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Ornstein-Uhlenbeck Type Processes on Wasserstein Space

Panpan Ren, Feng-Yu Wang

Abstract

Let $\mathcal P_2$ be the space of probability measures on $\R^d$ having finite second moment, and consider the Riemannian structure on $\mathcal P_2$ induced by the intrinsic derivative on the $L^2$-tangent space. By using stochastic analysis on the tangent space, we construct an Ornstein$-$Uhlenbeck (OU) type Dirichlet form on $\mathcal P_2$ whose generator is formally given by the intrinsic Laplacian with a drift. The log-Sobolev inequality holds and the associated Markov semigroup is $L^2$-compact. Perturbations of the OU Dirichlet form are also studied.

Ornstein-Uhlenbeck Type Processes on Wasserstein Space

Abstract

Let be the space of probability measures on having finite second moment, and consider the Riemannian structure on induced by the intrinsic derivative on the -tangent space. By using stochastic analysis on the tangent space, we construct an OrnsteinUhlenbeck (OU) type Dirichlet form on whose generator is formally given by the intrinsic Laplacian with a drift. The log-Sobolev inequality holds and the associated Markov semigroup is -compact. Perturbations of the OU Dirichlet form are also studied.
Paper Structure (13 sections, 8 theorems, 123 equations)

This paper contains 13 sections, 8 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Intrinsic Riemannian structure on the Wasserstein space
  3. Intrinsic derivative
  4. Intrinsic Laplacian
  5. Ornstein$-$Uhlenbeck process on Wasserstein space
  6. OU process on tangent space
  7. OU process on $\mathscr P_2$
  8. Generator as Laplacian with drift
  9. Remark 4.1.
  10. Perturbation of the OU process
  11. Acknowledgement.
  12. Conflict of interest statement.
  13. Data availability statement.

Key Result

Proposition 2.1

Let $f\in C^1(\mathscr P_2)$ such that then L holds, i.e. $f$ is $L$-differentiable.

Theorems & Definitions (16)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Proposition 3.1
  • Theorem 3.2
  • proof
  • ...and 6 more