The regularity of the linear drift in negatively curved spaces

TL;DR

This paper investigates the smoothness of the linear drift and stochastic entropy of Brownian motion on negatively curved manifolds, revealing their differentiability properties and critical points at symmetric metrics.

Contribution

It establishes the differentiability of linear drift and entropy along metric curves and formulates their derivatives, highlighting criticality at symmetric metrics.

Findings

Linear drift is $C^{k-2}$ differentiable along $C^k$ metric curves.

Stochastic entropy is $C^1$ differentiable along $C^3$ metric curves.

Derivatives of drift and entropy are critical at symmetric metrics.

Abstract

We show the linear drift of the Brownian motion on the universal cover of a closed connected Riemannian manifold is $C^{k-2}$ differentiable along any $C^{k}$ curve in the manifold of $C^k$ metrics with negative sectional curvature. We also show that the stochastic entropy of the Brownian motion is $C^1$ differentiable along any $C^{3}$ curve of $C^3$ metrics with negative sectional curvature. We formulate the first derivatives of the linear drift and entropy, respectively, and show they are critical at locally symmetric metrics.