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Erkan Nane

Publications and source records attributed to Erkan Nane.

At least 19 recordsLinked to original sources
arXiv2025

Space-time fractional SPDEs with locally Lipschitz coefficients: well-posedness

In this article, we study the space-time SPDE $$ \partial_t^\beta u=-(-\Delta)^{\alpha/2} u+I_t^{1-\beta}[b(u)+\sigma(u)\dot{W}],$$ where $u=u(t,x)$ is defined for $(t,x)\in\mathbb{R}_+\times \mathbb{R},$ $\beta\in(0,1), \alpha\in(0,2)$ and $\dot{W}$ denotes a space-time white noise. It has long been conjectured that this equation has a unique solution with finite moments under the minimal assumptions of locally Lipschitz coefficients $b$ and $\sigma$ with linear growth. We prove that this SPDE is well-posed under the assumptions that the initial condition $u_0$ is bounded and measurable, and the functions $b$ and $\sigma$ are locally Lipschitz and have at-most linear growth and some conditions on the Lipschitz constants on the truncated versions of $b$ and $\sigma$. Our results generalize the work of Foondun et al.(2025) to a space-time fractional setting.

math.PR↗
arXiv2024

Propagation of high peaks for the space-time fractional stochastic partial differential equations

We study the space-time nonlinear fractional stochastic heat equation driven by a space-time white noise, \begin{align*} \partial_t^\beta u(t,x)=-(-\Delta)^{\alpha/2}u(t,x)+I_t^{1-\beta}\Big[\sigma(u(t,x))\dot{W}(t,x)\Big],\ \ t>0, \ x\in \mathbb{R} , \end{align*} where $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is a globally Lipschitz function and the initial condition is a measure on $\mathbb{R}.$ Under some growth conditions on $\sigma,$ we derive two important properties about the moments of the solution: (i) For $p\geq 2,$ the $p^{\text{th}}$ absolute moment of the solution to the equation above grows exponentially with time. (ii) Moreover, the distances to the origin of the farthest high peaks of these moments grow exactly exponentially with time. Our results provide an extension of the work of Chen and Dalang (Stoch PDE: Anal Comp (2015) 3:360-397) to a time-fractional setting. We also show that condition (i) holds when we study the same equation for $x\in\mathbb{R}^d.$

math.PR↗
arXiv2022

Level of noises and long time behavior of the solution for space-time fractional SPDE in bounded domains

In this paper we study the long time behavior of the solution to a certain class of space-time fractional stochastic equations with respect to the level $λ$ of a noise and show how the choice of the order $β\in (0, \,1)$ of the fractional time derivative affects the growth and decay behavior of their solution. We consider both the cases of white noise and colored noise. Our results extend the main results in "M. Foondun, \textit{Remarks on a fractional-time stochastic equation}, Proc. Amer. Math. Soc. 149 (2021), 2235-2247" to fractional Laplacian as well as higher dimensional cases.

math.PR↗
arXiv2022

Pathwise Blowup of space-time fractional SPDEs

The finite time blowup in the almost sure sense of a class of space-time fractional stochastic partial differential equations is discussed. Both the cases of white noise and colored noise are considered. The sufficient and necessary condition between the blowup and Osgood condition is obtained when the spatial domain is bounded. And the sufficient condition for the blowup is obtained when the spatial domain is the whole space. The results in this paper could be regarded as extensions to some results in Foondun and Nualart, 2021.

math.PR↗
arXiv2020

A uniqueness determination of the fractional exponents in a three-parameter fractional diffusion

In this article, we consider the space-time Fractional (nonlocal) diffusion equation $$\partial_t^βu(t,x)={\mathtt{L}_D^{α_1,α_2}} u(t,x), \ \ t\geq 0, \ x\in D, $$ where $\partial_t^β$ is the Caputo fractional derivative of order $β\in (0,1)$ and the differential operator ${\mathtt{L}_D^{α_1,α_2}}$ is the generator of a Lévy process, sum of two symmetric independent $α_1-$stable and $α_2-$stable processes and ${D}$ is the open unit interval in $\mathbb{R}$. We consider a nonlocal inverse problem and show that the fractional exponents $β$ and $α_i, \ i=1,2$ are determined uniquely by the data $u(t, 0) = g(t),\ 0 < t < T.$ The uniqueness result is a theoretical background for determining experimentally the order of many anomalous diffusion phenomena, which are important in many fields, including physics and environmental engineering. We also discuss the numerical approximation of the inverse problem as a nonlinear least-squares problem and explore parameter sensitivity through numerical experiments.

math.AP↗
arXiv2020

Moment bounds of a class of stochastic heat equations driven by space-time colored noise in bounded domains

