Critical large deviations of one-dimensional annealed Brownian motion in a Poissonian potential
The aim of this interesting and deep article is to show a large deviation principle for \(t^{-1/3}Z_t\), where \(Z_t\) is the position at time \(t\) of an annealed one-dimensional Brownian motion moving in a soft repulsive Poissonian potential. This last process can be defined as follows. Let \(Z_.\) be the canonical Brownian motion and \({\mathbb{P}}\) the law of a Poisson point process of constant density \(\nu>0\), on the space \(\Omega\) of simple point measures on \({\mathbb{R}}\) denoting their elements by \(\omega\). The annealed weighted measure is \[ Q_t=\frac{1}{S_t} \exp \Biggl\{-\int_0^t V(Z_s,\omega)ds\Biggr\} P_0(dw){\mathbb{P}}(d\omega) \] where \(S_t\) is the normalizing constant and \(V\) is the Poissonian potential. The main result in this work is that under \(Q_t\), \(t^{-1/3}Z_t\) obeys a large deviation principle at rate \(t^{1/3}\) and with rate function \(J_1(.)\), that is \[ \limsup_{t\rightarrow\infty}t^{-1/3}\log Q_t(Z_t\in t^{1/3} A)\leq -\inf_{y\in A}J_1(A) \] where \(A \subset {\mathbb{R}}\) is closed, a dual inequality is also verified for \(\Theta\) open. The author defines the rate function in Theorem 1 and in its definition, among other functions and constants, appears the annealed Lyapunov exponent. This result should be compared with the case \(d\geq 2\) proved by Sznitman, where \(t^{-d/(d+2)}Z_t\) satisfies also a large deviation at rate \(t^{d/(d+2)}\) with rate function \(J_d(y)=\beta_0(y)\), \( y\in \mathbb{R}^d\), where \(\beta_0(y)\) denotes the \(d\)-dimensional annealed Lyapunov exponent. In the present paper \(d=1\), the rate function is more involved and it can be considered as singular. It is defined piecewise in three different regions. The article also provides an application of this large deviation principle to the long time behavior of one-dimensional annealed Brownian motion with constant drift \(h\), a model studied in the physical literature.
scientific article; zbMATH DE number 1130484