Semimartingales and geometric inequalities on locally symmetric manifolds

Abstract

We generalise, to complete, connected and locally symmetric Riemannian manifolds, the construction of coupled semimartingales X and Y given in Le and Barden (J Lond Math Soc 75:522–544, 2007). When such a manifold has non-negative curvature, this makes it possible for the stochastic anti-development of the corresponding semimartingale \({\rm exp}_{X_t} \big(\alpha\,{\rm exp}^{-1}_{X_t}(Y_t)\big)\) to be a time-changed Brownian motion with drift when X and Y are. As an application, we use the latter result to strengthen, and extend to locally symmetric spaces, the results of Le and Barden (J Lond Math Soc 75:522–544, 2007) concerning an inequality involving the solutions of the parabolic equation \(\frac{\partial\psi} {\partial t} = \frac{1}{2}\Delta\psi - h\,\psi\) with Dirichlet boundary condition and an inequality involving the first eigenvalues of the Laplacian, both on three related convex sets.