We consider the fractional stochastic heat type equation \begin{align*} \frac{\partial}{\partial t} u_t(x)=-(-Δ)^{α/2}u_t(x)+ξσ(u_t(x))\dot{F}(t,x),\ \ \ x\in D, \ \ t>0, \end{align*} with nonnegative bounded initial condition, where $α\in (0,2]$, $ξ>0$ is the noise level, $σ:\mathbb{R}\rightarrow\mathbb{R}$ is a globally Lipschitz function satisfying some growth conditions and the noise term behaves in space like the Riez kernel and is possibly correlated in time and $D$ is the unit open ball centered at the origin in $\mathbb{R}^d$. When the noise term is not correlated in time, we establish a change in the growth of the solution of these equations depending on the noise level $ξ$. On the other hand when the noise term behaves in time like the fractional Brownian motion with index $H\in (1/2,1)$, We also derive explicit bounds leading to a well-known intermittency property.

math.PR↗
arXiv2020

The nonlinear fractional diffusion equations with Nagumo-type sources and perturbed orders

We consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable $t$, the fractional order of the self-adjoint positive definite unbounded operator in a Hilbert space and a singular nonlinear source. These equations are generalizations of some well-known fractional equation such as the fractional Cahn-Allen equation, the fractional Burger equation, the fractional Cahn-Hilliard equation, the fractional Kuramoto-Sivashinsky equation, etc. We study both the initial value and the final value problem. Under some suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. For $t=0$, we show that the final value problem is instable and deduce that the problem is ill-posed. A regularization method is proposed to recover the initial data from the inexact fractional orders and the final data. By some regularity assumptions of the exact solutions of the problems, we obtain an error estimate of Hölder type.

math.AP↗
arXiv2020

Finite Time Blowup of Solutions to SPDEs with Bernstein Functions of the Laplacian

The blowup in finite time of solutions to SPDEs \begin{equation*} \partial_tu_t(x)=-ϕ(-Δ)u_t(x) +σ(u_t(x))\dotξ(t,x), \quad t>0,x\in\mathbb{R}^d, \end{equation*} { is} investigated, where $\dotξ$ could be either a white noise or a colored noise and $ϕ:(0,\infty)\to (0,\infty)$ is a Bernstein function. The sufficient conditions on $σ$, $\dotξ$ and the initial value that imply the non-existence of the global solution are discussed. The results in this paper generalise those in ``Foondun, M., Liu, W. and Nane, E. Some non-existence results for a class of stochastic partial differential equations. J. Differential Equations, 266 (5) (2019), 2575--2596.'', where the fractional Laplacian case was considered, i.e. $ϕ(-Δ)=(-Δ)^{α/2}$ ($1<α<2$).

math.PR↗
arXiv2019

Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation $\partial_t^βu= -\big(-Δ\big)^{α/2}u-\big(-Δ\big)^{γ/2}u$

In this article, we consider the space-time fractional (nonlocal) equation characterizing the so-called "double-scale" anomalous diffusion $$\partial_t^βu(t, x) = -(-Δ)^{α/2}u(t,x) - (-Δ)^{γ/2}u(t,x) \ \ t> 0, \ -1<x<1, $$ where $\partial_t^β$ is the Caputo fractional derivative of order $β\in (0,1)$ and $0<α\leq γ<2.$ We consider a nonlocal inverse problem and show that the fractional exponents $β$, $α$ and $γ$ are determined uniquely by the data $u(t, 0) = g(t), \ 0 < t \leq T.$ The existence of the solution for the inverse problem is proved using the quasi-solution method which is based on minimizing an error functional between the output data and the additional data. In this context, an input-output mapping is defined and its continuity is established. The uniqueness of the solution for the inverse problem is proved by means of eigenfunction expansion of the solution of the forward problem and some basic properties of fractional Laplacian. A numerical method based on discretization of the minimization problem, namely the steepest descent method and a least squares approach, is proposed for the solution of the inverse problem. The numerical method determines the fractional exponents simultaneously. Finally, numerical examples with noise-free and noisy data illustrate applicability and high accuracy of the proposed method.

math.AP↗
arXiv2019

Asymptotic behavior of solution and non-existence of global solution to a class of conformable time-fractional stochastic equation

Consider the following class of conformable time-fractional stochastic equation $$T_{α,t}^a u(x,t)=λσ(u(x,t))\dot{W}_t,\,\,\,\,x\in\mathbb{R},\,t\in[a,\infty), \,\,0<α<1,$$ with a non-random initial condition $u(x,0)=u_0(x),\,x\in\mathbb{R}$ assumed to be non-negative and bounded, $T_{α,t}^a$ is a conformable time - fractional derivative, $σ:\mathbb{R}\rightarrow\mathbb{R}$ is globally Lipschitz continuous, $\dot{W}_t$ a generalized derivative of Wiener process and $λ>0$ is the noise level. Given some precise and suitable conditions on the non-random initial function, we study the asymptotic behaviour of the solution with respect to the time parameter $t$ and the noise level parameter $λ$. We also show that when the non-linear term $σ$ grows faster than linear, the energy of the solution blows-up at finite time for all $α\in (0,1)$.

