# "Convergence speed" results for the Langevin process

The Langevin process is defined by the following stochastic differential equation:

$$\dot X = - \nabla \phi + \sqrt 2 dW_t$$

Its equilibrium distribution is the following:

$$p_\infty (x) \propto \exp( - \phi(x) )$$

(unless I've messed up a constant or two ^^)

Now consider the following: we initialize a particle at position $x_0$ at time $t=0$ and we look at the sequence of probability distributions describing where the particle could be at time $t$

$$x \rightarrow p(x ; x_0, t)$$

This family respects the Fokker-Planck forward equation (and the Kolmogorov backward equation). If I'm not mistaken, (with some small additional assumptions) as $t \rightarrow \infty$, $p(x; x_0, t) \rightarrow p_\infty$.

I'm interested in further characterizations of the family of probability distributions $p(x; x_0, t)$. Can we give total variation bounds on the convergence to $p_\infty$ ? Or any other metric for that matter (I'm particularly interested in the Wasserstein-1 distance) ?

Furtheremore: intuitively, $p(x; x_0 + \epsilon, t)$ and $p(x; x_0, t)$ should be close: are there any known characterizations of that fact ?

Finally, a simple question: does the family $p(x; x_0, t)$ have a name ?

The convergence of the time-dependent probability distribution $p(x;x_0,t)$ to its long-time limit was studied, also in connection to the Wasserstein distance, in On the trend to equilibrium for the Fokker-Planck equation:
• Thank you very much. This answers a lot of my questions. Would you happen to know whether the Brascamp-Lieb inequality (which extends the Poincaré inequality) can also be used to prove convergence speed results of a similar form ? It reads, for $p\propto \exp(-\phi(x))$ a log-concave probability distribution: $var_p(S(x)) \leq E_p( S'(x)^2 / \phi ''(x))$ Jan 4 '16 at 16:08