By Feng-Yu Wang

ISBN-10: 9814452645

ISBN-13: 9789814452649

Stochastic research on Riemannian manifolds with out boundary has been good verified. notwithstanding, the research for reflecting diffusion tactics and sub-elliptic diffusion procedures is way from whole. This e-book comprises contemporary advances during this course besides new principles and effective arguments, that are an important for extra advancements. Many effects contained right here (for instance, the formulation of the curvature utilizing derivatives of the semigroup) are new between latest monographs even within the case with out boundary.

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

**Read Online or Download Analysis for Diffusion Processes on Riemannian Manifolds : Advanced Series on Statistical Science and Applied Probability PDF**

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

**Sample text**

This completes the proof. 2 31 Shift Harnack inequality Let P (x, dy) be a transition probability on a Banach space E. Let f (y)P (x, dy), f ∈ Bb (Rd ) P f (x) = Rd be the associated Markov operator. Let Φ : [0, ∞) → [0, ∞) be a strictly increasing and convex continuous function. 10) for some x, e ∈ E and constant CΦ (x, e) ≥ 0. Obviously, if Φ(r) = rp for some p > 1 then this inequality reduces to the shift Harnack inequality with power p, while when Φ(r) = er it becomes the shift log-Harnack inequality.

2. Let E be a Banach space. Let e ∈ E, δe ∈ (0, 1) and βe ∈ C((δe , ∞) × E; [0, ∞)). Then the following assertions are equivalent. (1) For any positive f ∈ Cb1 (E), |P (∇e f )| ≤ δ P (f log f ) − (P f ) log P f + βe (δ, ·)P f, δ ≥ δe . (2) For any positive f ∈ Bb (E), r ∈ (0, δ1e ) and p ≥ 1 1−rδe , (P f )p ≤ P {f p (re + ·)} 1 exp 0 pr p−1 βe , · + sre ds . 1 + (p − 1)s r + r(p − 1)s Proof. 1. To prove (1) from (2), we let z, e ∈ E be fixed and assume that P (∇e f )(z) ≥ 0 (otherwise, simply use −e to replace August 1, 2013 18:21 World Scientific Book - 9in x 6in 36 ws-book9x6 Analysis for Diffusion Processes on Riemannian Manifolds e).

E (6) If µ is an invariant probability measure of P , then sup f ∈Bb+ (E),µ(Φ(f ))≤1 P f (x) ≤ 1 , x ∈ E. −Ψ(x,y) µ(dy) e E Proof. Since (6) is obvious, below we prove (1)-(5) respectively. (1) Let f ∈ Bb (E) be positive. 1) to 1 + εf in place of f for ε > 0, we have Φ(1 + εP f (x)) ≤ {P Φ(1 + εf )(y)}eΨ(x,y) , x, y ∈ E, ε > 0. 2) for small ε > 0. Letting y → x we obtain εP f (x) ≤ ε lim inf P f (y) + o(ε). y→x Thus, P f (x) ≤ lim inf y→x P f (y) holds for all x ∈ E. Similarly, changing the roles of x and y we obtain P f (y) ≥ lim supx→y P f (x) for any y ∈ E.

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

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