Fokker-Planck equation on a manifold. Effective diffusion and spectrum (Q1904294)

Fix a \(C^\infty\) \(n\)-dimensional compact Riemannian manifold \(M\) and a \(C^\infty\) function \(U\) on it; consider the diffusion \((\theta_t, V_t)\) on \(O(M) \times \mathbb{R}^n\) determined by the s.d.e.: \[ \begin{cases} dV_t & = - \biggl( \gamma V_t + \theta^{-1}_t \cdot \nabla U \bigl( \pi (\theta_t) \bigr) \biggr) dt + \sigma dW_t, \\ d \theta_t & = L_{V_t} (\theta_t) dt. \end{cases} \] Here \(L\) denotes the horizontal vector field, \(\pi\) is the projection on \(M\), \(\gamma\) and \(\sigma\) are positive constants, and \(W_t\) is the Brownian motion in \(\mathbb{R}^n\). Let \(X_t\) be the element of the tangent bundle \(TM\) having coordinates \(V_t\) in the frame \(\theta_t\). This is the kind of generalized Ornstein-Uhlenbeck process (with external potential \(U)\) considered in this article. The author carefully establishes for this homogeneous Feller process \(X\) living on \(TM\) the classical properties of the diffusions living on \(M\): he calculates its generator \(A\) and its invariant probability measure, he shows that the convergence to equilibrium holds, at exponential rate, and that the spectrum of \(A\) is discrete; and finally he establishes a central limit theorem for the integral along the \(\pi (X)\)-paths of any regular closed differential 1-form on \(M\).