Estimates on the Transition Densities of Girsanov Transforms of Symmetric Stable Processes

Song, Renming

doi:10.1007/s10959-006-0023-4

Cited by 6 publications

(14 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since the killing measure of X is zero and 1 + F (x, y) is bounded from below and above by positive constants, we conclude that X is conservative. By [17,Theorem 2.7], X has continuous transition densitiesp(t, x, y), t > 0, x, y ∈ R d .…”

Section: Isotropic Stable Lévy Processesmentioning

confidence: 99%

“…Since We note that F ∈ I(C, β) if, and only, if F is symmetric and |F (x, y)| C(|x−y| β ∧1) for some constant C > 0. It is proved in [17,Example p. 492] that I(C, β) ⊂ I 2 (X) if β > α/2. Similarly, we have the following simple result.…”

Section: Isotropic Stable Lévy Processesmentioning

confidence: 99%

“…We will write X = ( X t , M, M t , P x ) to denote this process. The process X is called a purely discontinuous Girsanov transform of X, see [4,17] and Section 2 for further details. Since L F t > 0, we have d P x|M t ∼ dP x|M t for all t 0.…”

Section: Introductionmentioning

confidence: 99%

“…[4,17]). (a) The class J(X) consists of all bounded, symmetric functions F : R d × R d → R which vanish on the diagonal and satisfy lim (X s− , y)|N(X s− , dy) dH s = 0.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Absolute continuity and singularity of probability measures induced by a purely discontinuous Girsanov transform of a stable process

Schilling

Vondraček

2016

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

In this paper we study mutual absolute continuity and singularity of probability measures on the path space which are induced by an isotropic stable Lévy process and the purely discontinuous Girsanov transform of this process. We also look at the problem of finiteness of the relative entropy of these measures. An important tool in the paper is the question under which circumstances the a.s. finiteness of an additive functional at infinity implies the finiteness of its expected value.2010 Mathematics Subject Classification: Primary 60J55; Secondary 60G52, 60J45, 60H10.

show abstract

Section: Isotropic Stable Lévy Processesmentioning

confidence: 99%

Section: Isotropic Stable Lévy Processesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…[4,17]). (a) The class J(X) consists of all bounded, symmetric functions F : R d × R d → R which vanish on the diagonal and satisfy lim (X s− , y)|N(X s− , dy) dH s = 0.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Absolute continuity and singularity of probability measures induced by a purely discontinuous Girsanov transform of a stable process

Schilling

Vondraček

2016

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Let

{true(- \overset{Δ}{true} true)}^{α / 2}

be the generator of

\overset{X}{true}

. Then

{true(- \overset{Δ}{true} true)}^{α / 2}

is formally given by

\begin{matrix} true \\ right {true(- \overset{Δ}{true} true)}^{α / 2} = {false(- normalΔ false)}^{α / 2} + d boldF - boldF 1 . \end{matrix}

It is known that a PCAF of X can be regarded as a PCAF of

\overset{X}{true}

([, Lemma 2.2]). Thus we see from [, Lemma 4.9] and [, Lemma 3.2] that

\begin{matrix} true \\ right U_{μ + F} (x) & left = 1 - E_{x} ([], e^{- A_{\infty}^{μ} - \sum_{t > 0} F false(X_{t -}, X_{t} false)}) \\ left = 1 - {\tilde{boldE}}_{x} ([], e^{- A_{\infty}^{μ} - \int_{0}^{\infty} F 1 false(X_{t} false) d t}) \\ left = {\tilde{boldE}}_{x} ([] \int_{0}^{\infty} e^{- A_{t}^{μ} - \int_{0}^{t} F 1 false(X_{s} false) d s} (d A_{t}^{μ} + F 1 (X_{t}) d t)) . \end{matrix}

Equation implies that

…”

Section: Scattering Lengthsmentioning

confidence: 99%

On a scattering length for additive functionals and spectrum of fractional Laplacian with a non‐local perturbation

Kim

Matsuura

2019

Mathematische Nachrichten

View full text Add to dashboard Cite

In this paper we study the scattering length for positive additive functionals of symmetric stable processes on ℝ . The additive functionals considered here are not necessarily continuous. We prove that the semi-classical limit of the scattering length equals the capacity of the support of a certain measure potential, thus extend previous results for the case of positive continuous additive functionals. We also give an equivalent criterion for the fractional Laplacian with a measure valued non-local operator as a perturbation to have purely discrete spectrum in terms of the scattering length, by considering the connection between scattering length and the bottom of the spectrum of Schrödinger operator in our settings. K E Y W O R D Sadditive functionals, Dirichlet forms, scattering lengths, Schrödinger operators M S C ( 2 0 1 0 ) Primary: 47D08, 60J45; Secondary: 31C15, 60G52, 60J55 www.mn-journal.org

show abstract

Weighted Poincaré Inequalities for Non-local Dirichlet Forms

Chen

Wang

2015

J Theor Probab

View full text Add to dashboard Cite

Let V be a locally bounded measurable function such that e −V is bounded and belongs to L 1 (dx), and let µ V (dx) := C V e −V (x) dx be a probability measure. We present the criterion for the weighted Poincaré inequality of the non-local Dirichlet formTaking ρ(r) = e −δr r −(d+α) with 0 < α < 2 and δ 0, we get some conclusions for general fractional Dirichlet forms, which can be regarded as a complement of our recent work [13], and an improvement of the main result in [8]. In this especial setting, concentration of measure for the standard Poincaré inequality is also derived.Our technique is based on the Lyapunov conditions for the associated truncated Dirichlet form, and it is considerably efficient for the weighted Poincaré inequality of the following non-local Dirichlet form, which is associated with symmetric Markov processes under Girsanov transform of pure jump type.

show abstract

Estimates on the Transition Densities of Girsanov Transforms of Symmetric Stable Processes

Abstract: In this paper, we first study a purely discontinuous Girsanov transform which is more general than that studied in Chen and Song [(2003)

Cited by 6 publications

References 14 publications

Absolute continuity and singularity of probability measures induced by a purely discontinuous Girsanov transform of a stable process

Absolute continuity and singularity of probability measures induced by a purely discontinuous Girsanov transform of a stable process

On a scattering length for additive functionals and spectrum of fractional Laplacian with a non‐local perturbation

Weighted Poincaré Inequalities for Non-local Dirichlet Forms

Contact Info

Product

Resources

About