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

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We consider the model of directed polymers in a random environment introduced by Petermann: the random walk is Rᵈ-valued and has independent gaussian N(0,Id)-increments, and the random media is a stationary centred Gaussian process (g(k,x), k ≥ 1, x ∈ Rᵈ) with covariance matrix cov(g(i,x),g(j,y))=δij Γ(x-y), where Γ is a bounded integrable function on Rᵈ. For this model, we establish an upper bound of the volume exponent in all dimensions d.