慢环境中的李雅普诺夫指数
Lyapunov exponents in a slow environment
摘要：
受缓慢变化的随机环境中的种群演化的启发，我们考虑有限体积上的一维Anderson模型，粘性为$ κ> 0 $：$$ \partial_{t} u(t,x) = κΔu(t,x) + ξ(t, x) u(t,x), \quad u(0, x) = u_{0}(x), \qquad t > 0, x \in \mathbb{T}. $$噪声$ ξ$在长度为$ τ>0 $的时间间隔上被选择为常数，并且在时间$ τ$之后独立采样。我们证明了李雅普诺夫指数$ λ(τ) $是积极的和近$ τ= 0 $遵循幂律，依赖于驱动噪声的规律。作为$ τ\to \infty $的李雅普诺夫指数收敛到相关的时间无关的安德森模型的平均顶部特征值。证明利用一个坚实的控制的投影分量的解决方案，并建立在Furstenberg-Khasminskii和Boué-Dupuis公式，以及Doob的H-变换和奇异随机偏微分方程的工具。
英文摘要：
Motivated by the evolution of a population in a slowly varying random environment, we consider the 1D Anderson model on finite volume, with viscosity $ κ> 0 $: $$ \partial_{t} u(t,x) = κΔu(t,x) + ξ(t, x) u(t,x), \quad u(0, x) = u_{0}(x), \qquad t > 0, x \in \mathbb{T}. $$ The noise $ ξ$ is chosen constant on time intervals of length $ τ>0 $ and sampled independently after a time $ τ$. We prove that the Lyapunov exponent $ λ(τ) $ is positive and near $ τ= 0 $ follows a power law that depends on the regularity on the driving noise. As $ τ\to \infty $ the Lyapunov exponent converges to the average top eigenvalue of the associated time-independent Anderson model. The proofs make use of a solid control of the projective component of the solution and build on the Furstenberg--Khasminskii and Boué--Dupuis formulas, as well as on Doob's H-transform and on tools from singular stochastic PDEs.
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