Exponential asymptotics for intersection local times of stable processes and random walks

Chen, Xia; Rosen, Jay

doi:10.1016/j.anihpb.2004.09.006

Cited by 16 publications

(26 citation statements)

References 12 publications

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“…Our second result is another application of Proposition , which is an extension of [, Theorem 1] in some sense. We will prove this in Section 5.…”

Section: Introduction and Main Resultsmentioning

confidence: 83%

“…When we need to clarify constants, we denote the above assumptions as

(boldA bold3; ρ, t_{0}, C)

(boldA bold4; μ, t_{0}, C)

(boldA; ρ, μ, t_{0}, C)

and

({boldA}^{'}; ρ, μ, t_{0}, C)

. Remark i)When E is compact, clearly (A2) and (A2′) are equivalent. ii)If

m (E) < \infty

, the ultra‐contractivity (A4) implies the tightness (A1); see for example. iii)In the proof of [, Theorem7], Chen and Rosen used a stronger condition than (A2′), namely the global Lipschitz continuity of

p_{t} false(\cdot, \cdot false)

. iv)[, Theorem 3.1] says that the ultra‐contractivity (A4) implies the existence of the bounded and continuous density on

E ∖ N

, where N is a properly exceptional set. …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The asymptotics of the logarithmic moment generating function

\lim_{t \to \infty} \frac{1}{t} \log E [\exp {θ ℓ_{t}^{normalIS} {(E)}^{1 / p}}] for θ > 0

is calculated in for the case of independent Brownian motions, and in for the case of independent stable processes. In these papers, they point out that the limit in is related to the best constants of the Gagliardo–Nirenberg type interpolation inequalities.…”

Section: Introductionmentioning

confidence: 99%

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Large deviations for intersection measures of some Markov processes

Mori

2019

Mathematische Nachrichten

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Consider an intersection measure ℓtIS of p independent (possibly different) m‐symmetric Hunt processes up to time t in a metric measure space E with a Radon measure m. We derive a Donsker–Varadhan type large deviation principle for the normalized intersection measure t−pℓtnormalIS on the set of finite measures on E as t→∞, under the condition that t is smaller than life times of all processes. This extends earlier work by W. König and C. Mukherjee [16], in which the large deviation principle was established for the intersection measure of p independent N‐dimensional Brownian motions before exiting some bounded open set D⊂double-struckRN. We also obtain the asymptotic behavior of logarithmic moment generating function, which is related to the results of X. Chen and J. Rosen [7] on the intersection measure of independent Brownian motions or stable processes. Our results rely on assumptions about the heat kernels and the 1‐order resolvents of the processes, hence include rich examples. For example, the assumptions hold for p∈Z with 2≤p

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“…Our second result is another application of Proposition , which is an extension of [, Theorem 1] in some sense. We will prove this in Section 5.…”

Section: Introduction and Main Resultsmentioning

confidence: 83%

“…When we need to clarify constants, we denote the above assumptions as

(boldA bold3; ρ, t_{0}, C)

(boldA bold4; μ, t_{0}, C)

(boldA; ρ, μ, t_{0}, C)

and

({boldA}^{'}; ρ, μ, t_{0}, C)

. Remark i)When E is compact, clearly (A2) and (A2′) are equivalent. ii)If

m (E) < \infty

p_{t} false(\cdot, \cdot false)

. iv)[, Theorem 3.1] says that the ultra‐contractivity (A4) implies the existence of the bounded and continuous density on

E ∖ N

, where N is a properly exceptional set. …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The asymptotics of the logarithmic moment generating function

\lim_{t \to \infty} \frac{1}{t} \log E [\exp {θ ℓ_{t}^{normalIS} {(E)}^{1 / p}}] for θ > 0

Section: Introductionmentioning

confidence: 99%

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Large deviations for intersection measures of some Markov processes

Mori

2019

Mathematische Nachrichten

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show abstract

“…It is proved in [16] that these constants are not degenerate when d < αq. We have then the following result.…”

Section: Introductionmentioning

confidence: 99%

“…In d = 1, they worked directly on the self-intersections of the limiting stable process, and used the existence and the regularity of their local times, thus reducing their result to α > d. In d ≥ 2, they first proved large deviations results for intersections of p-independent processes or random walks using a regularisation procedure in [13,16], and then transfered these results to large deviations for self-intersections. Here the assumption p = 2 is crucial since this tranfer is done via the bisection method introduced by Varadhan in [31], which does not work for p = 2.…”

Section: Introductionmentioning

confidence: 99%