math.PR↗
arXiv2019

Time-changed Stochastic Control Problem and its Maximum Principle Theory

This paper studies a time-changed stochastic control problem, where the underlying stochastic process is a Lévy noise time-changed by an inverse subordinator. We establish a maximum principle theory for the time-changed stochastic control problem. We also prove the existence and uniqueness of the corresponding time-changed backward stochastic differential equation involved in the stochastic control problem. Some examples are provided for illustration.

math.PR↗
arXiv2019

Space-time fractional stochastic partial differential equations with Lévy Noise

We consider non-linear time-fractional stochastic heat type equation $$\frac{\partial^βu}{\partial t^β}+ν(-Δ)^{α/2} u=I^{1-β}_t \bigg[\int_{\mathbb{R}^d}σ(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg]$$ and $$\frac{\partial^βu}{\partial t^β}+ν(-Δ)^{α/2} u=I^{1-β}_t \bigg[\int_{\mathbb{R}^d}σ(u(t,x),h) \stackrel{\cdot}{N }(t,x,h)\bigg]$$ in $(d+1)$ dimensions, where $α\in (0,2]$ and $d<\min\{2,β^{-1}\}α$, $ν>0$, $\partial^β_t$ is the Caputo fractional derivative, $-(-Δ)^{α/2} $ is the generator of an isotropic stable process, $I^{1-β}_t$ is the fractional integral operator, ${N}(t,x)$ are Poisson random measure with $\tilde{N}(t,x)$ being the compensated Poisson random measure. $σ:{\mathbb{R}}\to{\mathbb{R}}$ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in "M. Foondun and D. Khoshnevisan. Intermittence and nonlinear parabolic stochastic partial differential equations. \emph{ Electron. J. Probab.} {\bf14} (2009), 548--568" and " J. B. Walsh. An Introduction to Stochastic Partial Differential Equations, Écoled'été de Probabilités de Saint-Flour, XIV|1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, (1986), 265--439". Under the linear growth of $σ$, we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when $σ$ grows faster than linear.

math.PR↗
arXiv2019

On the infinite divisibility of distributions of some inverse subordinators

We consider the infinite divisibility of distributions of some well-known inverse subordinators. Using a tail probability bound, we establish that distributions of many of the inverse subordinators used in the literature are not infinitely divisible. We further show that the distribution of a renewal process time-changed by an inverse stable subordinator is not infinitely divisible, which in particular implies that the distribution of the fractional Poisson process is not infinitely divisible.

math.PR↗
arXiv2018

Blow-up results for space-time fractional stochastic partial differential equations

Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type, $$\partial^β_tu_t(x)=-ν(-Δ)^{α/2} u_t(x)+I^{1-β}_t[b(u)+ σ(u)\stackrel{\cdot}{F}(t,x)]$$ in $(d+1)$ dimensions, where $ν>0, β\in (0,1)$, $α\in (0,2]$. The operator $\partial^β_t$ is the Caputo fractional derivative while $-(-Δ)^{α/2} $ is the generator of an isotropic $α$-stable Lévy process and $I^{1-β}_t$ is the Riesz fractional integral operator. The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. These equations might be used as a model for materials with random thermal memory. We derive non-existence (blow-up) of global random field solutions under some additional conditions, most notably on $b$, $σ$ and the initial condition. Our results complement those of P. Chow in \cite{chow2}, \cite{chow1}, and Foondun et al. in \cite{Foondun-liu-nane}, \cite{foondun-parshad} among others.

math.PR↗
arXiv2018

Strong laws of large numbers for arrays of random variables and stable random fields

Strong laws of large numbers are established for random fields with weak or strong dependence. These limit theorems are applicable to random fields with heavy-tailed distributions including fractional stable random fields. The conditions for SLLN are described in terms of the $p$-th moments of the partial sums of the random fields, which are convenient to verify. The main technical tool in this paper is a maximal inequality for the moments of partial sums of random fields that extends the technique of Levental, Chobanyan and Salehi \cite{chobanyan-l-s} for a sequence of random variables indexed by a one-parameter.

math.PR↗
arXiv2018

Critical parameters for reaction-diffusion equations involving space-time fractional derivatives

We will look at reaction-diffusion type equations of the following type, $$\partial^β_tV(t,x)=-(-Δ)^{α/2} V(t,x)+I^{1-β}_t[V(t,x)^{1+η}].$$ We first study the equation on the whole space by making sense of it via an integral equation. Roughly speaking, we will show that when $0<η\leqη_c$, there is no global solution other than the trivial one while for $η>η_c$, non-trivial global solutions do exist. We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the time derivative induces a significant change in the behaviour of the solution.

math.AP↗
arXiv2018

Approximation of mild solutions of a semilinear fractional elliptic equation with random noise

We study for the first time the Cauchy problem for semilinear fractional elliptic equation. This paper is concerned with the Gaussian white noise model for the initial Cauchy data. We establish the ill-posedness of the problem. Then, under some assumption on the exact solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. Some convergence rates between the exact solution and the regularized solution is established in $L^2$ and $H^q$ norms.

math.AP↗