By Feng-Yu Wang

Stochastic research on Riemannian manifolds with out boundary has been good validated. besides the fact that, the research for reflecting diffusion techniques and sub-elliptic diffusion methods is way from entire. This e-book comprises fresh advances during this course besides new rules and effective arguments, that are the most important for extra advancements. Many effects contained right here (for instance, the formulation of the curvature utilizing derivatives of the semigroup) are new between present monographs even within the case with no boundary.

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

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August 1, 2013 34 18:21 World Scientific Book - 9in x 6in ws-book9x6 Analysis for Diffusion Processes on Riemannian Manifolds (1) For any strictly positive f ∈ Bb (E), |∇P f | ≤ δ P (f log f ) − (P f ) log P f + β(δ, ·)P f, δ > δ0 . (2) For any p > 1 and x, y ∈ E such that ρ(x, y) ≤ positive f ∈ Bb (E), p−1 pδ0 , and for any (P f )p (x) ≤ P f p (y) 1 exp 0 p−1 pρ(x, y) β , γ(s) ds , 1 + (p − 1)s ρ(x, y){1 + (p − 1)s} where γ : [0, 1] → E is a minimal geodesic from x to y with speed ρ(x, y). Proof.

The cost function ρ. e. ν = µ◦ −1 ), such that Π(dx, dy) := µ(dx)δx (dy) is an optimal coupling, where δx is the Dirac measure at x, then is called an optimal transportation map for the Lp -transportation cost. To fix (or estimate) the transportation cost, it is crucial to construct the optimal coupling or optimal map. Below we introduce some results on existence and construction of the optimal coupling/map. 2. Let (E, ρ) be a Polish space. Then for any µ, ν ∈ P(E) and any p ∈ [1, ∞), there exists an optimal coupling.

Now, we apply the above results to Dirichlet forms on H := L2 (µ) for a σ-finite complete measure space (E, B, µ). Let (E, D(E)) be a symmetric Dirichlet form in L2 (µ). 1) for B = Bφ := {g : |g| ≤ φ}, where φ > 0 is a fixed function in L2 (µ). In this case f B∗ = sup |µ(gf )| = µ(φ|f |). |g|≤φ In particular, if µ is finite we may take φ = 1 such that µ(φ|f |) = f The following result is taken from [Wang (2002a)]. 1. 3. Let r0 ≥ 0. e. there exists β : (r0 , ∞) → (0, ∞) such that µ(f 2 ) ≤ rE(f, f ) + β(r)µ(φ|f |)2 , r > r0 , f ∈ D(E).