Rough evolution equations

By Massimiliano Gubinelli, Samy Tindel. 2008

172 citations recorded by the source corpus.

We generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise. As an illustration, we discuss a class of linear and nonlinear 1d SPDEs driven by a space--time Gaussian noise with singular space covariance and Brownian time dependence.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/da7efd4f864e73d9b459684f3929cd92a7bec4d6

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In Applied Mathematics

In Finance

Citations in the corpus

A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion (2010, methodology): “The interested reader is sent to [20] for a complete account on the topic, or to [13,22] for a more detailed summary.”
Partial differential equations driven by rough paths (2008, methodology): “By combining ideas from rough path theory, in particular the construction of flows associated to rough differential equations (RDEs) and classical PDE theory we are able to show existence, uniqueness and a limit theorem for such rough partial differential equations (RPDEs) when the driving signal…”
Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency (2014, methodology): “structure of the noise, and just making sense of the equation oﬀers new challenges. Recent progresses for some speciﬁc Gaussian noises include [4, 27, 29] by means of stochastic analysis methods, and [10, 22, 17] using rough paths arguments. As mentioned above, we shall focus in this article on inte”
PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES (2012, background): “stic processes. But they rely on special features of the problem at hand in order to be able to apply the integration theory provided by the rough path machinery. − Deya, Lejay, Tindel, and Gubinelli [14, 15, 8] deal with SPDEs of the form ∂ tu(t,ξ)=∂ ξ 2u(t,ξ)+σ(u(t,ξ))η(t,ξ), where ξ ∈ [−π, π], η ”
A theory of regularity structures (2013, background): “rougher than white noise), but at the expense of requiring additional spatial regularity [19,53,96].”
A priori estimates for rough PDEs with application to rough conservation laws (2016, background): “E-mail addresses: Aurelien.Deya@univ-lorraine.fr (A. Deya), gubinelli@iam.uni-bonn.de (M. Gubinelli), hofmanov@math.tu-berlin.de (M. Hofmanová), stindel@purdue.edu (S. Tindel). https://doi.org/10.1016/j.jfa.2019.03.008 0022-1236/© 2019 Published by Elsevier Inc. Contents 1.”
Malliavin Calculus for Fractional Delay Equations (2009, methodology): “Observe now that the definition of our stochastic integrals weighted by the heat semigroup will require the introduction of a small variant of those Hölder spaces (see [7, 10] for further details): we define O C .”
Non-linear rough heat equations (2009, background): “The associated proofs can be found in [25].”
A Wong-Zakai theorem for stochastic PDEs (2014, methodology): “1.3 As will be clear, our proof automatically yields a situation analogous to that arising when building a solution theory to ﬁnite -dimensional SDEs using the theory of rough paths [LCL07] (see also [GT10] for an earlier attempt in this direction in the context of evolution equations): one obtains ”