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 ?