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Refined existence and regularity results for a class of semilinear dissipative SPDEs

Marinelli, C; Scarpa, L; (2020) Refined existence and regularity results for a class of semilinear dissipative SPDEs. Infinite Dimensional Analysis, Quantum Probability and Related Topics , 23 (2) , Article 2050014. 10.1142/S0219025720500149.

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Abstract

We prove the existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the pth moment of solutions depends on the integrability of the initial datum, in the whole range p∈]0,∞[. Lipschitz continuity of the solution map in pth moment is established, under a Lipschitz continuity assumption on the diffusion coefficient, in the even larger range p∈[0,∞[. A key role is played by an Itô formula for the square of the norm in the variational setting for processes satisfying minimal integrability conditions, which yields pathwise continuity of solutions. Moreover, we show how the regularity of the initial datum and of the diffusion coefficient improves the regularity of the solution and, if applicable, of the invariant measures.

Type: Article
Title: Refined existence and regularity results for a class of semilinear dissipative SPDEs
DOI: 10.1142/S0219025720500149
Publisher version: https://doi.org/10.1142/S0219025720500149
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Stochastic evolution equations, singular drift, variational approach, monotonicity methods, invariant measures
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10117155
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