By Feng-Yu Wang

ISBN-10: 9814452645

ISBN-13: 9789814452649

Stochastic research on Riemannian manifolds with no boundary has been good tested. even though, the research for reflecting diffusion techniques and sub-elliptic diffusion strategies is way from entire. This booklet comprises contemporary advances during this path in addition to new rules and effective arguments, that are an important for additional advancements. Many effects contained the following (for instance, the formulation of the curvature utilizing derivatives of the semigroup) are new between current monographs even within the case with no boundary.

Readership: Graduate scholars, researchers and execs in chance idea, differential geometry and partial differential equations.

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**Extra info for Analysis for Diffusion Processes on Riemannian Manifolds : Advanced Series on Statistical Science and Applied Probability**

**Sample text**

The proof is fundamental. Since it is easy to see that the class C(µ, ν) is tight, for a sequence of couplings {Πn }n≥1 such that lim Πn (ρp ) = Wpρ (µ, ν)p , n→∞ there is a weakly convergent subsequence, whose weak limit gives an optimal coupling. As for the optimal map, let us simply mention a result of McCann [McCann (1995)] for E = Rd , see [Villani (2009a)] and references within for extensions and historical remarks. 3. Let E = Rd , ρ(x, y) = |x − y|, and p = 2. Then for any two absolutely continuous probability measures µ(dx) := f (x)dx and ν(dx) := g(x)dx such that f > 0, there exists a unique optimal map, which is given by T = ∇V for a convex function V solving the equation f = g(∇V )det∇ac ∇V in the distribution sense, where ∇ac is the gradient for the absolutely continuous part of a distribution.

Let r0 ≥ 0. Then the following statements are equivalent: (1) σess (−L) ⊂ [r0−1 , ∞). 1) holds. 1) holds for some β : (r0 , ∞) → (0, ∞). Recall that a linear operator on a Banach space is called compact, if it sends bounded sets into relatively compact sets. Let Pt = etL . It is well known that Pt is compact for some/all t > 0 if and only if σess (L) = ∅. 2. The following statements are equivalent to each other: (1) σess (L) = ∅. 1) holds for r0 = 0, some compact set B and some β : (0, ∞) → (0, ∞).

0 . Since µ0 is P -invariant, we have gP ∗ 1 dµ0 = g dµ0 , g ∈ Bb (E). e. e. x ∈ E the measure P ∗ (x, ·) is a probability measure. On the other hand, since µ1 is P -invariant, we have (P ∗ f )g dµ0 = E f P g dµ0 = E = P g dµ1 E f g dµ0 , g ∈ Bb (E). e. Therefore, P∗ E 1 dµ0 = f +1 E 1 dµ0 = f +1 1 E P ∗f +1 dµ0 . s. s. e. x. t. s. s. since f is a probability density function. 1) to f = n ∧ Φ−1 p(x, ·) p(y, ·) and letting n → ∞, we obtain the desired inequality. (5) Let rΦ−1 (r) be convex for r ≥ 0.

### Analysis for Diffusion Processes on Riemannian Manifolds : Advanced Series on Statistical Science and Applied Probability by Feng-Yu Wang

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