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Stochastic partial differential equations: a rough paths view on weak solutions via Feynman–Kac
Joscha Diehl1; Peter K. Friz2; Wilhelm Stannat3
1 University of California San Diego, University of California San Diego, Dept of Mathematics - MC 0112, 9500 Gilman Drive La Jolla, CA 92093, USA
2 TU Berlin, Institut für Mathematik, MA 7-2, Strasse des 17. Juni 136, 10623 Berlin, Germany ifriz@math.tu-berlin.deWeierstrass-Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117 Berlin, Germany
3 TU Berlin, Institut für Mathematik, MA 7-2, Strasse des 17. Juni 136, 10623 Berlin, Germany
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 4, pp. 911-947.

We discuss regular and weak solutions to rough partial differential equations (RPDEs), thereby providing a (rough path-)wise view on important classes of SPDEs. In contrast to many previous works on RPDEs, our definition gives honest meaning to RPDEs as integral equations, based on which we are able to obtain existence, uniqueness and stability results. The case of weak “rough” forward equations, may be seen as robustification of the (measure-valued) Zakai equation in the rough path sense. Feynman–Kac representation for RPDEs, in formal analogy to similar classical results in SPDE theory, play an important role.

Nous discutons des solutions régulières et faibles d’équations aux dérivées partielles rugueuses (EDPR), fournissant ainsi un point de vue « chemins rugueux » sur des classes importantes d’ EDPS. Contrairement à de nombreux travaux antérieurs sur le sujet, notre définition donne un sens honnête aux EDPR en tant qu’équations intégrales, sur la base duquel nous sommes en mesure d’obtenir l’existence, l’unicité et la stabilité des résultats. Le cas d’équations forward faibles « rugueuses » peut être vu comme une robustification de l’équation de Zakai à valeurs mesure, au sens des chemins rugueux. Des représentations de type Feynman–Kac pour EDPR, par analogie formelle avec les résultats classiques similaires dans la théorie des EDPS, jouent un rôle important.

Published online:
DOI: 10.5802/afst.1556
Classification: 60H15
Keywords: stochastic partial differential equations, Zakai equation, Feynman–Kac formula, rough partial differential equations, rough paths

Joscha Diehl 1; Peter K. Friz 2; Wilhelm Stannat 3

1 University of California San Diego, University of California San Diego, Dept of Mathematics - MC 0112, 9500 Gilman Drive La Jolla, CA 92093, USA
2 TU Berlin, Institut für Mathematik, MA 7-2, Strasse des 17. Juni 136, 10623 Berlin, Germany ifriz@math.tu-berlin.deWeierstrass-Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117 Berlin, Germany
3 TU Berlin, Institut für Mathematik, MA 7-2, Strasse des 17. Juni 136, 10623 Berlin, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Joscha Diehl; Peter K. Friz; Wilhelm Stannat. Stochastic partial differential equations: a rough paths view on weak solutions via Feynman–Kac. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 4, pp. 911-947. doi : 10.5802/afst.1556. https://afst.centre-mersenne.org/articles/10.5802/afst.1556/

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