Quenched critical large deviations for Brownian motion in a Poissonian potential
The author considers a \(d\)-dimensional Brownian motion \(\{Z_t \}_{t \geq 0}\) moving in a random potential \(V\). The potential \(V(x, \omega)\equiv (W* \omega) (x)\), \(x\in \mathbb{R}^d\), consists of a bounded, nonnegative, compactly supported shape function \(W\) convoluted with the cloud configuration \(\omega= \sum_i \delta_{x_i}\) of a Poisson point process of constant intensity. For every \(t\geq 0\) and configuration \(\omega\), a weighted path measure \(Q_{t, \omega}\) is defined by introducing a weight proportional to \(\exp(- \int^t_0 V(Z_s, \omega)ds)\) for every Brownian path. Building on his previous work, the author proves for almost all cloud configurations \(\omega\), that the measures \(\{Q_{t, \omega} Y_t^{-1} \}_{t> 0}\) satisfy a large deviation principle on \(\mathbb{R}^d\) with scale \(t(\log t)^{-2/ d}\) as \(t\to \infty\), where \(Y_t \equiv (1/t) (\log t)^{2/d} Z_t\) is the correspondingly scaled position of the Brownian motion at time \(t\). This result is applied to determine the off-diagonal asymptotic behaviour of the Schrödinger heat kernel and the quenched large deviations of Brownian motion with drift in the above random potential \(V\).
scientific article; zbMATH DE number 847458