The DOI matches its Crossref record.
The Variational Formulation of the Fokker--Planck Equation
Open accessAbstract
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.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
No references could be resolved yet.
Research works printed but not linked: 24
- P. Billingsley, Probability and measure , John Wiley, New York, 1986.
- 10.1002/cpa.3160440402 · DOI · Paper page
- Luis Caffarelli, Allocation maps with general cost functions, Lecture Notes in Pure and Appl. Math., Vol. 177, Dekker, New York, 1996, 29–3597f:49055
- 10.1103/RevModPhys.15.1 · DOI · Paper page
- 10.1007/BF01448839 · DOI · Paper page
- 10.1137/S0036141094261847 · doi:10.1137/s0036141094261847
- 10.1007/978-3-662-02452-2 · DOI · Paper page
- Gangbo Wilfrid, 1995, C. R. Acad. Sci. Paris Sér. I Math., 321, 1653
- 10.1007/BF02392620 · DOI
- 10.1307/mmj/1029003026 · DOI · Paper page
- 10.1088/0951-7715/8/4/007 · doi:10.1088/0951-7715/8/4/007
- Richard Jordan, David Kinderlehrer, An extended variational principle, Lecture Notes in Pure and Appl. Math., Vol. 177, Dekker, New York, 1996, 187–20096m:49079
- 10.1016/S0167-2789(97)00093-6 · DOI
- R. Jordan, D. Kinderlehrer, and F. Otto, The route to stability through the Fokker–Planck dynamics , Proceedings of the First U.S.–China Conference on Differential Equations and Applications, to appear.
- 10.1007/BF02181242 · doi:10.1007/bf02181242
- 10.1137/0523001 · doi:10.1137/0523001
- 10.1016/S0031-8914(40)90098-2 · DOI · Paper page
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and quasi–linear equations of parabolic type , American Mathematical Society, Providence, RI, 1968. · DOI · Paper page
- 10.1007/s002050050073 · doi:10.1007/s002050050073
- 10.1006/jdeq.1996.0155 · doi:10.1006/jdeq.1996.0155
- Rachev Svetlozar, 1991, Probability metrics and the stability of stochastic models
- H. Risken, The Fokker‐Planck equation: Methods of solution and applications , 2nd ed., Springer‐Verlag, Berlin, Heidelberg, 1989.
- 10.1137/1022024 · doi:10.1137/1022024
- Roy Choudhury S., 1995, J. Egyptian Math. Soc., 3, 49
Cited by
Denoising Diffusion Implicit Models
Song, J., Meng, C., & Ermon, S. (2020) · arXiv · doi:10.48550/arxiv.2010.02502 · Cited by 2,507
Schiebinger, G., Shu, J., Tabaka, M., Cleary, B., Subramanian, V., Solomon, A., Gould, J., Liu, S., Lin, S., Berube, P., et al. (2019) · Cell · doi:10.1016/j.cell.2019.01.006 · Cited by 829
Scaling algorithms for unbalanced optimal transport problems
Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F.-X. (2018) · Mathematics of Computation · doi:10.1090/mcom/3303 · Cited by 336
Wasserstein Adversarial Examples via Projected Sinkhorn Iterations
Wong, E., Schmidt, F. R., & Kolter, J. Z. (2019) · arXiv · doi:10.48550/arxiv.1902.07906 · Cited by 145
“Proximal operators with respect to Wasserstein instead of Euclidean distances are used in gradient flow problems (Jordan et al., 1998; Peyré, 2015), but are not relevant to our setting.”
MethodOptimal transport for single-cell and spatial omics
Bunne, C., Schiebinger, G., Krause, A., Regev, A., & Cuturi, M. (2024) · Nature Reviews Methods Primers · doi:10.1038/s43586-024-00334-2 · Cited by 71
Metadata
- Authors
- Jordan, R., Kinderlehrer, D., & Otto, F.
- Venue
- SIAM Journal on Mathematical Analysis
- Published
- 1998-01
- Volume
- 29
- Issue
- 1
- Pages
- 1-17
- ISSN
- 0036-1410 (ISSN-L)
- 1095-7154
- OpenAlex
- W2071048859
- Cited by
- 1,603
- References
- 0
- FWCI
- 11.40
- Type
- journal article
- Language
- en
- Open access
- green
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: The Variational Formulation of the Fokker--Planck Equation. https://app.cochat.ai/papers/10.1137/s0036141096303359
- The paper
- Jordan, R., Kinderlehrer, D., & Otto, F. (1998). The Variational Formulation of the Fokker--Planck Equation. SIAM Journal on Mathematical Analysis. https://doi.org/10.1137/s0036141096303359
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
