Showing papers by "Erkan Nane published in 2019"
TL;DR: In this article, it was shown that there is no non-existence of global random field solutions under some additional conditions, most notably on σ and the initial condition, under the assumption that σ is Gaussian.
22 citations
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TL;DR: In this article, the authors consider a class of conformable time-fractional stochastic equation with noise level and show that the energy of the solution of the non-linear term grows faster than linear, and the energy blow-up at finite time for all α ∈ (0, 1) for all β ∈ 0, 1.
Abstract: Consider the following class of conformable time-fractional stochastic equation $$T_{\alpha,t}^a u(x,t)=\lambda\sigma(u(x,t))\dot{W}_t,\,\,\,\,x\in\mathbb{R},\,t\in[a,\infty), \,\,0 0$ is the noise level. Given some precise and suitable conditions on the non-random initial function, we study the asymptotic behaviour of the solution with respect to the time parameter $t$ and the noise level parameter $\lambda$. We also show that when the non-linear term $\sigma$ grows faster than linear, the energy of the solution blows-up at finite time for all $\alpha\in (0,1)$.
7 citations
TL;DR: In this article, the authors considered non-linear time-fractional stochastic heat type equation and proved the existence and uniqueness of mild solutions to this equation under a linear growth with respect to the time.
Abstract: We consider non-linear time-fractional stochastic heat type equation $$\frac{\partial^\beta u}{\partial t^\beta}+
u(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg]$$ and $$\frac{\partial^\beta u}{\partial t^\beta}+
u(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{N }(t,x,h)\bigg]$$
in $(d+1)$ dimensions, where $\alpha\in (0,2]$ and $d 0$, $\partial^\beta_t$ is the Caputo fractional derivative, $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process, $I^{1-\beta}_t$ is the fractional integral operator, ${N}(t,x)$ are Poisson random measure with $\tilde{N}(t,x)$ being the compensated Poisson random measure. $\sigma:{\mathbb{R}}\to{\mathbb{R}}$ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in "M. Foondun and D. Khoshnevisan. Intermittence and nonlinear parabolic stochastic partial differential equations. \emph{ Electron. J. Probab.} {\bf14} (2009), 548--568" and " J. B. Walsh. An Introduction to Stochastic Partial Differential Equations, Ecoled'ete de Probabilites de Saint-Flour, XIV|1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, (1986), 265--439". Under the linear growth of $\sigma$, we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when $\sigma$ grows faster than linear.
2 citations
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TL;DR: In this article, the authors consider the space-time fractional (nonlocal) equation characterizing the double-scale anomalous diffusion and show that the fractional exponents are determined uniquely by the data $u(t, 0) = g(t), \ 0 < t \leq T.$.
Abstract: In this article, we consider the space-time fractional (nonlocal) equation characterizing the so-called "double-scale" anomalous diffusion $$\partial_t^\beta u(t, x) = -(-\Delta)^{\alpha/2}u(t,x) - (-\Delta)^{\gamma/2}u(t,x) \ \ t> 0, \ -1
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TL;DR: In this paper, a maximum principle theory for the time-changed stochastic control problem was established for the problem of controlling a Levy noise timechanged by an inverse subordinator.
Abstract: This paper studies a time-changed stochastic control problem, where the underlying stochastic process is a Levy 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.