Let $f\geq 0$ be a Lipschitz function and let $(L\_t)\_{t\geq 0}$ be an $\alpha$-stable Lévy process ($0<\alpha<2$, possibly multivariate). Consider the process given by $$dX\_t=-\nabla f(X\_t)dt+\sigma dL\_t$$ where $\sigma>0$ is a constant. This SDE is reminiscient of Langevin Dynamics, where we usually let the process be driven by Brownian Motion instead of a Lévy process.


I am interested in seeing which results from the "Brownian setting" extend to the setting with $\alpha$-stable Lévy processes as described above.


Most importantly, my main question is: **Does $X\_t$ admit a stationary distribution?**
When the process is driven by Brownian Motion, [it is known that there exists a stationary distribution](https://math.stackexchange.com/questions/4200043/langevin-equation-and-convergence-to-stationary-solutions-free-energy-sde-fpe), for which we can also find a closed form expression for the density.


Example: Ornstein-Uhlenbeck Process
===================================


If we consider the case $f(x)=\frac12 |x|^2$ we get the Ornstein-Uhlenbeck process driven by the Lévy process $L$. There it is known, see e.g. [Topics in Infinitely Divisible Distributions and Lévy Processes](https://link.springer.com/book/10.1007/978-3-030-22700-5) Theorem 2.17, that the process admits a stationary distribution.



Stationary Distribution of Langevin Dynamics driven by Lévy Process
There are several generalizations of the Brownian case results to general L'evy processes. For instance, under a classical log-concave assumption on $U$ we have a convergence to equilibrium in Wassertein distance to a unique stationary distribution. More precisely, consider the following SDE


$$
dX^x\_t=-\nabla U(X^x\_t) dt +dN\_t, \quad X^x\_0=x \in \mathbb R^n
$$
where $N$ is any L'evy process such that $\mathbb{E} \left( \sup\_{t\in [0,T]} | N\_t |^p \right) <+\infty$ for some $p>1$. For stable processes one therefore asks that $\alpha>1$. Denote $P\_t$ the semigroup of $X\_t$.


**Theorem:** *Assume that there exists $a>0$ such that $\nabla^2 U \ge a$ (uniformly
in the sense of quadratic forms). Then, there exists a unique
probability measure $\mu$ in the Wasserstein space
$\mathcal{P}\_p(\mathbb R^n)$ such that for every $t \ge 0$, $\mu P\_t =
\mu$. Moreover, for every $t \ge 0$, and $\nu \in
\mathcal{P}\_p(\mathbb R^n)$ one has,
$$
W\_p ( \nu P\_t, \mu ) \le e^{-at} W\_p ( \nu , \mu ).
$$
Therefore $X\_t$ converges exponentially fast to the invariant distribution in the Wasserstein distance $W\_p$.*


This can be proved using similar arguments as in Theorem 4.9 in the paper [Transport inequalities for Markov kernels and their applications](https://arxiv.org/pdf/2004.02050.pdf) which treated the case $p=2$.


Here are the main steps.


Let $J\_t=\frac{\partial X\_t^x}{\partial x}$ be the first variation process. Let $f$ be a $C^1$ and bounded Lipschitz function.
Since $P\_tf(x)=\mathbb{E}( f(X\_t^x))$, by the chain rule we have
$$
\nabla P\_t f (x)=\mathbb{E}\left( J\_t^\* \nabla f(X\_t^x)\right).
$$
Therefore, by Holder inequality,
$$
| \nabla P\_t f (x) | \le \mathbb{E}\left( | J\_t^\*|^p\right)^{1/p}
\mathbb{E}\left( | \nabla f(X\_t^x) |^q\right)^{1/q}.
$$
where $q$ is the conjugate exponent of $p$.
Observe that
$$
dJ\_t=-\nabla^2 U(X^x\_t) J\_t dt, \quad J\_0=\mathbf{Id}\_{\mathbb R^n}.
$$
From the assumption $\nabla^2 U \ge a$ this yields
$$
| J\_t^\*| \le e^{-at }.
$$
One concludes $\mathbb{E}\left( | J\_t^\*|^p\right) \le e^{-pat }$ and therefore
$$
| \nabla P\_t f (x) | \le e^{-at} P\_t (| \nabla f |^q)(x)^{1/q}.
$$
By Kuwada duality, this yields that for every $\nu\_0,\nu\_1 \in \mathcal{P}\_p(\mathbb
R^n)$,
$$
W\_p ( \nu\_0 P\_t, \nu\_1 P\_t ) \le e^{-at} W\_p ( \nu\_0 , \nu\_1 ).
$$


A fixed point argument in the Wasserstein space allows then to conclude.


