Erkan Nane - Academia.edu (original) (raw)

Papers by Erkan Nane

Stochastic Processes and their Applications, 2015

In a continuous time random walk (CTRW), each random jump follows a random waiting time. CTRW sca... more In a continuous time random walk (CTRW), each random jump follows a random waiting time. CTRW scaling limits are time-changed processes that model anomalous diffusion. The outer process describes particle jumps, and the non-Markovian inner process (or time change) accounts for waiting times between jumps. This paper studies fractal properties of the sample functions of a timechanged process, and establishes some general results on the Hausdorff and packing dimensions of its range and graph. Then those results are applied to CTRW scaling limits.

Let p ∈ (0, ∞) be a constant and let {ξn} ⊂ L p (Ω, F, P) be a sequence of random variables. For ... more Let p ∈ (0, ∞) be a constant and let {ξn} ⊂ L p (Ω, F, P) be a sequence of random variables. For any integers m, n ≥ 0, denote Sm,n = P m+n k=m ξ k . It is proved that, if there exist a nondecreasing function ϕ : R+ → R+ (which satisfies a mild regularity condition) and an appropriately chosen integer a ≥ 2 such that ∞ X n=0 sup k≥0

We introduce a class of time dependent random fields on compact Riemannian monifolds. These are r... more We introduce a class of time dependent random fields on compact Riemannian monifolds. These are represented by time-changed Brownian motions. These processes are time-changed diffusion, or the stochastic solution to the equation involving the Laplace-Beltrami operator and a time-fractional derivative of order β ∈ (0, 1). The time dependent random fields we present in this work can therefore be realized through composition and can be viewed as random fields on randomly varying manifolds.

We consider time fractional stochastic heat type equation

Page 1. α-Time Fractional Brownian Motion: PDE Connections and Local Times Erkan Nane Auburn Univ... more Page 1. α-Time Fractional Brownian Motion: PDE Connections and Local Times Erkan Nane Auburn University Dongsheng Wu University of Alabama in Huntsville Yimin Xiao ∗ Michigan State University March 10, 2011 Abstract ...

Lately, many phenomena in both applied and abstract mathematics and related disciplines have been... more Lately, many phenomena in both applied and abstract mathematics and related disciplines have been expressed in terms of high order and fractional PDEs. Recently, Allouba introduced the Brownian-time Brownian sheet (BTBS) and connected it to a new system of fourth order interacting PDEs. The interaction in this multiparameter BTBS-PDEs connection is novel, leads to an intimately-connected linear system variant of the celebrated Kuramoto-Sivashinsky PDE, and is not shared with its one-timeparameter counterpart. It also means that these PDEs systems are to be solved for a family of functions, a feature exhibited in well known fluids dynamics models. On the other hand, the memory-preserving interaction between the PDE solution and the initial data is common to both the single-and the multi-parameter Brownian-time PDEs. Here, we introduce a new -even in the one-parameter case -proof that combines stochastic analysis with analysis and fractional calculus to simultaneously link BTBS to a new system of temporally half-derivative interacting PDEs as well as to the fourthorder system proved earlier and differently by Allouba. We then introduce a general class of random fields we call inverse-stable-Lévy-time Brownian sheets (ISLTBSs), and we link them to β-fractional-time-derivative systems of interacting PDEs for 0 < β < 1. When β = 1/ν, ν ∈ {2, 3, . . .}, our proof also connects an ISLTBS to a system of memory-preserving ν-Laplacian interacting PDEs. Memory is expressed via a sum of temporally-scaled k-Laplacians of the initial data, k = 1, . . . , ν − 1. Using a Fourier-Laplace-transform-fractional-calculus approach, we give a conditional equivalence result that gives a necessary and sufficient condition for the equivalence between the fractional and the high order systems. In the one-parameter case this condition automatically holds.

This paper develops strong solutions and stochastic solutions for the tempered fractional diffusi... more 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.

Statistics & Probability Letters, 2013

Continuous time random walks impose a random waiting time before each particle jump. Scaling limi... more 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.

Transactions of the American Mathematical Society, 2009

A Brownian time process is a Markov process subordinated to the absolute value of an independent ... more 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.

