The Regularity of the Linear Drift in Negatively Curved Spaces

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