Abstract
We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix $\gamma_Z$ of $Z := (u(s, y), u(t, x) - u(s, y))$, where $u$ is the solution to system of $d$ non-linear stochastic heat equations in spatial dimension $k \geq 1$. We also obtain the optimal exponents for the $L^p$-modulus of continuity of the increments of the solution and of its Malliavin derivatives. These lead to optimal lower bounds on hitting probabilities of the process $\{u(t, x): (t, x) \in [0, \infty[ \times \mathbb{R}\}$ in the non-Gaussian case in terms of Newtonian capacity, and improve a result in Dalang, Khoshnevisan and Nualart [\textit{Stoch PDE: Anal Comp} \textbf{1} (2013) 94--151].
Highlights
Introduction and main resultsConsider the following system of stochastic partial differential equations: ∂∂t ui(t, x) = 2 ∆xui(t, x) +d σij(u(t, x))Fj(t, x) + bi(u(t, x)), j=1 (1.1)for 1 ≤ i ≤ d, t ∈ [0, T ] and x ∈ Rk (k ≥ 1), where u := (u1, . . . , ud) with initial conditions u(0, x) = 0 for all x ∈ Rk, and the ∆x denotes the Laplacian in the spatial variable x
The d-dimensional driving noise Fis white in time and with a spatially homogeneous covariance given by the Riesz kernel f (x) = x −β
The potential theory for u has been developed by Dalang, Khoshnevisan and Nualart [6]
Summary
Consider the following system of stochastic partial differential equations:. for 1 ≤ i ≤ d, t ∈ [0, T ] and x ∈ Rk (k ≥ 1), where u := (u1, . . . , ud) with initial conditions u(0, x) = 0 for all x ∈ Rk, and the ∆x denotes the Laplacian in the spatial variable x. 1 driven by a d-dimensional space-time white noise, this type of question was studied in Dalang, Khoshnevisan and Nualart [4] and [5], in which the lower bounds on hitting probabilities in the Gaussian case (additive noise) and non-Gaussian case (multiplicative noise) are not identical. This discrepancy has been filled recently by Dalang and Pu [7], in which we have obtained the optimal lower bounds on hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension 1.
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