We consider a diffusion process X in a random Lévy potential $\mathbb{V}$ which is a solution of the informal stochastic differential equation
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\begin{eqnarray*}\cases{dX_{t}=d\beta_{t}-\frac{1}{2}\mathbb{V}'(X_{t})\,dt,\cr X_{0}=0,}\end{eqnarray*}
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(β B. M. independent of $\mathbb{V}$ ). We study the rate of convergence when the diffusion is transient under the assumption that the Lévy process $\mathbb{V}$ 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 exists 0<κ<1 such that $\mathbf{E}[e^{\kappa\mathbb{V}_{1}}]=1$ , then Xt/tκ converges to some nondegenerate distribution. These results are in a way analogous to those obtained by Kesten–Kozlov–Spitzer for the transient random walk in a random environment.