Regularization by noise for stochastic Hamilton–Jacobi equations

Abstract

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.

Nonlinear diffusion equations with nonlinear gradient noise

“…Only in recent years, in a series of works [LPS13, LPS14, GS17a, GS17b, GS17c, GS15] a kinetic approach to (simpler versions of) (1.1) was developed based on rough path methods, cf. also [HKRSs18,GPS15,GG18,GGLS18]. for numerical methods and regularity/qualitative properties of the solutions.…”

Path-dependent convex conservation laws

“…Although the results are similar, both relying on the strict convexity of the flux but with the one in [22] restricted to f = u 2 /2, the proofs are different. We work at the level of conservation laws and use the method of generalized characteristics.…”

“…where z is a continuous path, and F is a nonlinear function meeting the standard assumptions from the theory of viscosity solutions of fully nonlinear degenerate parabolic PDEs. An L ∞ -bound on the second derivative D 2 v is established in [22]. In the special case…”

Path-by-path regularization by noise for scalar conservation laws

“…It is therefore not clear from [7] if the regularizing effect of Brownian motion can be characterized in terms of its path properties. A partial answer has been given by the results of [6] which imply, as a special case, that the entropy solution to (1.2) satisfies a path-by-path estimate of the form…”

“…In fact, for P-almost every fixed realization of the solution u(·, ω) there will be times t > 0 where shocks appear and thus x → u(t, x, ω) is not Lipschitz continuous. However, the type of regularizing effects used in [7] and in [6] are of different nature. While [7] relies on averaging techniques and thus on an increased speed of averaging due to Brownian scaling, the effect in [6] relies on (strict) convexity of the flux function and dependence of the direction of the flux on w. This leads to the two main questions addressed in this work: First, to classify the properties of the paths of the Brownian motion leading to the regularizing effect observed in [7] and to thus obtain a better understanding of the interplay of the deterministic and stochastic averaging in this case.…”