×

Critical large deviations of one-dimensional annealed Brownian motion in a Poissonian potential. (English) Zbl 0911.60014

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.

MSC:

60F10 Large deviations
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bolthausen, E. (1994). Localization of a two dimensional random walk with an attractive path interaction. Ann. Probab. 22 875-918. · Zbl 0819.60028
[2] Deuschel, J. D. and Stroock, D. W. (1989). Large Deviations. Academic Press, Boston. · Zbl 0675.60086
[3] Donsker, M. D. and Varadhan, S. R. S. (1975). Asymptotics for the Wiener sausage. Comm. Pure Appl. Math. 28 525-565. · Zbl 0351.60070
[4] Eisele, T. and Lang, R. (1987). Asymptotics for the Wiener sausage with drift. Probab. Theory Related Fields 74 125-140. · Zbl 0586.60076
[5] Grassberger, P. and Procaccia, I. (1982). Diffusion and drift in a medium with randomly distributed traps. Phys. Rev. A 26 3686-3688.
[6] Povel, T. (1995). On weak convergence of conditional survival measure of one dimensional Brownian motion with drift. Ann. Appl. Probab. 5 222-238. · Zbl 0822.60094
[7] Schmock, U. (1990). Convergence of the normalized one dimensional Wiener sausage path measures to a mixture of Brownian taboo processes. Stochastics Stochastics Rep. 29 171-183. · Zbl 0697.60032
[8] Sznitman, A. S. (1991). On long excursions of Brownian motion among Poisonian obstacles. In Stochastic Analysis (M. Barlow and N. Bingham, eds.) 353-375. Cambridge Univ. Press. · Zbl 0760.60027
[9] Sznitman, A. S. (1991). On the confinement property of two dimensional Brownian motion among Poissonian obstacles. Comm. Pure Appl. Math. 44 1137-1170. · Zbl 0753.60075
[10] Sznitman, A. S. (1993). Brownian motion in a Poisson potential. Probab. Theory Related Fields 97 447-477. · Zbl 0794.60110
[11] Sznitman, A. S. (1994). Brownian motion and obstacles. In First European Congress of Mathematics (A. Joseph, F. Mignot, F. Murat, B. Prum and R. Rentschler, eds.) 225-248. Birkhäuser, Basel. · Zbl 0815.60077
[12] Sznitman, A. S. (1995). Annealed Lyapounov exponents and large deviations in a Poissonian potential I, II. Ann. Sci. École Norm. Sup. 4 28 345-390. · Zbl 0826.60018
[13] Sznitman, A. S. (1997). Capacity and principal eigenvalues: the method of enlargement of obstacles revisited. Ann. Probab. 25 1180-1209. · Zbl 0885.60063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.