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Splitting for some classes of homeomorphic and coalescing stochastic flows

M. B. Vovchanskyi

Abstract

The splitting scheme (the Kato-Trotter formula) is applied to stochastic flows with common noise of the type introduced by Th.E.~Harris. The case of possibly coalescing flows with continuous infinitesimal covariance is considered and the weak convergence of the corresponding finite-dimensional motions is established. As applications, results for the convergence of the associated pushforward measures and dual flows are given. Similarities between splitting and the Euler-Maruyama scheme yield estimates of the speed of the convergence under additional regularity assumptions.

Splitting for some classes of homeomorphic and coalescing stochastic flows

Abstract

The splitting scheme (the Kato-Trotter formula) is applied to stochastic flows with common noise of the type introduced by Th.E.~Harris. The case of possibly coalescing flows with continuous infinitesimal covariance is considered and the weak convergence of the corresponding finite-dimensional motions is established. As applications, results for the convergence of the associated pushforward measures and dual flows are given. Similarities between splitting and the Euler-Maruyama scheme yield estimates of the speed of the convergence under additional regularity assumptions.
Paper Structure (10 sections, 33 theorems, 201 equations)

This paper contains 10 sections, 33 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Existence of Harris flows with non-trivial drift
  3. Harris flows as solutions to SDEs
  4. The splitting scheme and the example of the Brownian web
  5. Weak convergence
  6. Convergence of pushforward measures
  7. Convergence of dual flows
  8. Estimates for flows with Holder continuous $\sqrt{1-\varphi}$
  9. Existence of Harris flows with drift
  10. Proofs of Propositions \ref{['prop:2.1.1']} and \ref{['prop:2.1.2']}

Key Result

Theorem 2.1

Suppose $\varphi\in \Phi_*$ and $a$ is measurable and of linear growth. Then the Harris flow $X^{\varphi, a}$ exists and is unique in distribution.

Theorems & Definitions (57)

  • Remark 1.1
  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.1
  • ...and 47 more