Linearization of nonlinear Fokker-Planck equations and applications

doi:10.1016/j.jde.2022.03.021

	Linearization of nonlinear Fokker-Planck equations and applications
	Ren, Panpan 1,2; Roeckner, Michael 3,4; Wang, Feng-Yu 2,5
刊名	JOURNAL OF DIFFERENTIAL EQUATIONS
	2022-06-15
卷号	322 页码:1-37
关键词	Nonlinear Fokker-Planck equation McKean-Vlasov stochastic differential equation Diffusion process Ergodicity Feynman-Kac formula
ISSN号	0022-0396
DOI	10.1016/j.jde.2022.03.021
英文摘要	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.
WOS研究方向	Mathematics
语种	英语
出版者	ACADEMIC PRESS INC ELSEVIER SCIENCE
WOS记录号	WOS:000792897400001
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/61282]
专题	中国科学院数学与系统科学研究院
通讯作者	Roeckner, Michael
作者单位	1.Univ Oxford, Math Inst, Woodstock Rd, Oxford OX2 6GG, England 2.Swansea Univ, Dept Math, Bay Campus, Swansea SA1 8EN, W Glam, Wales 3.Bielefeld Univ, Dept Math, D-33501 Bielefeld, Germany 4.Chinese Acad Sci, Acad Math & Syst Sci, Beijing, Peoples R China 5.Tianjin Univ, Ctr Appl Math, Tianjin 300072, Peoples R China
推荐引用方式 GB/T 7714	Ren, Panpan,Roeckner, Michael,Wang, Feng-Yu. Linearization of nonlinear Fokker-Planck equations and applications[J]. JOURNAL OF DIFFERENTIAL EQUATIONS,2022,322:1-37.
APA	Ren, Panpan,Roeckner, Michael,&Wang, Feng-Yu.(2022).Linearization of nonlinear Fokker-Planck equations and applications.JOURNAL OF DIFFERENTIAL EQUATIONS,322,1-37.
MLA	Ren, Panpan,et al."Linearization of nonlinear Fokker-Planck equations and applications".JOURNAL OF DIFFERENTIAL EQUATIONS 322(2022):1-37.