Analysis for Diffusion Processes on Riemannian Manifolds : by Feng-Yu Wang

By Feng-Yu Wang

Stochastic research on Riemannian manifolds with no boundary has been good tested. even though, the research for reflecting diffusion procedures and sub-elliptic diffusion approaches is much from entire. This publication comprises fresh 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 present monographs even within the case with out boundary.

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

Show description

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

Similar stochastic modeling books

Markov Chains and Stochastic Stability

Meyn and Tweedie is again! The bible on Markov chains commonly country areas has been cited so far to mirror advancements within the box due to the fact 1996 - a lot of them sparked by way of ebook of the 1st version. The pursuit of extra effective simulation algorithms for complicated Markovian types, or algorithms for computation of optimum guidelines for managed Markov versions, has opened new instructions for examine on Markov chains.

Selected Topics in Integral Geometry

The miracle of vital geometry is that it is usually attainable to get well a functionality on a manifold simply from the information of its integrals over yes submanifolds. The founding instance is the Radon rework, brought first and foremost of the 20 th century. in view that then, many different transforms have been discovered, and the overall conception was once constructed.

Uniform Central Limit Theorems

This vintage paintings on empirical methods has been significantly elevated and revised from the unique version. whilst samples turn into huge, the chance legislation of huge numbers and valuable restrict theorems are bound to carry uniformly over vast domain names. the writer, an stated professional, provides a radical remedy of the topic, together with the Fernique-Talagrand majorizing degree theorem for Gaussian techniques, a longer therapy of Vapnik-Chervonenkis combinatorics, the Ossiander L2 bracketing critical restrict theorem, the GinГ©-Zinn bootstrap significant restrict theorem in chance, the Bronstein theorem on approximation of convex units, and the Shor theorem on premiums of convergence over decrease layers.

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

Example text

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. 5. 10) for all x, e ∈ E := Rd and some non-negative measurable function CΦ on Rd × Rd . Then Φ(P f )(x) ≤ sup f ∈Bb+ (Rd ), Rd Φ◦f (x)dx≤1 1 , x ∈ Rd . t. the Lebesgue measure such that 1 p(x, y)Φ−1 (p(x, y))dy ≤ Φ−1 Rd Rd e−CΦ (x,e) de . 12) (2) If Φ(r) = rp for some p > 1, then 1 p p(x, y) p−1 dy ≤ ( Rd Proof.

For P (x, · − e)-null set A, let f = 1A . 10) implies that Φ(P (x, A)) ≤ 0; hence P (x, A) = 0 since Φ(r) > 0 for r > 0. t. P (x, · − e). Next, let Φ(r) = rΨ(r). 10) for f (y) = Ψ(n ∧ p(x, e; y)) and noting that Ψ(n ∧ p(x, e; y)) P (x, dy) P f (x) = E ≥ Φ(n ∧ p(x, e; y))P (x, dy − e), E we obtain Φ(n ∧ p(x, e; y))P (x, dy − e) ≤ Ψ−1 eCΦ (x,e) . E Then the proof is completed by letting n → ∞. 5 Harnack inequality and derivative estimate In this section, we consider the relationship between Harnack inequalities and derivative estimates of Markov operators on a geodesic space.

T. µ. t. the invariant probability measure is strictly positive. t. µ satisfies p(x, ·)Φ−1 E p(x, ·) dµ ≤ Φ−1 (eΨ(x,y) ), x, y ∈ E, p(y, ·) −1 where Φ (∞) := ∞ by convention. t. µ satisfies p(x, ·)p(y, ·)dµ ≥ e−Ψ(x,y) , x, y ∈ 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.

Download PDF sample

Rated 4.21 of 5 – based on 23 votes