Sb. 22:129–149, 1974) and Veretennikov (Theory Probab. Appl. 24:354–366, 1979) in the case of one-parameter SDEs with singular drift.