Intermittency properties in a hyperbolic Anderson problem
Lower and upper estimates of moments of solutions of the Cauchy problem \(u_{tt}=\Delta u+u\dot F\) on \(\mathbb R^3\) with constant initial conditions \(u(0,x)=u_0>0\), \(u_t(0,x)=\tilde u_0>0\) driven by a spatially homogeneous Wiener process \(\dot F\) with covariance \(\mathbb E\dot F(t,x)\dot F(s,y)=\delta_0(t-s)f(x-y)\) are proved under an assumption of continuity of \(f\) (to derive upper estimates) and a strict positivity at infinity of \(f\) (to derive lower estimates). Consequently, the upper and the lower Lyapunov exponents \(\bar\lambda_n\), \(\underline\lambda_n\) of the solutions defined as limes superior and limes inferior of \(t^{-1}\log\mathbb E|u(t,x)|^n\) as \(t\to\infty\) respectively satisfy \(\bar\lambda_n\leq Cn^{4/3}\) and \(\underline\lambda_{2n}\geq c(2n)^{4/3}\) for every \(n\in\mathbb N\), and thus a kind of intermittency holds for even Lyapunov exponents. If an equation \(u_t=\frac 12\Delta u+u\dot F\) on \(\mathbb R^d\), \(d\geq 1\) with a constant initial condition \(u(0,x)=u_0>0\) is considered instead, analogous upper and lower bounds are proved for moments of the solutions. In this case, however, the Lyapunov exponents \(\lambda_n=\lim_{t\to\infty}t^{-1}\log\mathbb Eu^n(t,x)\) exist and satisfy \(c(2n)^2\leq\lambda_{2n}\leq C(2n)^2\) for \(n\in\mathbb N\) and some \(c>0\), \(C>0\). A Feynman-Kac formula for moments derived by the authors and Roger Tribe is a key ingredient in the proofs.
scientific article; zbMATH DE number 5758875
scientific article