Stochastics and Dynamics, 2013

Lately, many phenomena in both applied and abstract mathematics and related disciplines have been... more Lately, many phenomena in both applied and abstract mathematics and related disciplines have been expressed in terms of high order and fractional PDEs. Recently, Allouba introduced the Brownian-time Brownian sheet (BTBS) and connected it to a new system of fourth order interacting PDEs. The interaction in this multiparameter BTBS-PDEs connection is novel, leads to an intimately-connected linear system variant of the celebrated Kuramoto-Sivashinsky PDE, and is not shared with its one-timeparameter counterpart. It also means that these PDEs systems are to be solved for a family of functions, a feature exhibited in well known fluids dynamics models. On the other hand, the memory-preserving interaction between the PDE solution and the initial data is common to both the single-and the multi-parameter Brownian-time PDEs. Here, we introduce a new -even in the one-parameter case -proof that combines stochastic analysis with analysis and fractional calculus to simultaneously link BTBS to a new system of temporally half-derivative interacting PDEs as well as to the fourthorder system proved earlier and differently by Allouba. We then introduce a general class of random fields we call inverse-stable-Lévy-time Brownian sheets (ISLTBSs), and we link them to β-fractional-time-derivative systems of interacting PDEs for 0 < β < 1. When β = 1/ν, ν ∈ {2, 3, . . .}, our proof also connects an ISLTBS to a system of memory-preserving ν-Laplacian interacting PDEs. Memory is expressed via a sum of temporally-scaled k-Laplacians of the initial data, k = 1, . . . , ν − 1. Using a Fourier-Laplace-transform-fractional-calculus approach, we give a conditional equivalence result that gives a necessary and sufficient condition for the equivalence between the fractional and the high order systems. In the one-parameter case this condition automatically holds.

Stochastic Processes and their Applications, 2014

We introduce a class of isotropic time dependent random fields on the non-homogeneous sphere repr... more We introduce a class of isotropic time dependent random fields on the non-homogeneous sphere represented by a time-changed spherical Brownian motion of order ν ∈ (0, 1] with which some anisotrophies can be captured in Cosmology. This process is a time-changed rotational diffusion (TRD) or the stochastic solution to the equation involving the spherical Laplace operator and a time-fractional derivative of order ν. TRD is a diffusion on the non-homogeneous sphere and therefore, the spherical coordinates given by TRD represent the coordinates of a non-homogeneous sphere by means of which an isotopic random field is indexed. The time dependent random fields we present in this work is therefore realized through composition and can be viewed as isotropic random field on randomly varying sphere.

Statistics & Probability Letters, 2014

The stochastic solution to diffusion equations with polynomial coefficients is called a Pearson d... more The stochastic solution to diffusion equations with polynomial coefficients is called a Pearson diffusion. If the time derivative is replaced by a distributed fractional derivative, the stochastic solution is called a fractional Pearson diffusion. This paper develops a formula for the covariance function of a fractional Pearson diffusion in steady state, in terms of generalized Mittag-Leffler functions. That formula shows that fractional Pearson diffusions are long-range dependent, with a correlation that falls off like a power law, whose exponent equals the smallest order of the distributed fractional derivative. ∂ 2 ∂y 2 p 1 (x, t; y),

Statistics & Probability Letters, 2011

We consider time-changed Poisson processes, and derive the governing difference-differential equa... more We consider time-changed Poisson processes, and derive the governing difference-differential equations (DDE) these processes. In particular, we consider the time-changed Poisson processes where the the time-change is inverse Gaussian, or its hitting time process, and discuss the governing DDE's. The stable subordinator, inverse stable subordinator and their iterated versions are also considered as time-changes. DDE's 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 Poisson process time-changed by tempered stable subordinator. Our results extend and complement the results in Baeumer et al. and Beghin et al. in several directions.

Continuous time random walks impose a random waiting time before each particle jump. Scaling limi... more 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.

Proceedings of the American Mathematical Society, 2014

Fractional derivatives can be used to model time delays in a diffusion process. When the order of... more 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. In some special cases a distributed order derivative can be used to model ultra-slow diffusion. We extend the results of Baeumer and Meerschaert in the single order fractional derivative case to the distributed order fractional derivative case. In particular, we develop strong analytic solutions of distributed order fractional Cauchy problems.

Proceedings of the American Mathematical Society, 2012

Fractional Cauchy problems replace the usual first order time derivative by a fractional derivati... more Fractional Cauchy problems replace the usual first order time derivative by a fractional derivative. A tempered fractional derivative provides an attractive alternative, and is closely connected to the tempered stable process. This paper develops classical solutions and stochastic analogues for tempered fractional Cauchy problems in a bounded domain D ⊂ R d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse tempered stable subordinator, whose tail index corresponds to the order of the tempered fractional time derivative.

Journal of Mathematical Analysis and Applications, 2014

Journal of Mathematical Analysis and Applications, 2012

Fractional diffusion equations replace the integer-order derivatives in space and time by their f... more 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.

Journal of Mathematical Analysis and Applications, 2011

Fractional derivatives can be used to model time delays in a diffusion process. When the order of... more 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.

