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We first provide a stochastic formula for the Carathéodory distance in terms of general Markovian couplings and prove a comparison result between the Carathéodory distance and the complete Kähler metric with a negative lower curvature bound using the Kendall–Cranston coupling. This probabilistic approach gives a version of the Schwarz lemma on complete noncompact Kähler manifolds with a further decomposition Ricci curvature into the orthogonal Ricci curvature and the holomorphic sectional curvature, which cannot be obtained by using Yau–Royden's Schwarz lemma. We also prove coupling estimates on quaternionic Kähler manifolds. As a by‐product, we obtain an improved gradient estimate of positive harmonic functions on Kähler manifolds and quaternionic Kähler manifolds under lower curvature bounds.

 

We consider sub-Laplacians in open bounded sets in a homogeneous Carnot group and study their spectral properties. We prove that these operators have a pure point spectrum and prove the existence of the spectral gap. In addition, we give applications to the small ball problem for a hypoelliptic Brownian motion and the large time behavior of the heat content in a regular domain.

 

A small ball problem and Chung’s law of iterated logarithm for a hypoelliptic Brownian motion in Heisenberg group are proven. In addition, bounds on the limit in Chung’s law are established. 

Abstract Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential U allowing for singularities. By modifying the direct approach to convergence in L 2 pioneered by Hérau and developed by Dolbeault et al , we show that the dynamics converges exponentially fast to equilibrium in the topologies L 2 (d μ ) and L 2 ( W * d μ ), where μ denotes the invariant probability measure and W * is a suitable Lyapunov weight. In both norms, we make precise how the exponential convergence rate depends on the friction parameter γ in Langevin dynamics, by providing a lower bound scaling as min( γ , γ −1 ). The results hold for usual polynomial-type potentials as well as potentials with singularities such as those arising from pairwise Lennard-Jones interactions between particles.