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Most probable flows for Kunita SDEs

Erlend Grong, Stefan Sommer

Abstract

We identify most probable flows for Kunita Brownian motions, i.e. stochastic flows with Eulerian noise and deterministic drifts. Such stochastic processes appear for example in fluid dynamics and shape analysis modelling coarse scale deterministic dynamics together with fine-grained noise. We treat this infinite dimensional problem by equipping the underlying domain with a Riemannian metric originating from the noise. The resulting most probable flows are compared with the non-perturbed deterministic flow, both analytically and experimentally by integrating the equations with various choice of noise structures.

Most probable flows for Kunita SDEs

Abstract

We identify most probable flows for Kunita Brownian motions, i.e. stochastic flows with Eulerian noise and deterministic drifts. Such stochastic processes appear for example in fluid dynamics and shape analysis modelling coarse scale deterministic dynamics together with fine-grained noise. We treat this infinite dimensional problem by equipping the underlying domain with a Riemannian metric originating from the noise. The resulting most probable flows are compared with the non-perturbed deterministic flow, both analytically and experimentally by integrating the equations with various choice of noise structures.
Paper Structure (14 sections, 4 theorems, 40 equations, 1 figure)

This paper contains 14 sections, 4 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Outline
  4. Stochastic flows and measures on path space
  5. Stochastic flows
  6. Path probability and Onsager-Machlup functionals
  7. Most probable flows
  8. Local coordinate representations for numerical implementation
  9. Perturbed right-invariant geodesic flows
  10. Reduction and EPDiff equations
  11. Stochastic Euler-Poincaré flows
  12. Energy minimizing flows
  13. Most probable flows
  14. Visualization of perturbed flows

Key Result

Theorem 3.1

Let $\nabla^t$ be the Levi-Civita connection of $g_t$ and let $z_t$ be the vector field in Loperator. With slight abuse of notation, we define adjoint operators $\dot g_t^*, (\nabla^t \xi)^* : TM \to TM$ by for vector fields $\xi$, $\xi_1$, $\xi_2$. Then $\varphi_t$ is a most probable flow if and only if it solves the equation with $w_t$ satisfying

Figures (1)

  • Figure 1: Red curves: deterministic flow (zero noise) with $u_t$ solving \ref{['OptU']}; blue curves: MPP trajectories; green curves: stochastic trajectories. Left column: flow field $u_t$ at $t=0$ with noise centers (green points); center left column: forward integration of MPP equations for 40 landmarks evenly distributed at horizontal $y=-0.5$ line; center right column: initial value problem solved for each landmark between the end points of the deterministic landmark trajectories; right column: stochastic EPDiff sample path (noise amplitude downscaled for visualization). Top row: a single noise field at $(0,0)$; middle row: two noise fields at $(-0.5,0)$ and $(0.5,0)$; bottom row: grid of $7^2$ noise fields.

Theorems & Definitions (10)

  • Theorem 3.1: Most probable flows
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Example 3.5: Brownian background noise
  • Theorem 4.1: holmVariationalPrinciplesStochastic2015arnaudonGeometricFrameworkStochastic2019
  • Theorem 4.2
  • proof
  • Remark 4.3