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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]
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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]
ISSN:00221236
DOI:10.1016/j.jfa.2008.05.005