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Fokker–Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces

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Title: Fokker–Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces
Authors: Bogachev, V.I.1 vibogach@mail.ru, Da Prato, G.2, Röckner, M.3,4
Source: Journal of Functional Analysis. Feb2009, Vol. 256 Issue 4, p1269-1298. 30p.
Subjects: Fokker-Planck equation, Hilbert space, Stochastic partial differential equations, Semigroups (Algebra), Reaction-diffusion equations, Parabolic differential equations
Abstract: Abstract: We consider a Kolmogorov operator in a Hilbert space H, related to a stochastic PDE with a time-dependent singular quasi-dissipative drift , defined on a suitable space of regular functions. We show that is essentially m-dissipative in the space , , where and the family is a solution of the Fokker–Planck equation given by . As a consequence, the closure of generates a Markov -semigroup. We also prove uniqueness of solutions to the Fokker–Planck equation for singular drifts F. Applications to reaction–diffusion equations with time-dependent reaction term are presented. This result is a generalization of the finite-dimensional case considered in [V. Bogachev, G. Da Prato, M. Röckner, Existence of solutions to weak parabolic equations for measures, Proc. London Math. Soc. (3) 88 (2004) 753–774], [V. Bogachev, G. Da Prato, M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (3) (2008) 397–418], and [V. Bogachev, G. Da Prato, M. Röckner, W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. London Math. Soc. 39 (2007) 631–640] to infinite dimensions. [Copyright &y& Elsevier]
Copyright of Journal of Functional Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Fokker–Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces
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  Data: Abstract: We consider a Kolmogorov operator in a Hilbert space H, related to a stochastic PDE with a time-dependent singular quasi-dissipative drift , defined on a suitable space of regular functions. We show that is essentially m-dissipative in the space , , where and the family is a solution of the Fokker–Planck equation given by . As a consequence, the closure of generates a Markov -semigroup. We also prove uniqueness of solutions to the Fokker–Planck equation for singular drifts F. Applications to reaction–diffusion equations with time-dependent reaction term are presented. This result is a generalization of the finite-dimensional case considered in [V. Bogachev, G. Da Prato, M. Röckner, Existence of solutions to weak parabolic equations for measures, Proc. London Math. Soc. (3) 88 (2004) 753–774], [V. Bogachev, G. Da Prato, M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (3) (2008) 397–418], and [V. Bogachev, G. Da Prato, M. Röckner, W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. London Math. Soc. 39 (2007) 631–640] to infinite dimensions. [Copyright &y& Elsevier]
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  Data: <i>Copyright of Journal of Functional Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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