Theory of Probability and Mathematical Statistics
Temporal properties of the stochastic fractional heat equation with spatially-colored noise
Ran Wang and Yimin Xiao
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Abstract: Consider the stochastic partial differential equation ∂ut(x)/∂t=-(-Δ)α/2ut(x)+b(ut(x))+σ(ut(x))+\dot{F}(t,x), t≥0, x∈Rd, where -(-Δ)α/2 denotes the fractional Laplacian with power α/2∈(1/2,1], and the driving noise \dot{F} is a centered Gaussian field which is white in time and has a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation of the temporal gradient ut+ε(x)-ut(x) at any fixed t>0 and x∈Rd, as ε↓0. As applications, we deduce Khintchin’s law of iterated logarithm, Chung’s law of iterated logarithm, and a result on the q-variations of the temporal process {ut(x)}t≥0 of the solution, where x∈Rd is fixed.
Keywords: Stochastic heat equation, fractional Brownian motion, fractional Laplacian, law of iterated logarithm, q-variation
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