Regularization by noise for stochastic Hamilton-Jacobi equations

Gassiat P, Gess B. Regularization by noise for stochastic Hamilton-Jacobi equations. PROBABILITY THEORY AND RELATED FIELDS. 2019;173(3-4):1063-1098.We study regularizing effects of nonlinear stochastic perturbations for fully nonlinear PDE. More precisely, path-by-path L bounds for the second derivative of solutions to such PDE are shown. These bounds are expressed as solutions to reflected SDE and are shown to be optimal