Temporal asymptotics for fractional parabolic Anderson model
نویسندگان
چکیده
In this paper, we consider fractional parabolic equation of the form ∂u ∂t = −(−∆) α 2 u+ uẆ (t, x), where −(−∆)α2 with α ∈ (0, 2] is a fractional Laplacian and Ẇ is a Gaussian noise colored in space and time. The precise moment Lyapunov exponents for the Stratonovich solution and the Skorohod solution are obtained by using a variational inequality and a Feynman-Kac type large deviation result for space-time Hamiltonians driven by α-stable process. As a byproduct, we obtain the critical values for θ and η such that E exp ( θ (∫ 1 0 ∫ 1 0 |r − s|0γ(Xr −Xs)drds )η) is finite, where X is d-dimensional symmetric α-stable process and γ(x) is |x|−β or dj=1 |xj|j .
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