Kavli Affiliate: Ran Wang
| First 5 Authors: Yuhui Guo, Jian Song, Ran Wang, Yimin Xiao,
| Summary:
We consider the following stochastic space-time fractional diffusion equation
with vanishing initial condition:$$ partial^{beta} u(t, x)=-
left(-Deltaright)^{alpha / 2} u(t, x)+ I_{0+}^{gamma}left[dot{W}(t,
x)right],quad tin[0,T],: x in mathbb{R}^d,$$ where $alpha>0$,
$betain(0,2)$, $gammain[0,1)$, $left(-Deltaright)^{alpha/2}$ is the
fractional/power of Laplacian and $dot{W}$ is a fractional space-time Gaussian
noise. We prove the existence and uniqueness of the solution and then focus on
various sample path regularity properties of the solution. More specifically,
we establish the exact uniform and local moduli of continuity and Chung-type
laws of the iterated logarithm. The small ball probability is also studied.
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