Time reversal of diffusion processes under a finite entropy condition

By Patrick Cattiaux, Giovanni Conforti, Ivan Gentil, Christian Léonard. Annales De L Institut Henri Poincare-probabilites Et Statistiques, 2021

82 citations recorded by the source corpus.

Motivated by entropic optimal transport, time reversal of diffusion processes is revisited. An integration by parts formula is derived for the carr\'e du champ of a Markov process in an abstract space. It leads to a time reversal formula for a wide class of diffusion processes in $ \mathbb{R}^n$ possibly with singular drifts, extending the already known results in this domain. The proof of the integration by parts formula relies on stochastic derivatives. Then, this formula is applied to compute the semimartingale characteristics of the time-reversed $P^*$ of a diffusion measure $P$ provided that the relative entropy of $P$ with respect to another diffusion measure $R$ is finite, and the semimartingale characteristics of the time-reversed $R^*$ are known (for instance when the reference path measure $R$ is reversible). As an illustration of the robustness of this method, the integration by parts formula is also employed to derive a time-reversal formula for a random walk on a graph.

Subfields: Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/af1cabab06c085e5d14480196ec64d9580e8c67d

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Citations in the corpus

Riemannian Score-Based Generative Modeling (2022, methodology): “The extrinsic approach is particularly convenient since the SDE appearing in Proposition S12 can be seen as an SDE on the Euclidean space R p .”
Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling (2021, methodology): “Under conditions on f , it is well-known (see Haussmann and Pardoux (1986); Föllmer (1985); Cattiaux et al. (2021) for instance) that the reverse-time process with initialization Y 0 ∼ p T , where p t denotes the marginal density of X t .”