Relating the almost-sure Lyapunov exponent of a parabolic SPDE and its coefficients' spatial regularity
The paper deals with lower bound on the large-time exponential rate of increasing of the solution to the stochastic parabolic equation with linear multiplicative potential: \[ \frac{\partial u}{\partial t}(t,x)=\frac{k}{2} \frac{\partial^2 u}{\partial x^2}(t,x)+u(t,x)\frac{\partial }{\partial t}W(t,x),\quad u(0,x)=u_0(x), \] where \(x\in \mathbb{R},~t \in \mathbb{R}^+\), \(k\) is a small positive constant, \(u_0\) is a function bounded below, \(W\) is a centered spatial homogeneous Gaussian random field on \(\mathbb{R}^+\times \mathbb{R}\) with covariance structure given by \( E[W(t,x)W(s,y)]=\min(t,s)Q(x-y). \) It is supposed that \(Q\) is precisely \(\alpha\)-Hölder-continuous at 0. In the main result, the authors prove that there exist constants \(C_\alpha>0\), \(k_0>0\) that depend only on \(Q\), such that for \(k<k_0\), and \(\varepsilon\) defined by \(k=\varepsilon^{\alpha+2}(\log \varepsilon^{-1})^{-2}\), almost surely \[ \liminf_{t\to \infty}\frac{1}{t}\log u(t,x)\geq C_\alpha \frac{\varepsilon^{\alpha}}{\log \varepsilon^{-1} }. \]
parabolic stochastic differential equations
scientific article; zbMATH DE number 2162531