【240710-5】FPK方程变分原理及自由能泛函_1998 The Variational Formulation of the Fokker–Planck Equation. Jordan, R., Kinderlehrer, D., & Otto, F. SIAM Journal on Mathematical Analysis, 29(1):1–17, January, 1998. ![【240710-5】FPK方程变分原理及自由能泛函_1998 The Variational Formulation of the Fokker–Planck Equation [link]](https://bibbase.org/img/filetypes/link.svg "link")
Paper doi abstract bibtex The Fokker–Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of time-dependent systems in which randomness plays a role. In this paper, we are concerned with Fokker–Planck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, we construct a time discrete, iterative variational scheme whose solutions converge to the solution of the Fokker–Planck equation. The major novelty of this iterative scheme is that the time-step is governed by the Wasserstein metric on probability measures. This formulation enables us to reveal an appealing, and previously unexplored, relationship between the Fokker–Planck equation and the associated free energy functional. Namely, we demonstrate that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy with respect to the Wasserstein metric.
@article{jordan_240710-5fpk_1998_1998,
title = {【240710-5】{FPK方程变分原理及自由能泛函}\_1998 {The} {Variational} {Formulation} of the {Fokker}--{Planck} {Equation}},
volume = {29},
issn = {0036-1410, 1095-7154},
url = {http://epubs.siam.org/doi/10.1137/S0036141096303359},
doi = {10.1137/S0036141096303359},
abstract = {The Fokker–Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of time-dependent systems in which randomness plays a role. In this paper, we are concerned with Fokker–Planck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, we construct a time discrete, iterative variational scheme whose solutions converge to the solution of the Fokker–Planck equation. The major novelty of this iterative scheme is that the time-step is governed by the Wasserstein metric on probability measures. This formulation enables us to reveal an appealing, and previously unexplored, relationship between the Fokker–Planck equation and the associated free energy functional. Namely, we demonstrate that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy with respect to the Wasserstein metric.},
language = {en},
number = {1},
urldate = {2024-12-10},
journal = {SIAM Journal on Mathematical Analysis},
author = {Jordan, Richard and Kinderlehrer, David and Otto, Felix},
month = jan,
year = {1998},
keywords = {/unread},
pages = {1--17},
}
Downloads: 0
{"_id":"5Kgfw7J2KeyAhg5PE","bibbaseid":"jordan-kinderlehrer-otto-2407105fpk1998thevariationalformulationofthefokkerplanckequation-1998","author_short":["Jordan, R.","Kinderlehrer, D.","Otto, F."],"bibdata":{"bibtype":"article","type":"article","title":"【240710-5】FPK方程变分原理及自由能泛函_1998 The Variational Formulation of the Fokker–Planck Equation","volume":"29","issn":"0036-1410, 1095-7154","url":"http://epubs.siam.org/doi/10.1137/S0036141096303359","doi":"10.1137/S0036141096303359","abstract":"The Fokker–Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of time-dependent systems in which randomness plays a role. In this paper, we are concerned with Fokker–Planck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, we construct a time discrete, iterative variational scheme whose solutions converge to the solution of the Fokker–Planck equation. The major novelty of this iterative scheme is that the time-step is governed by the Wasserstein metric on probability measures. This formulation enables us to reveal an appealing, and previously unexplored, relationship between the Fokker–Planck equation and the associated free energy functional. Namely, we demonstrate that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy with respect to the Wasserstein metric.","language":"en","number":"1","urldate":"2024-12-10","journal":"SIAM Journal on Mathematical Analysis","author":[{"propositions":[],"lastnames":["Jordan"],"firstnames":["Richard"],"suffixes":[]},{"propositions":[],"lastnames":["Kinderlehrer"],"firstnames":["David"],"suffixes":[]},{"propositions":[],"lastnames":["Otto"],"firstnames":["Felix"],"suffixes":[]}],"month":"January","year":"1998","keywords":"/unread","pages":"1–17","bibtex":"@article{jordan_240710-5fpk_1998_1998,\n\ttitle = {【240710-5】{FPK方程变分原理及自由能泛函}\\_1998 {The} {Variational} {Formulation} of the {Fokker}--{Planck} {Equation}},\n\tvolume = {29},\n\tissn = {0036-1410, 1095-7154},\n\turl = {http://epubs.siam.org/doi/10.1137/S0036141096303359},\n\tdoi = {10.1137/S0036141096303359},\n\tabstract = {The Fokker–Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of time-dependent systems in which randomness plays a role. In this paper, we are concerned with Fokker–Planck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, we construct a time discrete, iterative variational scheme whose solutions converge to the solution of the Fokker–Planck equation. The major novelty of this iterative scheme is that the time-step is governed by the Wasserstein metric on probability measures. This formulation enables us to reveal an appealing, and previously unexplored, relationship between the Fokker–Planck equation and the associated free energy functional. Namely, we demonstrate that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy with respect to the Wasserstein metric.},\n\tlanguage = {en},\n\tnumber = {1},\n\turldate = {2024-12-10},\n\tjournal = {SIAM Journal on Mathematical Analysis},\n\tauthor = {Jordan, Richard and Kinderlehrer, David and Otto, Felix},\n\tmonth = jan,\n\tyear = {1998},\n\tkeywords = {/unread},\n\tpages = {1--17},\n}\n\n\n\n","author_short":["Jordan, R.","Kinderlehrer, D.","Otto, F."],"key":"jordan_240710-5fpk_1998_1998","id":"jordan_240710-5fpk_1998_1998","bibbaseid":"jordan-kinderlehrer-otto-2407105fpk1998thevariationalformulationofthefokkerplanckequation-1998","role":"author","urls":{"Paper":"http://epubs.siam.org/doi/10.1137/S0036141096303359"},"keyword":["/unread"],"metadata":{"authorlinks":{}},"html":""},"bibtype":"article","biburl":"https://bibbase.org/zotero/Cantabile1228","dataSources":["svSjxFw6DzYd5XZdH"],"keywords":["/unread"],"search_terms":["240710","fpk","1998","variational","formulation","fokker","planck","equation","jordan","kinderlehrer","otto"],"title":"【240710-5】FPK方程变分原理及自由能泛函_1998 The Variational Formulation of the Fokker–Planck Equation","year":1998}