Homogenization of a stochastically forced Hamilton-Jacobi equation

Hölder regularity of Hamilton-Jacobi equations with stochastic forcing

“…We first give a parameter-dependent variant of Kolmogorov's continuity criterion. Its statement and proof are very similar to that in [14].…”

“…The proof of Lemma A.1 can be found in [14]. The arguments for Lemma A.2 are similar, but some further details are needed to account for the use of Itô's formula and the interaction between B and W .…”

“…These issues were considered by Seeger [14] for the equation (1.1) with H independent of (x, t) and convex in p. In this paper, we further extend the regularity results from [14] to apply also to second-order equations and with more complicated (x, t)-dependence for F and H. To do so, we follow [3] and prove that the equations exhibit an improvement of oscillation effect at all sufficiently small scales, which is a consequence only of the structure of the equation. The main difference with [3] is the addition of the random forcing term f (x) • dB(t) which obliges to revisit the analysis of [3] in a substantial way.…”