Sample Path Properties of the Local Times of Strongly Symmetric Markov Processes Via Gaussian Processes

Marcus, Michael B.; Rosen, Jay

doi:10.1214/aop/1176989524

Cited by 81 publications

(69 citation statements)

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“…A major problem in this field is to decide whether the mapping (r, t) → L(r, t) has a continuous version. This has been solved in a remarkable paper by Barlow [4], see also [3] and Marcus and Rosen [114] in the symmetric case. Proposition 8.1 provides a simple expression for the Laplace exponent Φ of the inverse local time, which is explicit in terms of the characteristic exponent Ψ.…”

Section: Local Time At a Fixed Pointmentioning

confidence: 93%

Subordinators: Examples and Applications

Bertoin

1999

Lecture Notes in Mathematics

237

294

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Section: Local Time At a Fixed Pointmentioning

confidence: 93%

Subordinators: Examples and Applications

Bertoin

1999

Lecture Notes in Mathematics

237

294

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“…The distribution of the local times for a Borel right process can be fully characterized by certain associated Gaussian processes; results of this flavor go by the name of Isomorphism Theorems. Several versions have been developed by Ray [44] and Knight [33], Dynkin [18,17], Marcus and Rosen [40,41], Eisenbaum [19] and Eisenbaum, Kaspi, Marcus, Rosen and Shi [20]. In what follows, we present the second Ray-Knight theorem in the special case of a continuous-time random walk.…”

Section: Lemma 113 (Matthews Bound) For Everymentioning

confidence: 99%

Cover times, blanket times, and majorizing measures

Ding

Lee

Peres

2011

Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing

115

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We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand's theory of majorizing measures. In particular, we show that the cover time of any graph G is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on G, scaled by the number of edges in G.This allows us to resolve a number of open questions. We give a deterministic polynomialtime algorithm that computes the cover time to within an O(1) factor for any graph, answering a question of Aldous and Fill (1994). We also positively resolve the blanket time conjectures of Winkler and Zuckerman (1996), showing that for any graph, the blanket and cover times are within an O(1) factor. The best previous approximation factor for both these problems was O((log log n)2 ) for n-vertex graphs, due to Kahn, Kim, Lovász, and Vu (2000).

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“…If there is a modification of the local time such that it is continuous in (x, t), we say that X has a jointly continuous local time. Necessary and suffcient conditions for the joint continuity of the local times of Lévy processes have been proved by Barlow and Hawkes [2], Barlow [1], and by Marcus and Rosen [19] using different methods.…”

Section: Proof For the Lower Boundmentioning

confidence: 99%

Uniform Dimension Results for the Inverse Images of Symmetric Lévy Processes

2019

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We prove uniform Hausdorff and packing dimension results for the inverse images of a large class of real-valued symmetric Lévy processes. Our main result for the Hausdorff dimension extends that of Kaufman (1985) for Brownian motion and that of Song, Xiao, and Yang (2018) for α-stable Lévy processes with 1 < α < 2. Along the way we also prove an upper bound for the uniform modulus of continuity of the local times of these processes.

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Sample Path Properties of the Local Times of Strongly Symmetric Markov Processes Via Gaussian Processes

Cited by 81 publications

References 0 publications

Subordinators: Examples and Applications

Subordinators: Examples and Applications

Cover times, blanket times, and majorizing measures

Uniform Dimension Results for the Inverse Images of Symmetric Lévy Processes

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