Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise

Carlo Marinelli; Lluís Quer-Sardanyons

doi:10.1007/s11118-021-09914-3

Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise

Carlo Marinelli^*, Lluís Quer-Sardanyons

^*Corresponding author for this work

University College London (UCL)

Research output: Contribution to journal › Article › Research › peer-review

2 Citations (Scopus)

1 Downloads (Pure)

Abstract

We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L²(G), where G is an open bounded domain in ℝ^d with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

Original language	English
Pages (from-to)	243-261
Number of pages	19
Journal	Potential Analysis
Volume	57
Issue number	2
DOIs	https://doi.org/10.1007/s11118-021-09914-3
Publication status	Published - Aug 2022

Keywords

Malliavin calculus
Reaction-diffusion equations
Stochastic PDEs

Access to Document

10.1007/s11118-021-09914-3

https://ddd.uab.cat/record/274605

Cite this

@article{c9be12553ad447e9b5d2a6dcbc2f2535,

title = "Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise",

abstract = "We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L2(G), where G is an open bounded domain in ℝd with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.",

keywords = "Malliavin calculus, Reaction-diffusion equations, Stochastic PDEs",

author = "Carlo Marinelli and Llu{\'i}s Quer-Sardanyons",

note = "Funding Information: The authors thank the anonymous referees for a careful reading of the manuscript. The first-named author is sincerely grateful to Prof. S. Albeverio for several very pleasant stays at the Interdisziplin{\"a}res Zentrum f{\"u}r Komplexe Systeme, Universit{\"a}t Bonn, where most of the work for this paper was done. The second-named author is supported by the grants MTM2015-67802P and PGC2018-097848-B-I00 (Ministerio de Econom{\'i}a y Competitividad). Funding Information: The authors thank the anonymous referees for a careful reading of the manuscript. The first-named author is sincerely grateful to Prof. S. Albeverio for several very pleasant stays at the Interdisziplin{\"a}res Zentrum f{\"u}r Komplexe Systeme, Universit{\"a}t Bonn, where most of the work for this paper was done. The second-named author is supported by the grants MTM2015-67802P and PGC2018-097848-B-I00 (Ministerio de Econom{\'i}a y Competitividad). Publisher Copyright: {\textcopyright} 2021, The Author(s).",

year = "2022",

month = aug,

doi = "10.1007/s11118-021-09914-3",

language = "English",

volume = "57",

pages = "243--261",

journal = "Potential Analysis",

issn = "0926-2601",

number = "2",

}

TY - JOUR

T1 - Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise

AU - Marinelli, Carlo

AU - Quer-Sardanyons, Lluís

N1 - Funding Information: The authors thank the anonymous referees for a careful reading of the manuscript. The first-named author is sincerely grateful to Prof. S. Albeverio for several very pleasant stays at the Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn, where most of the work for this paper was done. The second-named author is supported by the grants MTM2015-67802P and PGC2018-097848-B-I00 (Ministerio de Economía y Competitividad). Funding Information: The authors thank the anonymous referees for a careful reading of the manuscript. The first-named author is sincerely grateful to Prof. S. Albeverio for several very pleasant stays at the Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn, where most of the work for this paper was done. The second-named author is supported by the grants MTM2015-67802P and PGC2018-097848-B-I00 (Ministerio de Economía y Competitividad). Publisher Copyright: © 2021, The Author(s).

PY - 2022/8

Y1 - 2022/8

N2 - We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L2(G), where G is an open bounded domain in ℝd with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

AB - We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L2(G), where G is an open bounded domain in ℝd with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

KW - Malliavin calculus

KW - Reaction-diffusion equations

KW - Stochastic PDEs

UR - http://www.scopus.com/inward/record.url?scp=85103654908&partnerID=8YFLogxK

U2 - 10.1007/s11118-021-09914-3

DO - 10.1007/s11118-021-09914-3

M3 - Article

AN - SCOPUS:85103654908

SN - 0926-2601

VL - 57

SP - 243

EP - 261

JO - Potential Analysis

JF - Potential Analysis

IS - 2

ER -

Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Modelos estocásticos y aplicaciones

Cite this

Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Projects

Modelos estocásticos y aplicaciones

Cite this