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We consider a diffusion process X in a random Lévy potential V which is a solution of the informal stochastic differential equation dX_t=dβ_t-1/2V'(X_t) dt, X_0=0, (β B. M. independent of V). We study the rate of convergence when the diffusion is transient under the assumption that the Lévy process V does not possess positive jumps. We generalize the previous results of Hu--Shi--Yor for drifted Brownian potentials. In particular, we prove a conjecture of Carmona: provided that there existsdoi:10.1214/009117907000000123 fatcat:rj7sbodpbraaldj5xze5nwomlm