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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.
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
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 l
Consider the following equation $$partial_t u_t(x)=frac{1}{2}partial _{xx}u_t(x)+lambda sigma(u_t(x))dot{W}(t,,x)$$ on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentiall
In this paper, we establish concentration inequalities both for functionals of the whole solution on an interval [0, T ] of an additive SDE driven by a fractional Brownian motion with Hurst parameter H $in$ (0, 1) and for functionals of discrete-time
It is known that the couple formed by the two dimensional Brownian motion and its Levy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not