6533b871fe1ef96bd12d0b91
RESEARCH PRODUCT
Parallel translations, Newton flows and Q-Wiener processes on the Wasserstein space
Hao Dingsubject
Newton's methodÉquation de Dean-KawasakiParallel translationTransport optimalTransport parallèleTransport parallèle stochastiqueDean-Kawasaki equationDistance de WassersteinOptimal transportStochastic parallel translation[MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM]Wasserstein distanceMéthode de Newtondescription
- We extend the definition of Lott’s Levi-Civita connection to the Wasserstein space of probability measures having density and divergence. We give an extension of a vector field defined along an absolutely curve onto the whole space so that parallel translations can be introduced as done in differential geometry. In the case of torus, we prove the well-posedness of Lott’s equation for parallel translations.- We prove the well-posedness of the Newton flow equation on the Wasserstein space and show the connections between the relaxed Newton flow equation and the Keller-Segel equation.- We establish an intrinsic formalism for Itô stochastic calculus on the Wasserstein space throughout three kinds of functionals. We construct the weak and strong form of stochastic partial differential equations for stochastic parallel translations, the well-posedness is also proved in the case of torus. Non-degenerated diffusion process are constructed using the eigenfunctions of the Laplacian.- We construct a new interactive particle model approximation to the solution to the regularized martingale problem of the diffusive Dean-Kawasaki equation on the one-dimensional torus under a weaker condition on the spatial correlation intensity of the noise than the classical one.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2022-01-01 |