Non-linear rough heat equations

Abstract

This article is devoted to define and solve an evolution equation of the form dy t = y t dt + d X t (y t ), where stands for the Laplace operator on a space of the form L p (R n ), and X is a finite dimensional noisy nonlinearity whose typical form is given by) is a γ -Hölder function generating a rough path and each f i is a smooth enough function defined on L p (R n ). The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.

Perturbed linear rough differential equations

“…Of course, dealing with unbounded family of operators, for example for solving Stochastic Partial Differential Equations, is much more intricate and needs specific treatments. The reader is referred to the quickly growing literature on these subjects [11,12,[31][32][33], ... The variation of constant principle is also linked to Volterra equations which have been studied in the rough path context by A. Deya [23,24].…”

Stochastic differential equations with non-negativity constraints driven by fractional Brownian motion

“…As is natural, afterwards has been considered some of the possible generalizations of the diffusion processes. For instance, in the literature we can find now papers about PDEs [18,3,5,12], Volterra equations [6,7,2] or systems with delay [10,9,16,14].…”

Numerical Schemes for Rough Parabolic Equations

“…The first one, due to Friz, Caruana, Oberhauser and Diehl ([2, 12, 11, 10]), finds its inspiration in the viscositysolution theory for (ordinary) PDEs, and which efficiently combines with the rough paths stability results. The second one, developped by Gubinelli, Tindel and the author ( [16,8,7]) on the one hand and Teichmann ([34]) on the other, takes the mild formulation of PDEs as the basic model, and then tries to take profit of the semigroup regularizing properties in order to cope with time roughness. In this sense, the latter approach happens to be quite close to the stochastic infinite-dimensional theory by Da Prato and Zabczyk [4] (among others), and it shares many characteristics with the recent works of Jentzen, Kloeden and Röckner [21,24,26].…”

“…The aim of this paper is to remedy the problem by introducing easily-implementable approximation algorithms. To be more specific, we intend to follow the mild formulation of [16,8,7] and show that this formalism can be combined with the classical discretization procedures for (Wiener) SPDEs.…”

“…This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0, 1) perturbed by a non-linear rough signal. It is the continuation of [8,7], where the existence and uniqueness of a solution has been established. The approach combines rough paths methods with standard considerations on discretizing stochastic PDEs.…”