An Itô Formula for rough partial differential equations and some applications

Abstract

We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form ∂ t u−A t u−f = (Ẋ t (x)•∇+Ẏ t (x))u on [0, T ]×R d. To do so, we introduce a concept of "differential rough driver", which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces W k,p. We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e. when Y = 0), we use this framework to prove an Itô Formula (in the sense of a chain rule) for Nemytskii operations of the form u → F (u), where F is C 2 and vanishes at the origin. Our method is based on energy estimates, and a generalization of the Moser Iteration argument to prove boundedness of a dense class of solutions of parabolic problems as above. In particular, we avoid the use of flow transformations and work directly at the level of the original equation. We also show the corresponding chain rule for F (u) = |u| p with p ≥ 2, but also when Y = 0 and p ≥ 4. As an application of these results, we prove existence and uniqueness of a suitable class of L p-solutions of parabolic equations with multiplicative noise. Another related development is the homogeneous Dirichlet boundary problem on a smooth domain, for which a weak maximum principle is shown under appropriate assumptions on the coefficients.

A pathwise stochastic Landau-Lifshitz-Gilbert equation with application to large deviations

“…A rigorous notion of rough integral requires however to introduce the corresponding space of G-controlled paths, which is omitted here for simplicity of the presentation. In the context of unbounded rough drivers, we point out that related questions were addressed in [32,Sec. 3].…”

“…Since the argument seems to generalize well to other contexts, we present a general methodology (refered to as "rough standard machinery") which later permits to bootstrap. This method is built upon the formalism of "rough driver" developed in [3,12] and subsequently analyzed in [31,32]. In Section 5 we use these estimates to prove the continuity of the Itô-Lyons map, thereby finishing the proof of Theorem 2.14.…”

“…For the reasons alluded above, we build our setting upon the formalism of (unbounded) rough driver. These objects were introduced in [3] and further investigated in [12,31,32,36,33].…”

“…The objective of this paper is twofold. At first, we derive a robust pathwise formulation of (LLG) and obtain energy estimates for it, using tools from the variational approach to rough PDEs ( [12,31,36,37,32]). The combination of some of the techniques developed in these prior works with a "rough boostrap" argument (see Section 3.2) permit to show that energy estimates of the same kind are true at any level of spatial regularity, as permitted by the regularity of the data.…”

On a rough perturbation of the Navier–Stokes system and its vorticity formulation

“…⊗ := W n,2 (T 2 × T 2 ). Variations of the following result have already been proved in [12,18,19,21], so we omit the proof. PROPOSITION 5.1.…”