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Register a new userSpatial asymptotics and strong comparison principle for some fractional stochastic heat equations
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Consider the following stochastic heat equation, begin{align*} frac{partial u_t(x)}{partial t}=- u(-Delta)^{alpha/2} u_t(x)+sigma(u_t(x))dot{F}(t,,x), quad t>0, ; x in R^d. end{align*} Here $- u(-Delta)^{alpha/2}$ is the fractional Laplacian with $ u>0$ and $alpha in (0,2]$, $sigma: Rrightarrow R$ is a globally Lipschitz function, and $dot{F}(t,,x)$ is a Gaussian noise which is white in time and colored in space. Under some suitable additional conditions, we prove a strong comparison theorem and explore the effect of the initial data on the spatial asymptotic properties of the solution. This constitutes an important extension over a series of recent works.
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In this paper, we obtain upper and lower bounds for the moments of the solution to a class of fractional stochastic heat equations on the ball driven by a Gaussian noise which is white in time, and with a spatial correlation in space of Riesz kernel type. We also consider the space-time white noise case on an interval.
In this note we consider the parabolic Anderson model in one dimension with time-independent fractional noise $dot{W}$ in space. We consider the case $H<frac{1}{2}$ and get existence and uniqueness of solution. In order to find the quenched asymptotics for the solution we consider its Feynman-Kac representation and explore the asymptotics of the principal eigenvalue for a random operator of the form $frac{1}{2} Delta + dot{W}$.
This paper is devoted to proving the strong averaging principle for slow-fast stochastic partial differential equations with locally monotone coefficients, where the slow component is a stochastic partial differential equations with locally monotone coefficients and the fast component is a stochastic partial differential equations (SPDEs) with strongly monotone coefficients. The result is applicable to a large class of examples, such as the stochastic porous medium equation, the stochastic $p$-Laplace equation, the stochastic Burgers type equation and the stochastic 2D Navier-Stokes equation, which are the nonlinear stochastic partial differential equations. The main techniques are based on time discretization and the variational approach to SPDEs.
We consider stochastic heat equations with fractional Laplacian on $mathbb{R}^d$. Here, the driving noise is generalized Gaussian which is white in time but spatially homogenous and the spatial covariance is given by the Riesz kernels. We study the large-scale structure of the tall peaks for (i) the linear stochastic heat equation and (ii) the parabolic Anderson model. We obtain the largest order of the tall peaks and compute the macroscopic Hausdorff dimensions of the tall peaks for both (i) and (ii). These results imply that both (i) and (ii) exhibit multi-fractal behavior in a macroscopic scale even though (i) is not intermittent and (ii) is intermittent. This is an extension of a recent result by Khoshnevisan et al to a wider class of stochastic heat equations.
The averaging principle is established for the slow component and the fast component being two dimensional stochastic Navier-Stokes equations and stochastic reaction-diffusion equations, respectively. The classical Khasminskii approach based on time discretization is used for the proof of the slow component strong convergence to the solution of the corresponding averaged equation under some suitable conditions. Meanwhile, some powerful techniques are used to overcome the difficulties caused by the nonlinear term and to release the regularity of the initial value.
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