Large Time Behaviour of Moments of Fundamental Solutions of the Random Parabolic Equation

Abstract

Let V(x), x ∈ IR d , be a random ergodic field in IR d (potential) and P ±(t,x,y), t ≥ 0, x, y ∈ IR d be the fundamental solution of the equation $$ \frac{{\partial {P_\pm }}}{{\partial t}} = \Delta {P_\pm }\pm V(x){P_\pm },P|t = 0 = \delta (x - y) $$ (1.1) $$[tex] = - {H^ \pm }{P_ \pm }[/tex]$$ (1.2) where $$[tex]{H^ \pm } = - \Delta \pm V[/tex]$$ (1.3) is the Schrodinger opertator in L 2(IR) with a random potential V(x). Denote by E {...} the mathematical expectation corresponding to V. The simplest probabilistic characteristic of the random functions P ±(t,x,y) is $$[tex]E\left\{ {{P_ \pm }(t,0,0)} \right\} \equiv \tilde N(t)[/tex]$$ (1.4) According to [1], if $$[tex]E\left\{ {{e^{ \pm tV(0)}}} \right\} 0[/tex]$$ (1.5) then expectation (1.4) exists and $$[tex]{\tilde N_ \pm }(t) = \int_{ - \infty }^\infty {{e^{ - \lambda t}}{N_ \pm }(d\lambda )} [/tex]$$ where N± (dλ) is the measure on IR, whose distribution function N(λ = N α((-∞, λ]) is called integrated density of states (IDS) of the operators (1.3). The IDS plays an important role in the spectral theory of the Schrodinger operator with a random or almost periodic potential and the theory of disordered systems (see e.g. [2] and [3] resp. ). IDS can be definded as follows. Consider a family of finitely growing cubes Λ ⊂ IR d with the centers in the origin and sides parallel to the coordinate axes.