Upper bound of a volume exponent for directed polymers in a random environment

We consider the model of directed polymers in a random environment introduced by Petermann: the random walk is R-d-valued and has independent N(0, I-d)-increments, and the random media is a stationary centered Gaussian process (g(k. x). k greater than or equal to 1, x is an element of R-d) with covariance matrix cov(g(i, x), g(j, y)) = delta(ij) Gamma(x - y), where Gamma is a bounded integrable function on Rd. For this model, we establish an upper bound of the volume exponent in all dimensions d. (C) 2004 Elsevier SAS. All rights reserved.