By Feng-Yu Wang

Stochastic research on Riemannian manifolds with no boundary has been good proven. notwithstanding, the research for reflecting diffusion procedures and sub-elliptic diffusion tactics is way from entire. This publication comprises contemporary advances during this course in addition to new principles and effective arguments, that are the most important for extra 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 out boundary.

Readership: Graduate scholars, researchers and pros in likelihood conception, differential geometry and partial differential equations.

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

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F |(x) = lim sup y→x |f (x) − f (y)| . ρ(x, y) Obviously, |∇f | ≥ 0 and |f (x) − f (y)| ≤ ρ(x, y) ∇f operator on Bb (E). 1 ∞. 1. Let δ0 ≥ 0 and β ∈ C((δ0 , ∞) × E; [0, ∞)). The following two statements are equivalent. 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).

Pµ (X ∈ A) = 0 or 1 holds for µ ∈ P(E) and A ∈ T . (4) For any µ, ν ∈ P(E), Pµ = Pν holds on T . A weaker notion than the coupling property is the shift-coupling property. 6. The strong Markov process is said to have the shift coupling property, if for any x, y ∈ E there is a coupling (X, Y ) starting at (x, y) such that XT1 = YT2 holds for some finite stopping times T1 and T2 . Let I := A ∈ F(W ) : w ∈ A implies w(t + ·) ∈ A, t ≥ 0 be the shift-invariant σ-field. Below are some equivalent statements for the shift-coupling property (see [Aldous and Thorisson (1993); Cranston and Greven (1995); Thorisson (1994)]).

Therefore, for any x, y ∈ E and any coupling (X, Y ) starting at (x, y) with coupling time Tx,y , we have |Pt f (x) − Pt f (y)| = |E(f (Xt ) − f (Y˜t ))| ≤ osc(f )P(Tx,y > t), f ∈ Bb (E), where osc(f ) := sup f − inf f. This implies the following assertions, which are fundamentally crucial for applications of coupling in the study of Markov processes. e. Pt Bb (E) ⊂ Cb (E). (ii) Let µ be an invariant probability measure. If the coupling time Tx,y is measurable in (x, y), then νPt − µ var ≤2 P(Tx,y > t)Π(dx, dy), Π ∈ C(µ, ν) E×E holds for ν ∈ P(E).

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