Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations

Barbu V, Röckner M (2021)
Journal of Functional Analysis 280(7): 108926.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Barbu, Viorel; Röckner, MichaelUniBi
Abstract / Bemerkung
One proves the existence and uniqueness of a generalized (mild) solution for the nonlinear Fokker-Planck equation (FPE) u(t) - Delta(beta(u)) + div(D(x)b(u)u) = 0, t >= 0, x is an element of R-d, d not equal 2, u(0, .) = u(0), in R-d, where u(0) is an element of L-1 (R-d ), beta is an element of C-2 (R) is a nondecreasing function, b is an element of C-1, bounded, b >= 0, D is an element of L-infinity (R-d; R-d) with div D is an element of (L-2+L infinity)(R-d), and (div D)(-) is an element of L-infinity (R-d), beta strictly increasing, if b is not constant. Moreover, t -> u(t, u(0)) is a semigroup of contractions in L-1 (R-d), which leaves invariant the set of probability density functions in R-d. If div D >= 0, beta'(r) >= a vertical bar r vertical bar(alpha-)(1) and vertical bar beta(r)vertical bar <= Cr-alpha, alpha >= 1, d >= 3, then vertical bar u(t)vertical bar L-infinity <= Ct(-d/d+(alpha-1)d) vertical bar u(0)vertical bar(2/2+(m-1d)), t > 0, and if D is an element of L-2 (R-d;R-d) the existence extends to initial data u(0) in the space M-b of bounded measures in R-d. As a consequence for arbitrary initial laws, we obtain weak solutions to a class of McKean-Vlasov SDEs with coefficients which have singular dependence on the time marginal laws. (C) 2021 Elsevier Inc. All rights reserved.
Stichworte
Fokker-Planck equation; m-accretive; Measure as initial data; McKean-Vlasov stochastic differential equation
Erscheinungsjahr
2021
Zeitschriftentitel
Journal of Functional Analysis
Band
280
Ausgabe
7
Art.-Nr.
108926
ISSN
0022-1236
eISSN
1096-0783
Page URI
https://pub.uni-bielefeld.de/record/2952713

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Barbu V, Röckner M. Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations. Journal of Functional Analysis. 2021;280(7): 108926.
Barbu, V., & Röckner, M. (2021). Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations. Journal of Functional Analysis, 280(7), 108926. https://doi.org/10.1016/j.jfa.2021.108926
Barbu, V., and Röckner, M. (2021). Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations. Journal of Functional Analysis 280:108926.
Barbu, V., & Röckner, M., 2021. Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations. Journal of Functional Analysis, 280(7): 108926.
V. Barbu and M. Röckner, “Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations”, Journal of Functional Analysis, vol. 280, 2021, : 108926.
Barbu, V., Röckner, M.: Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations. Journal of Functional Analysis. 280, : 108926 (2021).
Barbu, Viorel, and Röckner, Michael. “Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations”. Journal of Functional Analysis 280.7 (2021): 108926.

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