Homogenization of pathwise Hamilton–Jacobi equations

Abstract

We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we consider Hamilton-Jacobi equations forced by a noise term with small amplitude, and, in increasing the strength of the noise, we observe a sharp transition at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity of the solutions of stochastically forced Hamilton-Jacobi equations, which are of independent interest.

Pathwise Solutions for Fully Nonlinear First- and Second-Order Partial Differential Equations with Multiplicative Rough Time Dependence

“…This leads to the question if it is possible to find, instead of λ|x − y| 2 , an initial datum for the doubled pde, which is still coercive, and, in the mean time, better adjusted to the structure of the doubled equation. This question was answered affirmatively for equations with quadratic Hamiltonians corresponding to Riemannian metrics in Friz, Gassiat, Lions and Souganidis [24], positively homogeneous and convex in p Hamiltonians in Seeger [77] and, for general, convex in p Hamiltonians in Lions and Souganidis [56].…”

The Neumann problem for fully nonlinear SPDE

“…A number of applications were also discussed in these works, among them the level-set formulation of the motion of hypersurfaces when the dynamics are perturbed by a stochastic noise (this includes in particular the case of stochastic mean curvature flow). More recent developments include well-posedness of the equation for more general (in particular, x-dependent) Hamiltonians [FGLS17, See18a,See18b]; analysis of qualitative behavior, for instance, long-time behavior, regularity/regularization by noise, and finite/infinite speed of propagation [Gas17,GG19,GGLS20,LS20b]; the construction of numerical schemes [See20]; and applications to stochastically perturbed mean curvature flow [SY04,LS20a].…”

“…On equations with x-dependent Hamiltonians. We focus in this paper on spatially homogeneous Hamiltonians H. The case of x-dependent H is much more demanding technically, even without boundary conditions, and requires some care on the assumptions, see for instance [FGLS17,See18a,Sou19,GGLS20]. Unlike in the x-independent case, where we can deal with any continuous signal ζ, the regularity of the signal plays a role in the x-dependent case (this is not surprising, since this is already the case for ODEs, as made very explicit in Lyons' rough path theory [Lyo98]).…”

Interpolation results for pathwise Hamilton-Jacobi equations

“…Indeed, more regularity and structural requirements are needed for the Hamiltonian, as is described in more detail in [40]. Some particular and instructive examples are explored in the works of Friz, Gassiat, Lions, and Souganidis [11], Lions and Souganidis [28], and Seeger [39]. If H is linear in Du, more general spatial dependence can be treated using either stochastic calculus or the theory of rough paths, as in Caruana, Friz, and Oberhauser [6] and Diehl, Friz, and Oberhauser [10].…”