Abstract
This introduction surveys a renormalisation group perspective on log-Sobolev inequalities and related properties of stochastic dynamics. We also explain the relationship of this approach to related recent and less recent developments such as Eldan’s stochastic localisation and the Föllmer process, the Boué–Dupuis variational formula and the Barashkov–Gubinelli approach, the transportation of measure perspective, and the classical analogues of these ideas for Hamilton–Jacobi equations which arise in mean-field limits.
Funding Statement
This work was supported by the European Research Council under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 851682 SPINRG).
Acknowledgments
We thank D. Chafai, R. Eldan, N. Gozlan, J. Lehec, Y. Shenfeld, and H.-T. Yau for various discussions related to the material of this introduction, encouragement, and for pointing out several additional references.
We thank the organisers of the following summer schools at which some of the material was presented: the One World Probability Summer School on “PDE and Randomness” in Bath/Zoom organised by Hendrik Weber and Andris Gerasimovics; the “Summer School on SPDE and Related Fields” in Beijing/Zoom organised by Hao Shen, Scott Smith, Rongchan Zhu, and Xiangchan Zhu; and the Summer School on “PDE and Randomness” at the Max Planck Institute for Mathematics in the Sciences organised by Rishabh Gvalani, Francesco Mattesini, Felix Otto, and Markus Tempelmayr. In particular, we also thank Jiwoon Park for leading the exercise classes at the last summer school.
Citation
Roland Bauerschmidt. Thierry Bodineau. Benoit Dagallier. "Stochastic dynamics and the Polchinski equation: An introduction." Probab. Surveys 21 200 - 290, 2024. https://doi.org/10.1214/24-PS27
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