Lacoin, Hubert Superdiffusivity for Brownian motion in a Poissonian potential with long range correlation. II: Upper bound on the volume exponent. (English. French summary) Zbl 1267.82147 Ann. Inst. Henri Poincaré, Probab. Stat. 48, No. 4, 1029-1048 (2012). Typical trajectories of the Brownian motion are those killed with a homogeneous rate and conditioned to survive till hitting a distant hyperplane. They stay in a tube centered along the direction orthogonal to this hyperplane of diameter \(\sqrt{L}\), whereby \(L\) is the distance between the source of the trajectories and the hyperplane. Adding an inhomogeneous term to the killing rate makes the transversal fluctuation of the trajectories superdiffusive, e.g., of amplitude \(L^{\xi }\) for some \(\xi \in (1/2, 1)\) (\(\xi \) is called the volume exponent). The pertinent inhomogeneity stems from a random (Poissonian) potential \(V\) representing a field of traps whose center locations are given by a Poisson point process, while their radii are distributed according to a common (heavy-tailed) distribution of an unbounded support. The presence of long-range spatial correlations is a crucial difference when compared with Poissonian models studied so far, c.f. a summary of that topic in [A. S. Sznitman, Brownian motion, obstacles and random media. Berlin: Springer (1998; Zbl 0973.60003)].The paper is a direct sequel to [ibid. 48, No. 4, 1010–1028 (2012; Zbl 1267.82146)], where a lower bound on the so-called volume exponent \(\xi \) for the superdiffusive case was established. The present analysis is focused on identifying an upper bound for the volume exponent \(\xi \). Reviewer: Piotr Garbaczewski (Opole) Cited in 2 Documents MSC: 82D60 Statistical mechanics of polymers 60K37 Processes in random environments 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics Keywords:Stretched polymer; quenched disorder; superdiffusivity; Brownian motion; Poissonian obstacles; correlation Citations:Zbl 0973.60003; Zbl 1267.82146 PDFBibTeX XMLCite \textit{H. Lacoin}, Ann. Inst. Henri Poincaré, Probab. Stat. 48, No. 4, 1029--1048 (2012; Zbl 1267.82147) Full Text: DOI arXiv Euclid References: [1] M. Balasz, J. Quastel and T. Seppäläinen. Fluctuation exponent of the KPZ/stochastic Burgers equation. J. Amer. Math. Soc. 24 (2011) 683-708. · Zbl 1227.60083 [2] K. Johansson. Transversal fluctuation for increasing subsequences on the plane. Probab. Theory Related Fields 116 (2000) 445-456. · Zbl 0960.60097 [3] H. Kesten. On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 (1993) 296-338. · Zbl 0783.60103 [4] H. Lacoin. Influence of spatial correlation for directed polymers. Ann. Probab. 39 (2011) 139-175. · Zbl 1208.82084 [5] H. Lacoin. Superdiffusivity for Brownian motion in a Poissonian potential with long range correlation I: Lower bound on the volume exponent. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 1010-1028. · Zbl 1267.82146 [6] O. Méjane. Upper bound of a volume exponent for directed polymers in a random environment. Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 299-308. · Zbl 1041.60079 [7] T. Seppäläinen. Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40 (2012) 19-73. · Zbl 1254.60098 [8] A. S. Sznitman. Shape theorem, Lyapounov exponents and large deviation for Brownian motion in Poissonian potential. Comm. Pure Appl. Math. 47 (1994) 1655-1688. · Zbl 0814.60022 [9] A. S. Sznitman. Distance fluctuations and Lyapounov exponents. Ann. Probab. 24 (1996) 1507-1530. · Zbl 0871.60088 [10] A. S. Sznitman. Brownian Motion, Obstacles and Random Media. Springer Monographs in Mathematics . Springer, Berlin, 1998. · Zbl 0973.60003 [11] M. Wütrich. Superdiffusive behavior of two-dimensional Brownian motion in a Poissonian potential. Ann. Probab. 26 (1998) 1000-1015. · Zbl 0935.60099 [12] M. Wütrich. Fluctuation results for Brownian motion in a Poissonian potential. Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 279-308. · Zbl 0909.60073 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.