Abstract
In this paper, we study the parabolic Anderson model of Skorohod type driven by a fractional Gaussian noise in time with Hurst parameter $H \in (0, 1/2)$. By using the Feynman-Kac representation for the $L^p(\Omega)$ moments of the solution, we find the upper and lower bounds for the moments.
Highlights
In this paper, we consider the following parabolic Anderson model of Skorohod type ∂u(t, x) = ∆u(t, x) + u(t, x) W (t, x), ∂t (1.1)where denotes the Wick product
We study the parabolic Anderson model of Skorohod type driven by a fractional Gaussian noise in time with Hurst parameter H ∈ (0, 1/2)
An alternative proof of Theorem 3.3 can be established by the chaos expansion of the solution to the stochastic partial differential equation (SPDE) (1.1) and the hypercontractivity property of fixed Wiener chaos
Summary
We consider the following parabolic Anderson model of Skorohod type. where denotes the Wick product. A similar equation in the Stratonovich sense, where the Wick product in (1.1) is replaced by the ordinary product, has been studied by Hu et al [5] and Chen et al [1] In these papers, it has been proved that under Hypotheses (H1) and (H2), the Stratonovich type equation with bounded initial condition has a unique solution, which admits a Feynman-Kac representation. By using the Feynman-Kac formula for the moments, we see that in comparison with the Stratonovich case, the exponent in our case contains an additional negative term This increases the difficulty to estimate lower bounds for the moments. We estimate the probability of the event that the supremum and the Hölder norm of the Brownian bridge (motion) are bounded above and below by appropriate constants This allows us to find a lower bound for the moments of the solution.
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