The regularity of the linear drift in negatively curved spaces

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.

The introduction has been rewritten, Proposition 4.11 has been modified and Section 4.3 and 4.5 are simplified accordingly