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

Bogachev VI, Da Prato G, Röckner M (2009)
Journal of Functional Analysis 256(4): 1269-1298.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
We consider a Kolmogorov operator L-0 in a Hilbert space H, related to a stochastic PDE with a time-dependent singular quasi-dissipative drill F = F(t,.): H -> H, defined on a Suitable space of regular functions. We show that L-0 is essentially m-dissipative in the space L-p(vertical bar 0, T vertical bar x H ; nu), p >= 1, where nu(dt, dx) = nu(t)(dx)dt and the family (nu(t))(t is an element of vertical bar 0,T vertical bar) is a solution of the Fokker-Planck equation given by L-0. As a consequence, the closure of L-0 generates a Markov C-0-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. Rockner, Existence of solutions to weak parabolic equations for measures, Proc. London Math. Soc. (3) 88 (2004) 753-774], [V. Bogachev, G. Da Prato, M. Rockner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (3) (2008) 397-418], and [V. Bogachev, G. Da Prato, M. Rockner, W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. London Math. Soc. 39 (2007) 631-640] to infinite dimensions. (C) 2008 Elsevier Inc. All rights reserved.
Erscheinungsjahr
Zeitschriftentitel
Journal of Functional Analysis
Band
256
Ausgabe
4
Seite(n)
1269-1298
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Bogachev VI, Da Prato G, Röckner M. Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces. Journal of Functional Analysis. 2009;256(4):1269-1298.
Bogachev, V. I., Da Prato, G., & Röckner, M. (2009). Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces. Journal of Functional Analysis, 256(4), 1269-1298. doi:10.1016/j.jfa.2008.05.005
Bogachev, V. I., Da Prato, G., and Röckner, M. (2009). Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces. Journal of Functional Analysis 256, 1269-1298.
Bogachev, V.I., Da Prato, G., & Röckner, M., 2009. Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces. Journal of Functional Analysis, 256(4), p 1269-1298.
V.I. Bogachev, G. Da Prato, and M. Röckner, “Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces”, Journal of Functional Analysis, vol. 256, 2009, pp. 1269-1298.
Bogachev, V.I., Da Prato, G., Röckner, M.: Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces. Journal of Functional Analysis. 256, 1269-1298 (2009).
Bogachev, Vladimir I., Da Prato, Giuseppe, and Röckner, Michael. “Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces”. Journal of Functional Analysis 256.4 (2009): 1269-1298.