Showing papers by "Palaniappan Vellaisamy published in 2011"
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 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 {discusses the relation between} the fractional Poisson process and Brownian time.
243 citations
TL;DR: In this article, strong solutions and stochastic analogues for distributed-order time-fractional diffusion equations on bounded domains, with Dirichlet boundary conditions, were provided.
137 citations
TL;DR: In this article, the authors considered time-changed Poisson processes and derived the governing difference-differential equations (DDEs) for these processes, and derived a new governing partial differential equation for the tempered stable subordinator of index 0 β 1.
51 citations
TL;DR: In this article, the authors show that the FNIG process emerges naturally as the limit of a random walk with correlated jumps separated by i.i.d. waiting times, and that the NIG process, a Brownian motion subordinated to an inverse Gaussian process, is the limit for random walks with uncorrelated jumps.
24 citations
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TL;DR: In this paper, the first-exit time process of an inverse Gaussian L\'evy process is considered and the one-dimensional distribution functions of the process are obtained, which are not infinitely divisible and the tail probabilities decay exponentially.
Abstract: The first-exit time process of an inverse Gaussian L\'evy process is considered. The one-dimensional distribution functions of the process are obtained. They are not infinitely divisible and the tail probabilities decay exponentially. These distribution functions can also be viewed as distribution functions of supremum of the Brownian motion with drift. The density function is shown to solve a fractional PDE and the result is also generalized to tempered stable subordinators. The subordination of this process to the Brownian motion is considered and the underlying PDE of the subordinated process is obtained. The infinite divisibility of the first-exit time of a $\beta$-stable subordinator is also discussed.
5 citations
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TL;DR: In this article, the first hitting time process of an inverse Gaussian process is considered and the density functions of one-dimensional distributions of the process are obtained, and it is shown that their distribution functions are not infinitely divisible and their tail probability decay exponentially.
Abstract: The first hitting time process of an inverse Gaussian process is considered. It is shown that this process is not Levy and has monotonically increasing continuous sample paths. The density functions of one-dimensional distributions of the process are obtained. Its distribution functions are not infinitely divisible and their tail probability decay exponentially. It is also shown that the hitting time process of a stable process with index 0 < β < 1 is not infinitely divisible. The density function is shown to solve a fractional partial differential equation. Subordination of the hitting time process to Brownian motion is considered and the underlying PDE of the subordinated process is derived.
3 citations