Erkan Nane

Fractional Cauchy problems on bounded domains

Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain DR d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordi- nator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brow- nian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time. 1. Introduction. In this paper, we extend the approach of Meerschaert and Scheffler ( 23) and Meerschaert et al. (24) to fractional Cauchy problems on bounded domains. Our methods involve eigenfunction expansions, killed Markov processes and inverse stable subordinators. In a recent related paper (7), we establish a connection between fractional Cauchy problems with index β = 1/2 on an unbounded domain, and iterated Brownian motion (IBM), defined as Zt = B(|Yt|), where B is a Brownian motion with values in R d and Y is an independent one-dimensional Brownian motion. Since IBM is also the stochastic solution to a Cauchy problem involving a fourth-order derivative in space (2, 14), that paper also establishes a connection between certain higher-order Cauchy problems and their time-fractional analogues. More generally, Baeumer, Meerschaert and Nane (7) shows a connection between fractional Cauchy problems with β = 1/2 and higher-order Cauchy problems that involve the square of the generator. In the present paper, we

Brownian subordinators and fractional Cauchy problems

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.

Distributed-order fractional diffusions on bounded domains

Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. This paper provides explicit strong solutions and stochastic analogues for distributed-order time-fractional diffusion equations on bounded domains, with Dirichlet boundary conditions.

Correlated continuous time random walks

Continuous time random walks impose a random waiting time before each particle jump. Scaling limits of heavy-tailed continuous time random walks are governed by fractional evolution equations. Space-fractional derivatives describe heavy-tailed jumps, and the time-fractional version codes heavy-tailed waiting times. This paper develops scaling limits and governing equations in the case of correlated jumps. For long-range dependent jumps, this leads to fractional Brownian motion or linear fractional stable motion, with the time parameter replaced by an inverse stable subordinator in the case of heavy-tailed waiting times. These scaling limits provide an interesting class of non-Markovian, non-Gaussian self-similar processes.

Space-time fractional diffusion on bounded domains

Abstract Fractional diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues. They are used in physics to model anomalous diffusion. This paper develops strong solutions of space–time fractional diffusion equations on bounded domains, as well as probabilistic representations of these solutions, which are useful for particle tracking codes.

Higher order PDE’s and iterated processes

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 PDEs for a rational. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.

Time-changed Poisson processes

We consider time-changed Poisson processes, and derive the governing difference–differential equations (DDEs) for these processes. In particular, we consider the time-changed Poisson processes where the time-change is inverse Gaussian, or its hitting time process, and discuss the governing DDEs. The stable subordinator, inverse stable subordinator and their iterated versions are also considered as time-changes. DDEs corresponding to probability mass functions of these time-changed processes are obtained. Finally, we obtain a new governing partial differential equation for the tempered stable subordinator of index 0<β<1, when β is a rational number. We then use this result to obtain the governing DDE for the mass function of the Poisson process time-changed by the tempered stable subordinator. Our results extend and complement the results in Baeumer et al. (2009) and Beghin and Orsingher (2009) in several directions.

Iterated Brownian motion in bounded domains in Rn

Let [tau]D(Z) be the first exit time of iterated Brownian motion from a domain started at z[set membership, variant]D and let Pz[[tau]D(Z)>t] be its distribution. In this paper we establish the exact asymptotics of Pz[[tau]D(Z)>t] over bounded domains as an extension of the result in [R.D. DeBlassie, Iterated Brownian motion in an open set, Ann. Appl. Probab. 14 (3) (2004) 1529-1558], for z[set membership, variant]D: We also study asymptotics of the life time of Brownian-time Brownian motion (BTBM), , where Xt and Yt are independent one-dimensional Brownian motions.

Iterated Brownian Motion in Parabola-Shaped Domains

AbstractIterated Brownian motion Zt serves as a physical model for diffusions in a crack. If τD(Z) is the first exit time of this processes from a domain D⊂ℝn, started at z∈D, then Pz[τD(Z)>t] is the distribution of the lifetime of the process in D. In this paper we determine the large time asymptotics of

Large deviations for local time fractional Brownian motion and applications

P_{z}[\tau _{P_{\alpha}}(Z)>t]