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Main Title: An Itô Formula for rough partial differential equations and some applications
Author(s): Hocquet, Antoine
Nilssen, Torstein
Type: Article
Abstract: We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form ∂ t u − A t u − f = ( X ̇ t ( x ) ⋅ ∇ + Y ̇ 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.
Subject(s): energy method
Itô formula
maximum principle
renormalized solutions
rough paths
rough PDEs
weak solutions
Issue Date: 20-Apr-2020
Date Available: 12-Mar-2021
Language Code: en
DDC Class: 510 Mathematik
Sponsor/Funder: TU Berlin, Open-Access-Mittel – 2020
DFG, 277012070, FOR 2402: Rough Paths, Stochastic Partial Differential Equations and Related Topics
Journal Title: Potential Analysis
Publisher: SpringerNature
Volume: 54
Publisher DOI: 10.1007/s11118-020-09830-y
Page Start: 331
Page End: 386
EISSN: 1572-929X
ISSN: 0926-2601
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik » FG Mathematische Stochastik / Stochastische Prozesse in den Neurowissenschaften
Appears in Collections:Technische Universität Berlin » Publications

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