A mild Itô formula for SPDEs

Da Prato, Giuseppe; Jentzen, Arnulf; Röckner, Michael

doi:10.1090/tran/7165

A mild Itô formula for SPDEs
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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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