A class of processes on the path space over a compact Riemannian manifold with unbounded diffusion

Löbus, Jörg-Uwe

doi:10.1090/s0002-9947-04-03439-7

Cited by 11 publications

(15 citation statements)

References 13 publications

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“…Since (A1) and (A3) are verified in Example 2.3, the damped Dirichlet form is also quasi-regular. We provide the following example such that A(γ) may be an unbounded operator, which can be viewed a generalization of that in [22] and [29].…”

Section: Functional Inequalitiesmentioning

confidence: 99%

“…we will show that for every l ∈ C ∞ 0 (R), l(ρ) ∈ D(E ), where C ∞ 0 (R) denotes the set of smooth functions on R with compact supports. Based on such property, we can construct more general Dirichlet forms with diffusion coefficients, which can be viewed as a generalization of those in [22] and [29]. In fact, let…”

Section: Introductionmentioning

confidence: 99%

“…The quasi-regularity of the O-U Dirichlet form on path space over a compact manifold was first shown in [9]. And the quasi-regularity of a class of Dirichlet forms with constant diffusion coefficients was established in [22]. We also want to remark that if in condition (A2) above, we replace the set F C b,loc by F C b , and the constant ε(R) is independent of R > 0 in (A3) , then the quasi-regularity of such Dirichlet form (E A , D(E A )) was shown in [29].…”

Section: Introductionmentioning

confidence: 99%

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Functional inequality on path space over a non-compact Riemannian manifold

Chen

2014

Journal of Functional Analysis

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We prove the existence of the O-U Dirichlet form and the damped O-U Dirichlet form on path space over a general non-compact Riemannian manifold which is complete and stochastically complete. We show a weighted log-Sobolev inequality for the O-U Dirichlet form and the (standard) log-Sobolev inequality for the damped O-U Dirichlet form. In particular, the Poincaré inequality (and the super Poincaré inequality) can be established for the O-U Dirichlet form on path space over a class of Riemannian manifolds with unbounded Ricci curvatures. Moreover, we construct a large class of quasi-regular local Dirichlet forms with unbounded random diffusion coefficients on path space over a general non-compact manifold.Suppose M is a n-dimensional non-compact complete connected Riemannian manifold, the Riemannian path space C o,T (M) over M is defined byC o,T (M) := {γ ∈ C([0, T ]; M) : γ(0) = o}, where T is a positive constant and o ∈ M. Let d M be the Riemannian distance on M, then C o,T (M) is a Polish space under the uniform distance d(γ, σ) := sup t∈[0,T ]d M (γ(t), σ(t)), γ, σ ∈ C o,T (M).

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Section: Functional Inequalitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Functional inequality on path space over a non-compact Riemannian manifold

Chen

2014

Journal of Functional Analysis

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“…Let

0 < λ_{1} \leq λ_{2} \leq ...

be a sequence of real numbers satisfying

\sum_{m = 0}^{\infty} 2^{- m} λ_{d 2^{m}} < \infty .

This paper is concerned with an Ornstein–Uhlenbeck type Dirichlet form

(E, D (E))

obtained by the closure of the positive symmetric bilinear form

E (F, G) = \int \sum_{i = 1}^{\infty} λ_{i} {⟨ D F, S_{i} ⟩}_{double-struckH} {⟨ D G, S_{i} ⟩}_{double-struckH} d ν, F, G \in Y,

L^{2} (ν)

. The Dirichlet form

(E, D (E))

is a particular case of the set of Dirichlet forms associated with a certain class of infinite dimensional processes with unbounded diffusion introduced and studied in , , and . In particular, it has been shown in , Proposition 4.2, that condition is necessary and sufficient for closability of

(E, Y)

L^{2} (ν)

.…”

Section: Introductionmentioning

confidence: 99%

“…We wish to investigate a fairly accessible representative of the relatively abstract class of infinite dimensional stochastic processes with unbounded diffusion introduced in [15], [24], [4], and [13]. Existence and representation of standard elements of the stochastic calculus such as quadratic variation and Itô formula may convince that these processes fit in the general concept of infinite dimensional stochastic processes.…”

Section: Introductionmentioning

confidence: 99%

Infinite dimensional Ornstein–Uhlenbeck processes with unbounded diffusion – Approximation, quadratic variation, and Itô formula

Karlsson

Löbus

2016

Mathematische Nachrichten

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The paper studies a class of Ornstein–Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron–Martin space. It is shown that the distributions of certain finite dimensional Ornstein–Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein–Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related Itô formula is presented.

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