Entropic Measure and Wasserstein Diffusion

By Max-K. von Renesse, Karl‐Theodor Sturm. 2007

120 citations recorded by the source corpus.

We construct a new random probability measure on the sphere and on the unit interval which in both cases has a Gibbs structure with the relative entropy functional as Hamiltonian. It satisfies a quasi-invariance formula with respect to the action of smooth diffeomorphism of the sphere and the interval respectively. The associated integration by parts formula is used to construct two classes of diffusion processes on probability measures (on the sphere or the unit interval) by Dirichlet form methods. The first one is closely related to Malliavin's Brownian motion on the homeomorphism group. The second one is a probability valued stochastic perturbation of the heat flow, whose intrinsic metric is the quadratic Wasserstein distance. It may be regarded as the canonical diffusion process on the Wasserstein space.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/9a2dd7e45f425148478fd9ffc524f05bd54e5d4f

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Citations in the corpus

Particle approximation of the Wasserstein diffusion (2007, methodology): “The actual construction of (μt)t≥0 in [18] uses abstract Dirichlet form methods without direct reference to physical intuition.”
{Euclidean, metric, and Wasserstein} gradient flows: an overview (2016, background): “It has been found (see [79]) that the condition of λ-geodesic convexity of the Entropy functional → ́ log( )dVol (where is absolutely continuous w.”
Bismut formula for Lions derivative of distribution dependent SDEs and applications (2018, methodology): “This directional derivative has been used in earlier references, see for instance [21] for the Wasserstein di usions constructed using the directional derivative on P2(S (1)), where S(1) is the unit circle.”
On the inverse implication of Brenier-Mccann theorems and the structure of (P 2 (M),W 2 ) (2011, methodology): “the entropic measure P built by vonRenesse-Sturm for any β > 0 in [30].”
Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, background): “5, the infinite-dimensional space in consideration is P(Rd), and even if one were to replace R by a compact subset, restrict the attention to the subspace of probability measures with finite second moments and equip the latter with the 2-Wasserstein distance, the associated metric measure spaces (se”