Randall Herrell

Bio: Randall Herrell is an academic researcher from University of Alabama in Huntsville. The author has contributed to research in topics: Gaussian & Heat equation. The author has an hindex of 1, co-authored 2 publications receiving 8 citations.

Sharp space-time regularity of the solution to stochastic heat equation driven by fractional-colored noise

Abstract: In this article, we study the following stochastic heat equation ∂tu(t,x)=Lu(t,x)+Ḃ, u(0,x)=0, 0≤t≤T, x∈Rd, where L is the generator of a Levy process X in Rd, B is a fractional-colored Gaussian n...

Sharp Space-Time Regularity of the Solution to Stochastic Heat Equation Driven by Fractional-Colored Noise

Abstract: In this paper, we study the following stochastic heat equation \[ \partial_tu=\mathcal{L} u(t,x)+\dot{B},\quad u(0,x)=0,\quad 0\le t\le T,\quad x\in\mathbb{R}d, \] where $\mathcal{L}$ is the generator of a Levy process $X$ taking value in $\mathbb{R}^d$, $B$ is a fractional-colored Gaussian noise with Hurst index $H\in\left(\frac12,\,1\right)$ for the time variable and spatial covariance function $f$ which is the Fourier transform of a tempered measure $\mu.$ After establishing the existence of solution for the stochastic heat equation, we study the regularity of the solution $\{u(t,x),\, t\ge 0,\, x\in\mathbb{R}^d\}$ in both time and space variables. Under mild conditions, we give the exact uniform modulus of continuity and a Chung-type law of iterated logarithm for the sample function $(t,x)\mapsto u(t,x)$. Our results generalize and strengthen the corresponding results of Balan and Tudor (2008) and Tudor and Xiao (2017).

To fixate or not to fixate in two-type annihilating branching random walks

Abstract: We study a model of competition between two types evolving as branching random walks on $\mathbb{Z}^d$. The two types are represented by red and blue balls respectively, with the rule that balls of different colour annihilate upon contact. We consider initial configurations in which the sites of $\mathbb{Z}^d$ contain one ball each, which are independently coloured red with probability $p$ and blue otherwise. We address the question of \emph{fixation}, referring to the sites eventually settling for a given colour, or not. Under a mild moment condition on the branching rule, we prove that the process will fixate almost surely for $p eq 1/2$, and that every site will change colour infinitely often almost surely for the balanced initial condition $p=1/2$.

Polarity of almost all points for systems of non-linear stochastic heat equations in the critical dimension.

Abstract: We study vector-valued solutions $u(t,x)\in\mathbb{R}^d$ to systems of nonlinear stochastic heat equations with multiplicative noise: \begin{equation*} \frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)+\sigma(u(t,x))\dot{W}(t,x). \end{equation*} Here $t\geq 0$, $x\in\mathbb{R}$ and $\dot{W}(t,x)$ is an $\mathbb{R}^d$-valued space-time white noise. We say that a point $z\in\mathbb{R}^d$ is polar if \begin{equation*} P\{u(t,x)=z\text{ for some $t>0$ and $x\in\mathbb{R}$}\}=0. \end{equation*} We show that in the critical dimension $d=6$, almost all points in $\mathbb{R}^d$ are polar.

Local nondeterminism and local times of the stochastic wave equation driven by fractional-colored noise

Abstract: We investigate the existence and regularity of the local times of the solution to a linear system of stochastic wave equations driven by a Gaussian noise that is fractional in time and colored in space. Using Fourier analytic methods, we establish strong local nondeterminism properties of the solution and the existence of jointly continuous local times. We also study the differentiability and moduli of continuity of the local times and deduce some sample path properties of the solution.

Local Nondeterminism and Local Times of the Stochastic Wave Equation Driven by Fractional-Colored Noise

Abstract: Abstract We investigate the existence and regularity of the local times of the solution to a linear system of stochastic wave equations driven by a Gaussian noise that is fractional in time and colored in space. Using Fourier analytic methods, we establish strong local nondeterminism of the solution and the existence of jointly continuous local times. We also study the differentiability and moduli of continuity of the local times and deduce some sample path properties of the solution.

Hölder continuity of mild solutions of space-time fractional stochastic heat equation driven by colored noise

Abstract: An initial value problem for space-time fractional stochastic heat equations driven by colored noise $$\partial _t^s u_t + \frac{ u }{2} (- \Delta )^{\alpha /2} u_t = \sigma (u_t) \dot{W}$$ has been discussed in this work. Here, $$\partial _t^s$$ and $$(- \Delta )^{\alpha /2}$$ stand for the Caputo’s fractional derivative of order $$s\in (0,1)$$ and the fractional Laplace operator of order $$\alpha \in (0,2)$$ , where the second one is also the generator of a strict stable process $$X_t$$ of order $$\alpha $$ with $$E e^{i\xi \cdot X_t}=e^{-t|\xi |^\alpha }$$ . The nonlinearity $$\sigma $$ is assumed to be Lipschitz continuous. A formulation of mild random field solutions is obtained due to the called space-time fractional Green functions G, H, where H contains a singular kernel. We focus on studying the spatially–temporally Holder continuity of mild random field solutions, which can be obtained by constructing relevant moment bounds for increments of the convolutions $$ H \otimes b(u) $$ and $$ H \circledast \sigma (u)$$ . Our techniques are based on connecting the space-time fractional Green functions G, H to the fundamental solution of $$\partial _t P_t + (-\Delta )^{\alpha /2}P_t = 0$$ , $$P_0=\delta _0$$ via the Wright-type function $${\mathcal {M}}_s$$ .