📝 SummarXiv

Abstract

By applying the theory of rough paths, Martin Hairer provided a notion of solution for a class of nonlinear stochastic partial differential equations (SPDEs) of Burgers type, driven by additive space-time white noise in one spatial dimension. These equations exhibit spatial roughness that is too severe for classical analytical techniques to handle. Hairer developed a pathwise framework for solutions when the spatial regularity of the solution lies in the range $(\frac{1}{3},\frac{1}{2})$. In this paper, we generalize Hairer's result by extending the spatial regularity to the range $(0, 1]$. More precisely, we establish the pathwise existence and uniqueness of mild (and, equivalently, weak) solutions to Burgers-type SPDEs under this spatial regularity regime.