New regularity results for Hamilton–Jacobi equations and long time behavior of pathwise (stochastic) viscosity solutions

Abstract

We present two new sharp regularity results (regularizing effect and propagation of regularity) for viscosity solutions of uniformly convex homogeneous (space independent) Hamilton–Jacobi equations. The estimates do not depend on the convexity constants of the Hamiltonians. The sharp propagation of regularity result holds in dimension larger than one without additional smoothness assumptions on the data if and only if the Hamiltonians is quadratic in the gradient, a very surprising fact in the theory of Hamilton–Jacobi equations. In turn, the estimates yield new intermittent stochastic regularization results for pathwise (stochastic) viscosity solutions of Hamilton–Jacobi equations with uniformly convex Hamiltonians and rough multiplicative time dependence. Finally, the intermittent estimates allow for the study of the long time behavior of the pathwise (stochastic) viscosity solutions.