Kavli Affiliate: Ran Wang
| First 5 Authors: Ran Wang, , , ,
| Summary:
Consider the stochastic partial differential equation $$
frac{partial }{partial t}u_t(mathbf{x})=
-(-Delta)^{frac{alpha}{2}}u_t(mathbf{x})
+bleft(u_t(mathbf{x})right)+sigmaleft(u_t(mathbf{x})right) dot F(t,
mathbf{x}), tge0, mathbf{x}in mathbb R^d, $$
where $-(-Delta)^{frac{alpha}{2}}$ denotes the fractional Laplacian with
the power $alpha/2in (1/2,1]$, and the driving noise $dot F$ is a centered
Gaussian field which is white in time and with a spatial homogeneous covariance
given by the Riesz kernel. We study the detailed behavior of the approximation
spatial gradient $u_t(mathbf{x})-u_t(mathbf{x}-varepsilon mathbf e)$ at any
fixed time $t>0$, as $varepsilondownarrow 0$, where $mathbf e$ is the unit
vector in $mathbb R^d$. As applications, we deduce the law of iterated
logarithm and the behavior of the $q$-variations of the solution in space.
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