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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 ?

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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:

We present connections between the problem of trend to equilibrium for the Fokker-Planck equation of statistical physics, and several inequalities from functional analysis, like logarithmic Sobolev or Poincare inequalities, together with some inequalities arising in the context of concentration of measures (Talagrand inequality and Wasserstein distance), or in the study of Gaussian isoperimetry.

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  • $\begingroup$ 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)) $ $\endgroup$ Jan 4 '16 at 16:08

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