Some Path Properties of Iterated Brownian Motion

Burdzy, Krzysztof

doi:10.1007/978-1-4612-0339-1_3

Cited by 100 publications

(117 citation statements)

References 5 publications

Supporting

Mentioning

116

Contrasting

Order By: Relevance

“…While, for d ≥ 2, the solution P 1/2 shows a pole at r → 0, the integral of the probability for any finite yet arbitrarily small region is finite, thus having a meaningful interpretation as a probability density. The probability density (31) actually admits an interesting interpretation in terms of an iterated Brownian motion (IBM) [44][45][46][47]. 6 This process is defined by a normal Brownian motion evaluated "in random time."…”

Section: B Diffusion Employing Fractional Derivativesmentioning

confidence: 99%

Probing the quantum nature of spacetime by diffusion

2013

View full text Add to dashboard Cite

Many approaches to quantum gravity have resorted to diffusion processes to characterize the spectral properties of the resulting quantum spacetimes. We critically discuss these quantum-improved diffusion equations and point out that a crucial property, namely positivity of their solutions, is not preserved automatically. We then construct a novel set of diffusion equations with positive semidefinite probability densities, applicable to asymptotically safe gravity, Hořava-Lifshitz gravity and loop quantum gravity. These recover all previous results on the spectral dimension and shed further light on the structure of the quantum spacetimes by assessing the underlying stochastic processes. Pointing out that manifestly different diffusion processes lead to the same spectral dimension, we propose the probability distribution of the diffusion process as a refined probe of quantum spacetime.

show abstract

Section: B Diffusion Employing Fractional Derivativesmentioning

confidence: 99%

Probing the quantum nature of spacetime by diffusion

2013

View full text Add to dashboard Cite

show abstract

“…Since the landmark paper of Burdzy [3] on iterated Brownian motion (IBM) the investigation of various IBM type processes became increasingly popular. To name a few: Arcones [1], Khoshnevisan and Lewis [13,14], Hu and Shi [11], Khoshnevisan et al [15], Shi [17,18].…”

Section: Introductionmentioning

confidence: 99%

Strassen theorems for a class of iterated processes

Csáki

Földes²,

Révész

1997

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Iterated Brownian motions arise naturally in a variety of problems in probability and mathematical statistics. For example, Funaki [F] used a modification of Z to give a probabilistic solution to the partial differential equation Burdzy [Bu1,Bu2] established the following local law of the iterated logarithm:…”

Section: Introductionmentioning

confidence: 99%

“…What is surprising is that known general techniques, e.g., metric entropy and majorizing measures, are not refined enough to prove the upper bound in Theorem 1. Indeed, some estimates of Burdzy [Bu1] can be used to show that there exists K > 0, such that for all t, ε > 0 and x > 1,…”

Section: Introductionmentioning

confidence: 99%