Fréchet differentiability of mild solutions to SPDEs with respect to the initial datum

We establish n -th-order Fréchet differentiability with respect to the initial datum of mild solutions to a class of jump diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz-continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicat...

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Bibliographic Details
Published in:Journal of evolution equations 2020-09, Vol.20 (3), p.1093-1130
Main Authors: Marinelli, Carlo, Scarpa, Luca
Format: Article
Language:English
Subjects:
Analysis
Hilbert space
Mathematical analysis
Mathematics
Mathematics and Statistics
Well posed problems
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Summary:We establish n -th-order Fréchet differentiability with respect to the initial datum of mild solutions to a class of jump diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz-continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicative) noise sources are a cylindrical Wiener process and a quasi-left-continuous integer-valued random measure. As preliminary steps, we prove well-posedness in the mild sense for this class of equations, as well as first-order Gâteaux differentiability of their solutions with respect to the initial datum, extending previous results by Marinelli, Prévôt, and Röckner in several ways. The differentiability results obtained here are a fundamental step to construct classical solutions to non-local Kolmogorov equations with sufficiently regular coefficients by probabilistic means.
ISSN:1424-3199
1424-3202
DOI:10.1007/s00028-019-00546-0