Stochastic Processes and their Applications, 2015

In a continuous time random walk (CTRW), each random jump follows a random waiting time. CTRW sca... more In a continuous time random walk (CTRW), each random jump follows a random waiting time. CTRW scaling limits are time-changed processes that model anomalous diffusion. The outer process describes particle jumps, and the non-Markovian inner process (or time change) accounts for waiting times between jumps. This paper studies fractal properties of the sample functions of a timechanged process, and establishes some general results on the Hausdorff and packing dimensions of its range and graph. Then those results are applied to CTRW scaling limits.

Let p ∈ (0, ∞) be a constant and let {ξn} ⊂ L p (Ω, F, P) be a sequence of random variables. For ... more Let p ∈ (0, ∞) be a constant and let {ξn} ⊂ L p (Ω, F, P) be a sequence of random variables. For any integers m, n ≥ 0, denote Sm,n = P m+n k=m ξ k . It is proved that, if there exist a nondecreasing function ϕ : R+ → R+ (which satisfies a mild regularity condition) and an appropriately chosen integer a ≥ 2 such that ∞ X n=0 sup k≥0

We introduce a class of time dependent random fields on compact Riemannian monifolds. These are r... more We introduce a class of time dependent random fields on compact Riemannian monifolds. These are represented by time-changed Brownian motions. These processes are time-changed diffusion, or the stochastic solution to the equation involving the Laplace-Beltrami operator and a time-fractional derivative of order β ∈ (0, 1). The time dependent random fields we present in this work can therefore be realized through composition and can be viewed as random fields on randomly varying manifolds.

We consider time fractional stochastic heat type equation

Page 1. α-Time Fractional Brownian Motion: PDE Connections and Local Times Erkan Nane Auburn Univ... more Page 1. α-Time Fractional Brownian Motion: PDE Connections and Local Times Erkan Nane Auburn University Dongsheng Wu University of Alabama in Huntsville Yimin Xiao ∗ Michigan State University March 10, 2011 Abstract ...

Lately, many phenomena in both applied and abstract mathematics and related disciplines have been... more Lately, many phenomena in both applied and abstract mathematics and related disciplines have been expressed in terms of high order and fractional PDEs. Recently, Allouba introduced the Brownian-time Brownian sheet (BTBS) and connected it to a new system of fourth order interacting PDEs. The interaction in this multiparameter BTBS-PDEs connection is novel, leads to an intimately-connected linear system variant of the celebrated Kuramoto-Sivashinsky PDE, and is not shared with its one-timeparameter counterpart. It also means that these PDEs systems are to be solved for a family of functions, a feature exhibited in well known fluids dynamics models. On the other hand, the memory-preserving interaction between the PDE solution and the initial data is common to both the single-and the multi-parameter Brownian-time PDEs. Here, we introduce a new -even in the one-parameter case -proof that combines stochastic analysis with analysis and fractional calculus to simultaneously link BTBS to a new system of temporally half-derivative interacting PDEs as well as to the fourthorder system proved earlier and differently by Allouba. We then introduce a general class of random fields we call inverse-stable-Lévy-time Brownian sheets (ISLTBSs), and we link them to β-fractional-time-derivative systems of interacting PDEs for 0 < β < 1. When β = 1/ν, ν ∈ {2, 3, . . .}, our proof also connects an ISLTBS to a system of memory-preserving ν-Laplacian interacting PDEs. Memory is expressed via a sum of temporally-scaled k-Laplacians of the initial data, k = 1, . . . , ν − 1. Using a Fourier-Laplace-transform-fractional-calculus approach, we give a conditional equivalence result that gives a necessary and sufficient condition for the equivalence between the fractional and the high order systems. In the one-parameter case this condition automatically holds.

This paper develops strong solutions and stochastic solutions for the tempered fractional diffusi... more 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.

Statistics & Probability Letters, 2013

Continuous time random walks impose a random waiting time before each particle jump. Scaling limi... more 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.

Transactions of the American Mathematical Society, 2009

A Brownian time process is a Markov process subordinated to the absolute value of an independent ... more 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.

