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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A mild Itô formula for SPDEs
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by Giuseppe Da Prato, Arnulf Jentzen and Michael Röckner PDF
Trans. Amer. Math. Soc. 372 (2019), 3755-3807 Request permission

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
  • MR Author ID: 53850
  • Arnulf Jentzen
  • Affiliation: Department of Mathematics, Seminar for Applied Mathematics, ETH Zurich, 8092 Zurich, Switzerland
  • MR Author ID: 824543
  • Michael Röckner
  • Affiliation: Faculty of Mathematics, Bielefeld University, 33501 Bielefeld, Germany
  • MR Author ID: 149365
  • 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”.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 3755-3807
  • MSC (2010): Primary 35R60; Secondary 60H15
  • DOI: https://doi.org/10.1090/tran/7165
  • MathSciNet review: 4009384