Lifetime asymptotics of iterated Brownian motion in $\mathbb{R}^{n}$

2010

Stoch. Dyn.

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Abstract. We study solutions of a class of higher order partial differential equations in bounded domains. These partial differential equations appeared first time in the papers of Allouba and Zheng [4], Baeumer, Meerschaert and Nane [10], Meerschaert, Nane and Vellaisamy [37], and Nane [42]. We express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index 0 < β < 1, or by the absolute value of a symmetric α-stable process with 0 < α ≤ 2, independent of the Markov process. In some special cases we represent the solutions by running composition of k independent Brownian motions, called k-iterated Brownian motion for an integer k ≥ 2. We make use of a connection between fractional-time diffusions and higher order partial differential equations established first by Allouba and Zheng [4] and later extended in several directions by Baeumer, Meerschaert and Nane [10].

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

STOCHASTIC SOLUTIONS OF A CLASS OF HIGHER ORDER CAUCHY PROBLEMS IN ℝ^d

2010

Stoch. Dyn.

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“…Burdzy [13,14] introduced the so-called iterated Brownian motion (IBM), A(t) = B 1 (B(t)), by replacing the time parameter in B 1 (t) by an independent one-dimensional Brownian motion B = {B(t), t ≥ 0}. His work inspired many researchers to explore the connections between IBM (or other iterated processes) and PDEs, to establish potential theoretical results and to study its sample path properties, see [1,31,32,34,35,36,40] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Laws of the iterated logarithm for a class of iterated processes

Statistics & Probability Letters

2009

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Let $X=\{X(t), t\geq 0\}$ be a Brownian motion or a spectrally negative stable process of index $1<\a<2$. Let $E=\{E(t),t\geq 0\}$ be the hitting time of a stable subordinator of index $0<\beta<1$ independent of $X$. We use a connection between $X(E(t))$ and the stable subordinator of index $\beta/\a$ to derive information on the path behavior of $X(E_t)$. This is an extension of the connection of iterated Brownian motion and (1/4)-stable subordinator due to Bertoin \cite{bertoin}. Using this connection, we obtain various laws of the iterated logarithm for $X(E(t))$. In particular, we establish law of the iterated logarithm for local time Brownian motion, $X(L(t))$, where $X$ is a Brownian motion (the case $\a=2$) and $L(t)$ is the local time at zero of a stable process $Y$ of index $1<\gamma\leq 2$ independent of $X$. In this case $E(\rho t)=L(t)$ with $\beta=1-1/\gamma$ for some constant $\rho>0$. This establishes the lower bound in the law of the iterated logarithm which we could not prove with the techniques of our paper \cite{MNX}. We also obtain exact small ball probability for $X(E_t)$ using ideas from \cite{aurzada}.Comment: 13 page

“…Bañuelos and DeBlassie [11] studied the distribution of the exit place for IBM in cones. Nane studied the lifetime asymptotics of IBM on bounded and unbounded domains [45,46,50], and generalized isoperimetric-type inequalities to IBM [49]. Remark 2.2.…”

Section: Introductionmentioning

confidence: 99%

Brownian subordinators and fractional Cauchy problems

Baeumer

Meerschaert

Trans. Amer. Math. Soc.

2009

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102

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.