Heat semigroup and singular PDEs

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• 作者：I. Bailleul^a ; ¹ ; ^{ismael.bailleul@univ-rennes1.fr" class="auth_mail" title="E-mail the corresponding author} ; F. Bernicot^b ; ² ; ³ ; ^{frederic.bernicot@univ-nantes.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Stochastic partial differential equations ; Paracontrolled calculus ; Heat semigroup ; Parabolic Anderson model equation
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：270
• 期：9
• 页码：3344-3452
• 全文大小：1297 K

文摘

We provide in this work a semigroup approach to the study of singular PDEs, in the line of the paracontrolled approach developed recently by Gubinelli, Imkeller and Perkowski. Starting from a heat semigroup, we develop a functional calculus and introduce a paraproduct based on the semigroup, for which commutator estimates and Schauder estimates are proved, together with their paracontrolled extensions. This machinery allows us to investigate singular PDEs in potentially unbounded Riemannian manifolds under mild geometric conditions. As an illustration, we study the generalized parabolic Anderson model equation and prove, under mild geometric conditions, its well-posed character in Hölders spaces, in small time on a potentially unbounded 2-dimensional Riemannian manifold, for an equation driven by a weighted noise, and for all times for the linear parabolic Anderson model equation in 2-dimensional unbounded manifolds. This machinery can be extended to an even more singular setting and deal with Sobolev scales of spaces rather than Hölder spaces.