Regularization by noise for stochastic Hamilton-Jacobi equations
  We study regularizing effects of nonlinear stochastic perturbations for fully nonlinear PDE. More precisely, path-by-path $L^{\infty}$ 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. 
Paul Gassiat, Benjamin Gess
math.PR math.AP