Abstract

AbstractWe give a construction allowing us to build local renormalized solutions to general quasilinear stochastic PDEs within the theory of regularity structures, thus greatly generalizing the recent results of [1, 5, 11]. Loosely speaking, our construction covers quasilinear variants of all classes of equations for which the general construction of [3, 4, 7] applies, including in particular one‐dimensional systems with KPZ‐type nonlinearities driven by space‐time white noise. In a less singular and more specific case, we furthermore show that the counterterms introduced by the renormalization procedure are given by local functionals of the solution. The main feature of our construction is that it allows exploitation of a number of existing results developed for the semilinear case, so that the number of additional arguments it requires is relatively small. © 2018 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc.

Highlights

  • Which extends [OW16] to the regime of regularity, which includes noises slightly better than space-time white noise in 1 dimension (similar to the setting of our example (1.1) below)

  • The generality in which we succeed in building local solution theories is, in some sense, optimal: loosely speaking, we show that if an equation can be solved with regularity structures and its solution has positive regularity, its quasilinear variants can be solved

  • The only disadvantage of our approach, compared to [BDH16, FG16, OW16], is that it is not obvious at all a priori why the counterterms generated by the renormalisation procedure should be local in the solution

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Summary

INTRODUCTION

Which extends [OW16] to the regime of regularity, which includes noises slightly better than space-time white noise in 1 dimension (similar to the setting of our example (1.1) below). The generality in which we succeed in building local solution theories is, in some sense, optimal: loosely speaking, we show that if an equation can be solved with regularity structures and its solution has positive regularity, its quasilinear variants can be solved (locally) We deal with both the analytic and the probabilistic side of the theory in the sense that we show that the general machinery developed in [CH16] can be exploited in order to produce random models that do precisely fit our needs. Another major advantage of our approach is that its formulation is such that it allows to leverage many existing results from the semilinear situation without requiring us to reinvent the wheel.

An equivalent formulation
Regularity structures with continuous parameter-dependence
A regularity structure
Admissible models
Constructing models
Lifting the operator I
A CONCRETE EXAMPLE
A concrete example
Full Text
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