Most probable flows for Kunita SDEs

TL;DR

This paper identifies the most probable stochastic flows for Kunita SDEs, combining analytical and experimental methods to compare these flows with deterministic ones in fluid dynamics and shape analysis.

Contribution

It introduces a novel approach to determine the most probable flows for infinite-dimensional Kunita SDEs using a Riemannian metric derived from noise.

Findings

Most probable flows differ from deterministic flows under various noise structures.

The approach effectively characterizes the influence of noise in stochastic flow models.

Experimental results validate the analytical predictions.

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.