Stochastic Modeling

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

By Feng-Yu Wang

Stochastic research on Riemannian manifolds with no boundary has been good verified. even if, the research for reflecting diffusion techniques and sub-elliptic diffusion procedures is much from whole. This ebook includes contemporary advances during this path besides new rules and effective arguments, that are an important for additional advancements. Many effects contained the following (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 likelihood conception, differential geometry and partial differential equations.

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

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

Example text

2) holds for all z, z ∈ E, f ≥ 0, f ∈ Bb (E). Proof. 2). Let γ : [0, 1] → E be a minimal geodesic such that γ(0) = z, γ(1) = z . 1), for any positive f ∈ Bb (E) and constant r > 0, we have d f P (γ(s)) ds 1 + rsf ≤ −rP ≤ f2 (γs ) + Cρ(z, z ) (1 + rsf )2 P f 1 + rsf C 2 ρ(z, z )2 . 4r So, P C 2 ρ(z, z )2 f . (z ) ≤ P f (z) + 1 + rf 4r Combining this with the fact that f rf 2 =f− ≥ f − rf 2 , 1 + rf 1 + rf 2 (γs ) August 1, 2013 18:21 World Scientific Book - 9in x 6in Preliminaries ws-book9x6 37 we obtain C 2 ρ(z, z )2 + rP f 2 (z ).

12) holds for some α if and only if lim t→∞ f µ((Pt f )2 ) = 0. sup 1∨ f ∞ ≤1 So, the second assertion follows from the first one. (b) Let f ∈ D(E) with E(f, f ) = 0. For any ε > 0 let fε = (|f | − ε)+ ∧ 1. We have E(fε , fε ) = 0 and by the Schwarz inequality fε 2 1 ≤ µ(f 2 )µ(|f | > ε) ≤ µ(f 2 )2 . 12) to fε we obtain µ(fε2 ) ≤ r(1+ε−2 µ(|f |)2 ) for all r > 0. This implies fε = 0 for all ε > 0 and thus, f = 0. 12) holds for some function α : (0, ∞) → (0, ∞). Otherwise, there exist some r > 0 and a sequence {fn } ⊂ D(E) such that 1 = µ(fn2 ) > nE(fn , fn ) + r( fn 1 ∨ fn 2 ∞) , n ≥ 1.

4) August 1, 2013 18:21 28 World Scientific Book - 9in x 6in ws-book9x6 Analysis for Diffusion Processes on Riemannian Manifolds where c > 0 is a constant. e. for any x = y and any s ∈ (0, 1), there exists a sequence {zn } ⊂ E such that ρ(x, zn ) → sρ(x, y) and ρ(zn , y) → (1 − s)ρ(x, y) as n → ∞. 2. Assume that (E, ρ) is a length space and let α1 , α2 > 1 be two constants. 4) holds for α = α1 , α2 , it holds also for α = α1 α2 . Proof. Let α1 (α2 − 1) α1 − 1 , or equivalently, 1 − s = , α1 α2 − 1 α1 α2 − 1 and let {zn } ⊂ E such that ρ(x, zn ) → sρ(x, y) and ρ(zn , y) → (1−s)ρ(x, y) as n → ∞.

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