Linearization of nonlinear Fokker-Planck equations and applications

Ren P, Röckner M, Wang F-Y (2022)
Journal of Differential Equations 322: 1-37.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Ren, Panpan; Röckner, MichaelUniBi; Wang, Feng-Yu
Abstract / Bemerkung
Let 9 be the space of probability measures on Rd. We associate a coupled nonlinear Fokker-Planck equation on Rd, i.e. with solution paths in 9, to a linear Fokker-Planck equation for probability measures on the product space Rd x 9, i.e. with solution paths in 9(Rd x 9). We explicitly determine the corresponding linear Kolmogorov operator L tilde t using the natural tangent bundle over 9 with corresponding gradient operator backward difference 9. Then it is proved that the diffusion process generated by L tilde t on Rd x 9 is intrinsically related to the solution of a McKean-Vlasov stochastic differential equation (SDE). We also characterize the ergodicity of the diffusion process generated by L tilde t in terms of asymptotic properties of the coupled nonlinear Fokker-Planck equation. Another main result of the paper is that the restricted well-posedness of the non-linear Fokker-Planck equation and its linearized version imply the (restricted) well-posedness of the McKean-Vlasov equation and that in this case the laws of the solutions have the Markov property. All this is done under merely measurability conditions on the coefficients in their measure dependence, hence in particular applies if the latter is of "Nemytskii-type". As a consequence, we obtain the restricted weak well-posedness and the Markov property of the so-called nonlinear distorted Brownian motion, whose associated nonlinear Fokker-Planck equation is a porous media equation perturbed by a nonlinear transport term. This realizes a programme put forward by McKean in his seminal paper of 1966 for a large class of nonlinear PDEs. As a further application we obtain a probabilistic representation of solutions to Schrodinger type PDEs on Rd x .92, through the Feynman-Kac formula for the corresponding diffusion processes. (c) 2022 Elsevier Inc. All rights reserved.
Stichworte
Nonlinear Fokker-Planck equation; McKean-Vlasov stochastic differential; equation; Diffusion process; Ergodicity; Feynman-Kac formula
Erscheinungsjahr
2022
Zeitschriftentitel
Journal of Differential Equations
Band
322
Seite(n)
1-37
ISSN
0022-0396
eISSN
1090-2732
Page URI
https://pub.uni-bielefeld.de/record/2963459

Zitieren

Ren P, Röckner M, Wang F-Y. Linearization of nonlinear Fokker-Planck equations and applications. Journal of Differential Equations . 2022;322:1-37.
Ren, P., Röckner, M., & Wang, F. - Y. (2022). Linearization of nonlinear Fokker-Planck equations and applications. Journal of Differential Equations , 322, 1-37. https://doi.org/10.1016/j.jde.2022.03.021
Ren, P., Röckner, M., and Wang, F. - Y. (2022). Linearization of nonlinear Fokker-Planck equations and applications. Journal of Differential Equations 322, 1-37.
Ren, P., Röckner, M., & Wang, F.-Y., 2022. Linearization of nonlinear Fokker-Planck equations and applications. Journal of Differential Equations , 322, p 1-37.
P. Ren, M. Röckner, and F.-Y. Wang, “Linearization of nonlinear Fokker-Planck equations and applications”, Journal of Differential Equations , vol. 322, 2022, pp. 1-37.
Ren, P., Röckner, M., Wang, F.-Y.: Linearization of nonlinear Fokker-Planck equations and applications. Journal of Differential Equations . 322, 1-37 (2022).
Ren, Panpan, Röckner, Michael, and Wang, Feng-Yu. “Linearization of nonlinear Fokker-Planck equations and applications”. Journal of Differential Equations 322 (2022): 1-37.

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