A Slow Transient Diffusion in a Drifted Stable Potential

We consider a diffusion process X in a random potential $${\mathbb{V}}$$ of the form $${\mathbb{V}_x = \mathbb{S}_x -\delta x}$$ , where $$\delta$$ is a positive drift and $$\mathbb{S}$$ is a strictly stable process of index $$\alpha\in (1,2)$$ with positive jumps. Then the diffusion is transient and $$X_t / \log^\alpha t$$ converges in law towards an exponential distribution. This behaviour contrasts with the case where $${\mathbb{V}}$$ is a drifted Brownian motion and provides an example of a transient diffusion in a random potential which is as “slow” as in the recurrent setting.