By Feng-Yu Wang

ISBN-10: 9814452645

ISBN-13: 9789814452649

Stochastic research on Riemannian manifolds with no boundary has been good validated. despite the fact that, the research for reflecting diffusion techniques and sub-elliptic diffusion tactics is much from whole. This e-book includes fresh advances during this path in addition to 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 latest monographs even within the case with no boundary.

Readership: Graduate scholars, researchers and execs in likelihood idea, 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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**Additional info for Analysis for Diffusion Processes on Riemannian Manifolds : Advanced Series on Statistical Science and Applied Probability**

**Example text**

Let (E, B) be a measurable space and ρ a non-negative measurable function on E × E. 1) E×E is also called the Lp -transportation cost between probability measures µ and ν induced by the cost function ρ. g. [Chen (1992)]). 1) that any coupling provides an upper bound of the transportation cost, while the following Kontorovich dual formula enables one to find lower bound estimates. 1 (Monge-Kontorovich dual formula). For p ≥ 1, let Cp = {(f, g) : f, g ∈ Bb (E), f (x) ≤ g(y) + ρ(x, y)p , x, y ∈ E}.

7). 7) to fn = 1 + npx,y , we arrive at P {log px,y }(x) ≤ P (log fn )(x) − log n ≤ log P fn (y) − log n + Ψ(x, y) = log n+1 + Ψ(x, y). 9). (3) Let Π ∈ C(f µ, µ). t. Π, we obtain µ (P ∗ f ) log P ∗ f = P log P ∗ f (x)Π(dx, dy) E×E log P P ∗ f (y)Π(dx, dy) + Π(Ψ) ≤ E×E = µ(log P P ∗ f ) + Π(Ψ) ≤ log µ(P P ∗ f ) + Π(Ψ) = Π(Ψ), where in the last two steps we have used the Jensen inequality and that µ is P P ∗ -invariant. This completes the proof. 2 31 Shift Harnack inequality Let P (x, dy) be a transition probability on a Banach space E.

10) α(r) = 2r inf ξ −1 (s exp[1 − s/r]), s>0 s where ξ −1 (t) := inf{r > 0 : ξ(r) ≤ t}. 6) holds for C = 2/δ and all f ∈ D(L) with Ψ(f ) < ∞. 11) for some positive h ∈ C[0, ∞). 11) holds for h = 1 provided L is self-adjoint. 1(2) below). 16. 11) holds. 5) with Φ = Ψ and ξ −1 (r) h(s)ds, r > 0. 5 Equivalence of irreducibility and weak Poincar´ e inequality Let (E, F, µ) be a σ-finite measure space. A Dirichlet form (E, D(E)) on L2 (µ) is called non-conservative if either 1 ∈ / D(E) or E(1, 1) > 0, while it is called irreducible if f ∈ D(E) with E(f, f ) = 0 implies f = 0.

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

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