Variation of iterated Brownian motion

Burdzy, Krzysztof

doi:10.1090/crmp/005/03

Cited by 48 publications

(32 citation statements)

References 8 publications

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“…An example of particular importance for us is iterated Brownian motion (IBM) [86][87][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102][103][104]. To illustrate iterated Brownian motion, we set α = 1 and consider the ordinary D-dimensional higher-order operator ∇ n , Eq.…”

Section: E Iterated Brownian Motion (β = 1 γ = 2 S = 0)mentioning

confidence: 99%

Diffusion in multiscale spacetimes

Calcagni

2013

Phys. Rev. E

140

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We study diffusion processes in anomalous spacetimes regarded as models of quantum geometry. Several types of diffusion equation and their solutions are presented and the associated stochastic processes are identified. These results are partly based on the literature in probability and percolation theory but their physical interpretation here is different since they apply to quantum spacetime itself. The case of multiscale (in particular, multifractal) spacetimes is then considered through a number of examples, and the most general spectral-dimension profile of multifractional spaces is constructed.

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Section: E Iterated Brownian Motion (β = 1 γ = 2 S = 0)mentioning

confidence: 99%

Diffusion in multiscale spacetimes

Calcagni

2013

Phys. Rev. E

140

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“…Wlth «°>*) = Mx)-Deheuvels and Mason [11], Burdzy ([6], [7]), Arcones [1], Hu and Shi [20], Shi [29], Hu, Pierre-Loti-Viaud, and Shi [21], Csaki; Csorgo, Foldes, and Revesz [9], Khoshnevisan and Lewis ( [22], [23]) and Csaki, Foldes and Revesz [10] have studied different properties of the iterated Brownian mo tion. We will prove that for each 0 < M < oo,…”

Section: The Process {B(b(t))\ T E R} Is Called An Iterated Brownian mentioning

confidence: 99%

The large deviation principle for stochastic processes. I

Arcones¹

2002

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“…The first class of processes uses Brownian motion as a subordinator [1,2,18,19,20,21]. The second class uses the inverse or hitting time process of a nondecreasing stable Lévy process as a subordinator [38,39,40,12,41].…”

Section: Introductionmentioning

confidence: 99%

Brownian subordinators and fractional Cauchy problems

Baeumer

Meerschaert

Nane

2009

Trans. Amer. Math. Soc.

102

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Abstract. A Brownian time process is a Markov process subordinated to the absolute value of an independent one-dimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involves subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. In this paper, we will show a close and unexpected connection between these two classes of processes and, consequently, an equivalence between these two families of partial differential equations.

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Variation of iterated Brownian motion

Cited by 48 publications

References 8 publications

Diffusion in multiscale spacetimes

Diffusion in multiscale spacetimes

The large deviation principle for stochastic processes. I

Brownian subordinators and fractional Cauchy problems

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