Rates of convergence of a transient diffusion in a spectrally negative L\'{e}vy potential

We consider a diffusion process $X$ in a random L\'{e}vy potential $\mathbb{V}$ which is a solution of the informal stochastic differential equation \begin{eqnarray*}\cases{dX_t=d\beta_t-{1/2}\mathbb{V}'(X_t) dt,\cr X_0=0,}\end{eqnarray*} ($\beta$ B. M. independent of $\mathbb{V}$). We study the rate of convergence when the diffusion is transient under the assumption that the L\'{e}vy 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 $0<\kappa<1$ such that $\mathbf{E}[e^{\kappa\mathbb{V}_1}]=1$, then $X_t/t^{\kappa}$ 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.