Showing papers by "Erkan Nane published in 2007"
TL;DR: In this paper, a class of stochastic processes based on symmetric α-stable processes, called α-time processes, for a ∈ (0,2j) was introduced, which are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric alpha-stable process.
Abstract: We introduce a class of stochastic processes based on symmetric α-stable processes, for a ∈ (0,2j. These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric α-stable process. We call them α-time processes. They generalize Brownian time processes studied in Allouba and Zheng (2001), Allouba (2002), (2003), and they introduce new interesting examples. We establish the connection of α-time processes to some higher order PDE's for a rational. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.
47 citations
01 Jan 2007
TL;DR: In this paper, the exact asymptotics of Pz(τD(Z) >t ) over bounded domains were established for z ∈ D (−t),t < 0.
Abstract: Let τD(Z) be the first exit time of iterated Brownian motion from a domain D ⊂ R n started at z ∈ D and let Pz(τD(Z) >t ) be its distribution. In this paper we establish the exact asymptotics of Pz(τD(Z) >t ) over bounded domains as an improvement of the results in DeBlassie (2004) (12) and Nane (2006) (24), for z ∈ D (−t) ,t <0.
14 citations
TL;DR: DeBlassie et al. as mentioned in this paper established the exact asymptotics of over bounded domains as an improvement of the results in Deblassie, Ann. Appl. 14 (2004) 1529-1558] and Nane, Stochastic Processes Appl. 116 (2006) 905-916], for the first eigenvalue of D and ψ.
Abstract: Let be the first exit time of iterated Brownian motion from a domain started at and let be its distribution. In this paper we establish the exact asymptotics of over bounded domains as an improvement of the results in DeBlassie (2004) [DeBlassie, Ann. Appl. Prob. 14 (2004) 1529–1558] and Nane (2006) [Nane, Stochastic Processes Appl. 116 (2006) 905–916], for where . Here λD is the first eigenvalue of the Dirichlet Laplacian in D , and ψ is the eigenfunction corresponding to λD . We also study lifetime asymptotics of Brownian-time Brownian motion, , where X t and Y t are independent one-dimensional Brownian motions, in several unbounded domains. Using these results we obtain partial results for lifetime asymptotics of iterated Brownian motion in these unbounded domains.
9 citations
TL;DR: In this paper, the scaling limits of continuous time random walks were shown to be equivalent to the hitting time process of a classical stable subordinator, and an equivalence between these two families of partial differential equations was established.
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, involve 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.
1 citations
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TL;DR: In this article, the equivalence of higher-order PDE's and fractional in-time Cauchy problems has been shown for iterated processes with Brownian subordinators.
Abstract: We survey the results in Nane (E. Nane, Higher order PDE's and iterated processes, Trans. American Math. Soc. (to appear)) and Baeumer, Meerschaert, and Nane (B. Baeumer, M.M. Meerschaert and E. Nane, Brownian subordinators and fractional Cauchy problems: Submitted (2007)) which deal with PDE connection of some iterated processes, and obtain a new probabilistic proof of the equivalence of the higher order PDE's and fractional in time PDE's.