Characterizing Gaussian flows arising from It\=o's stochastic differential equations

TL;DR

This paper introduces and characterizes a class of Gaussian flows derived from Itô's stochastic differential equations, demonstrating their unique solutions to certain SPDEs and expanding understanding of their properties.

Contribution

It provides a new characterization of Gaussian flows from SDEs and establishes their connection to specific SPDE solutions, extending previous work.

Findings

Gaussian flows are characterized and shown to be solutions to SPDEs

The transpose of the flow is proven to be the unique solution under extended conditions

Utilizes the Monotonicity inequality to establish uniqueness

Abstract

We introduce and characterize a class of flows, which turn out to be Gaussian. This characterization allows us to show, using the Monotonicity inequality, that the transpose of the flow, for an extended class of initial conditions, is the unique solution of the SPDE introduced in Rajeev and Thangavelu (2008).