Related papers: Stochastic fractional diffusion equations with Gau…
In this work we study the one-dimensional stochastic Kimura equation $\partial_{t}u\left(z,t\right)=z\partial_{z}^{2}u\left(z,t\right)+u\left(z,t\right)\dot{W}\left(z,t\right)$ for $z,t>0$ equipped with a Dirichlet boundary condition at…
We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index $H>1/2$, and has a homogeneous spatial covariance structure given by the Riesz kernel of order $\alpha$. The solution is interpreted…
We investigate the Cauchy problem for a semilinear spatio--temporal fractional diffusion equation with a time-dependent forcing term: \[ \partial_t^\alpha u + (-\Delta)^{\mathsf{s}} u = |u|^p + t^{\sigma}\,\mathbf{w}(x), \quad (t,x) \in…
We present uniqueness and existence in weighted Sobolev spaces of the equation $$ u_t=(au_{xx}+bu_x+cu)+ \xi |u|^{1+\lambda} {\dot{B}}, \quad\,\, t>0, \, x\in (0,1) $$ with initial data $u(0,\cdot)=u_0$ and zero boundary data. Here…
In this article, we consider the stochastic wave equation on the real line driven by a linear multiplicative Gaussian noise, which is white in time and whose spatial correlation corresponds to that of a fractional Brownian motion with Hurst…
We pursue the investigations initiated in [Aur{\'e}lien Deya: A non-linear wave equation with fractional perturbation (2017)] about a wave-equation model with quadratic perturbation and stochastic forcing given by a space-time fractional…
We investigate the fractional Hardy-H\'enon equation with fractional Brownian noise $$ \partial_tu(t)+(-\Delta)^{\theta/2} u(t)=|x|^{-\gamma} |u(t)|^{p-1}u(t)+\mu \, \partial_t B^H(t), $$ where $\theta>0$, $p>1$, $\gamma\geq 0$, $\mu…
We consider a reaction-diffusion equation of the type \[ \partial_t\psi = \partial^2_x\psi + V(\psi) + \lambda\sigma(\psi)\dot{W} \qquad\text{on $(0\,,\infty)\times\mathbb{T}$}, \] subject to a "nice" initial value and periodic boundary,…
In this paper Gaussian models of retarded and accelerated anomalous diffusion are considered. Stochastic differential equations of fractional order driven by single or multiple fractional Gaussian noise terms are introduced to describe…
This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation,…
We consider the quasi-linear stochastic wave and heat equations in $\mathbb{R}^d$ with $d\in \{1,2,3\}$ and $d\geq 1$, respectively, and perturbed by an additive Gaussian noise which is white in time and has a homogeneous spatial…
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…
In this paper, a fractional generalization of the wave equation that describes propagation of damped waves is considered. In contrast to the fractional diffusion-wave equation, the fractional wave equation contains fractional derivatives of…
Fractional (in time and in space) evolution equations defined on Dirichlet regular bounded open domains, driven by fractional integrated in time Gaussian spatiotemporal white noise, are considered here. Sufficient conditions for the…
For the stochastic partial differential equation $\frac{\partial u}{\partial t}=\mathcal L u +u\dot W$ where $\dot W$ is Gaussian noise colored in time and $\mathcal L$ is the infinitesimal generator of a Feller process $X$, we obtain…
We study the solution to a nonlinear stochastic heat equation in $d\geq 3$. The equation is driven by a Gaussian multiplicative noise that is white in time and smooth in space. For a small coupling constant, we prove (i) the solution…
Consider the stochastic partial differential equation $\partial_t u = Lu+\sigma(u)\xi$, where $\xi$ denotes space-time white noise and $L:=-(-\Delta)^{\alpha/2}$ denotes the fractional Laplace operator of index…
In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the stochastic heat equation with multiplicative noise and in any space dimension. The driving perturbation is a Gaussian noise…
In this paper the solutions $u_{\nu}=u_{\nu}(x,t)$ to fractional diffusion equations of order $0<\nu \leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations…
In this article, we study the stochastic wave equation on the entire space $\mathbb{R}^d$, driven by a space-time L\'evy white noise with possibly infinite variance (such as the $\alpha$-stable L\'evy noise). In this equation, the noise is…