Stochastics and Dynamics, 2013

Lately, many phenomena in both applied and abstract mathematics and related disciplines have been... more Lately, many phenomena in both applied and abstract mathematics and related disciplines have been expressed in terms of high order and fractional PDEs. Recently, Allouba introduced the Brownian-time Brownian sheet (BTBS) and connected it to a new system of fourth order interacting PDEs. The interaction in this multiparameter BTBS-PDEs connection is novel, leads to an intimately-connected linear system variant of the celebrated Kuramoto-Sivashinsky PDE, and is not shared with its one-timeparameter counterpart. It also means that these PDEs systems are to be solved for a family of functions, a feature exhibited in well known fluids dynamics models. On the other hand, the memory-preserving interaction between the PDE solution and the initial data is common to both the single-and the multi-parameter Brownian-time PDEs. Here, we introduce a new -even in the one-parameter case -proof that combines stochastic analysis with analysis and fractional calculus to simultaneously link BTBS to a new system of temporally half-derivative interacting PDEs as well as to the fourthorder system proved earlier and differently by Allouba. We then introduce a general class of random fields we call inverse-stable-Lévy-time Brownian sheets (ISLTBSs), and we link them to β-fractional-time-derivative systems of interacting PDEs for 0 < β < 1. When β = 1/ν, ν ∈ {2, 3, . . .}, our proof also connects an ISLTBS to a system of memory-preserving ν-Laplacian interacting PDEs. Memory is expressed via a sum of temporally-scaled k-Laplacians of the initial data, k = 1, . . . , ν − 1. Using a Fourier-Laplace-transform-fractional-calculus approach, we give a conditional equivalence result that gives a necessary and sufficient condition for the equivalence between the fractional and the high order systems. In the one-parameter case this condition automatically holds.

Stochastic Processes and their Applications, 2014

We introduce a class of isotropic time dependent random fields on the non-homogeneous sphere repr... more We introduce a class of isotropic time dependent random fields on the non-homogeneous sphere represented by a time-changed spherical Brownian motion of order ν ∈ (0, 1] with which some anisotrophies can be captured in Cosmology. This process is a time-changed rotational diffusion (TRD) or the stochastic solution to the equation involving the spherical Laplace operator and a time-fractional derivative of order ν. TRD is a diffusion on the non-homogeneous sphere and therefore, the spherical coordinates given by TRD represent the coordinates of a non-homogeneous sphere by means of which an isotopic random field is indexed. The time dependent random fields we present in this work is therefore realized through composition and can be viewed as isotropic random field on randomly varying sphere.

Statistics & Probability Letters, 2014

The stochastic solution to diffusion equations with polynomial coefficients is called a Pearson d... more The stochastic solution to diffusion equations with polynomial coefficients is called a Pearson diffusion. If the time derivative is replaced by a distributed fractional derivative, the stochastic solution is called a fractional Pearson diffusion. This paper develops a formula for the covariance function of a fractional Pearson diffusion in steady state, in terms of generalized Mittag-Leffler functions. That formula shows that fractional Pearson diffusions are long-range dependent, with a correlation that falls off like a power law, whose exponent equals the smallest order of the distributed fractional derivative. ∂ 2 ∂y 2 p 1 (x, t; y),

Statistics & Probability Letters, 2011

We consider time-changed Poisson processes, and derive the governing difference-differential equa... more We consider time-changed Poisson processes, and derive the governing difference-differential equations (DDE) these processes. In particular, we consider the time-changed Poisson processes where the the time-change is inverse Gaussian, or its hitting time process, and discuss the governing DDE's. The stable subordinator, inverse stable subordinator and their iterated versions are also considered as time-changes. DDE's 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 Poisson process time-changed by tempered stable subordinator. Our results extend and complement the results in Baeumer et al. and Beghin et al. in several directions.

Continuous time random walks impose a random waiting time before each particle jump. Scaling limi... more 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.

Proceedings of the American Mathematical Society, 2014

Fractional derivatives can be used to model time delays in a diffusion process. When the order of... more 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. In some special cases a distributed order derivative can be used to model ultra-slow diffusion. We extend the results of Baeumer and Meerschaert in the single order fractional derivative case to the distributed order fractional derivative case. In particular, we develop strong analytic solutions of distributed order fractional Cauchy problems.

Proceedings of the American Mathematical Society, 2012

Fractional Cauchy problems replace the usual first order time derivative by a fractional derivati... more Fractional Cauchy problems replace the usual first order time derivative by a fractional derivative. A tempered fractional derivative provides an attractive alternative, and is closely connected to the tempered stable process. This paper develops classical solutions and stochastic analogues for tempered fractional Cauchy problems in a bounded domain D ⊂ R d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse tempered stable subordinator, whose tail index corresponds to the order of the tempered fractional time derivative.

Journal of Mathematical Analysis and Applications, 2014

Journal of Mathematical Analysis and Applications, 2012

Fractional diffusion equations replace the integer-order derivatives in space and time by their f... more 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.

Journal of Mathematical Analysis and Applications, 2011

Fractional derivatives can be used to model time delays in a diffusion process. When the order of... more 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.