Complex Solutions to Bessel SDEs and SLEs

TL;DR

This paper studies complex-valued Bessel SDEs related to SLE traces, establishing existence, uniqueness, and continuity of solutions, and connecting these flows to SLE for certain parameters.

Contribution

It introduces and analyzes complex Bessel SDEs, proving strong solution existence, uniqueness, and continuity, and links these solutions to SLE processes for specific parameters.

Findings

Existence and uniqueness of strong solutions for complex Bessel SDEs with negative dimension.

Almost sure continuity of the associated stochastic flow.

Connection established between these flows and SLE$_{ ext{κ}}$ for κ<4.

Abstract

We consider a variant of Bessel SDE by allowing the solution to be complex valued. Such SDEs appear naturally while studying the trace of Schramm-Loewner-Evolutions (SLE). We establish the existence and uniqueness of the strong solution to such SDEs when the dimension is negative. We also consider the stochastic flow associated to such SDEs and prove that it is almost surely continuous. Our proofs are based on an improvement of the derivative estimate of Rohde-Schramm \cite{RS05}. We finally show the connection between such stochastic flows and SLE$_{\kappa}$ for $\kappa <4$.