Linearization of nonlinear Fokker-Planck equations and applications
文献类型：期刊论文
| 作者 | Ren, Panpan1,2; Roeckner, Michael3,4; Wang, Feng-Yu2,5 |
| 刊名 | JOURNAL OF DIFFERENTIAL EQUATIONS
 |
| 出版日期 | 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 |
| WOS记录号 | WOS:000792897400001 |
| 出版者 | ACADEMIC PRESS INC ELSEVIER SCIENCE |
| 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. |
入库方式： OAI收割
来源：数学与系统科学研究院
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