Probability distance inequalities on Riemannian manifolds and path spaces. (Q1425155)

Consider a Riemannian manifold \(M\), endowed with a probability measure \(\mu:= e^{V(x)}dx\), and a smooth field \(A\) of strictly positive definite, symmetric operators on \(TM\). The induced distance is \[ \rho_A(x,x'):= \sup\{| f(x)- f(x')|:f\in C^1(M),\langle A\nabla f,\nabla f\rangle\leq 1\}. \] Assume that the metric space \((M,\rho_A)\) is complete and that the different process associated with \(L:= \text{div}(A\nabla)+ A\nabla V\) does not explode. Fix \(p\in [1,2]\), and consider the \(L^p\)-Wasserstein distance: \[ W_p(\mu_1,\mu_2):= \inf_{\pi} \Biggl\{\int_{M^2} \rho_A(x, x')^p \pi(dx, dx')\Biggr\}^{1/p} \] between two probability measures on \(M\), where \(\pi\) denotes a probability measure on \(M^2\) having \(\mu_1\), \(\mu_2\) as marginal laws. Suppose that the following functional inequality (which interpolates between Poincaré, for \(p= 1\), and log-Sobolev, for \(p=2\), inequalities) holds on \((M,\mu)\): \[ {\mu(f^2)- \mu(f^p)^{2/p}\over 2-p}\leq C\mu(\langle A\nabla f,\nabla f\rangle),\quad\text{for (smooth) }f\geq 0. \] The main result is the following transportation cost inequality \[ W_p((P_t f)\mu,\mu)\leq pe^{-t/C}\times \sqrt{C\times{\mu(f^{2/p})- 1\over 2-p}},\quad\text{for }f\geq 0,\;\mu(f)= 1,\;t\geq 0 \] (for \(p= 2\): \(W_2((P_t f)\mu,\mu)\leq e^{-t/C}\times \sqrt{2\mu(f\log f)})\). Moreover, this result remains valid on the path space over a compact manifold, establishing in particular \((p=2)\) a Talagrand's inequality in this framework. Some concentration inequalities are deduced. As to the proof, it uses and improves the coupling method introduced by Otto and Villani, based on the solution \(\phi_s\) to: \(\phi_0= \text{Id}\), \({d\over ds}\phi= -(A\nabla\log P_{t+s}f)\circ\phi_s\), the coupling being \(\pi_s(dx, dx'):= P_t f(x)\mu(dx)\times \delta_{\phi_s(x)}(dx')\).