Showing papers by "Erkan Nane published in 2010"
Posted Content•
TL;DR: In this article, it was shown that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional poisson process with Mittag-Leffler waiting times, which unifies the two main approaches in the stochastic theory of time-fractional diffusion equations.
Abstract: The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also establishes an interesting connection between the fractional Poisson process and Brownian time.
22 citations
TL;DR: In this paper, the authors study solutions of a class of higher-order partial differential equations in bounded domains, and they 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.
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].
18 citations
Journal Article•
TL;DR: In this article, the authors studied asymptotic properties of stochastic processes with heavy-tailed distributions, such as linear fractional stable motion and harmonizable fractional stability.
Abstract: The study on strong law of large numbers has a long history and there is a vast body of references on this topic. This note is motivated by our interest in studying asymptotic properties of stochastic processes with heavy-tailed distributions. Typical examples of such processes are linear fractional stable motion and harmonizable fractional stable motion. See Samorodnitsky and Taqqu [7] and Embrechts and Maejima [4]. Let p ∈ (0,∞) be a constant and let {ξn} ⊂ Lp(Ω,F ,P) be a sequence of random variables. For any integers m,n ≥ 0, denote
8 citations
Posted Content•
22 Apr 2010
TL;DR: In this paper, strong solutions and stochastic solutions for the tempered fractional diffusion equation on bounded domains were developed, where a separation of variables, and eigenfunction expansions in time and space, were used to write strong solutions.
Abstract: This paper develops strong solutions and stochastic solutions for the tempered fractional diffusion equation on bounded domains. First the eigenvalue problem for tempered fractional derivatives is solved. Then a separation of variables, and eigenfunction expansions in time and space, are used to write strong solutions. Finally, stochastic solutions are written in terms of an inverse subordinator.
3 citations
Posted Content•
TL;DR: In this article, strong solutions and stochastic solutions for the tempered fractional diffusion equation on bounded domains were developed, where a separation of variables, and eigenfunction expansions in time and space, were used to write strong solutions.
Abstract: This paper develops strong solutions and stochastic solutions for the tempered fractional diffusion equation on bounded domains. First the eigenvalue problem for tempered fractional derivatives is solved. Then a separation of variables, and eigenfunction expansions in time and space, are used to write strong solutions. Finally, stochastic solutions are written in terms of an inverse subordinator.
1 citations