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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A mild Itô formula for SPDEs


Authors: Giuseppe Da Prato, Arnulf Jentzen and Michael Röckner
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 35R60; Secondary 60H15
DOI: https://doi.org/10.1090/tran/7165
Published electronically: June 10, 2019
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Abstract: This article introduces a certain class of stochastic processes,
which we suggest calling mild Itô processes, and a new, somehow mild, Itô-type formula for such processes. Examples of mild Itô processes are mild solutions of stochastic partial differential equations (SPDEs) and their numerical approximation processes. We illustrate the capacity of the mild Itô formula by several applications. In particular, we illustrate how the mild Itô formula can be used to derive improved a priori bounds for SPDEs, we demonstrate how the mild Itô formula can be employed to establish improved Hölder continuity properties for solutions of Kolmogorov partial differential equations (PDEs) in Hilbert spaces, and we illustrate how the mild Itô formula can be used to solve the weak convergence problem for numerical approximations of SPDEs with nonlinear diffusion coefficients.


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Additional Information

Giuseppe Da Prato
Affiliation: Scuola Normale Superiore di Pisa, 56126 Pisa, Italy

Arnulf Jentzen
Affiliation: Department of Mathematics, Seminar for Applied Mathematics, ETH Zurich, 8092 Zurich, Switzerland

Michael Röckner
Affiliation: Faculty of Mathematics, Bielefeld University, 33501 Bielefeld, Germany

DOI: https://doi.org/10.1090/tran/7165
Received by editor(s): June 16, 2016
Received by editor(s) in revised form: November 23, 2016
Published electronically: June 10, 2019
Additional Notes: This work has been partially supported by the Collaborative Research Centre $701$ “Spectral Structures and Topological Methods in Mathematics”, by the International Graduate School “Stochastics and Real World Models”, by the research project “Numerical solutions of stochastic differential equations with non-globally Lipschitz continuous coefficients” (all funded by the German Research Foundation), and by the BiBoS Research Center. The support of the Issac Newton Institute for Mathematical Sciences in Cambridge is also gratefully acknowledged, where this work was initiated during the special semester “Stochastic Partial Differential Equations”.
Article copyright: © Copyright 2019 American Mathematical Society