These notes are based on the tutorial I gave at the Geometric Methods in Optimization and Sampling Boot Camp at the Simons Institute in Berkeley.

Suppose we wish to obtain samples from some probability measure on . If has a sufficiently well-behaved density with respect to the Lebesgue measure, i.e., , then we can use the (overdamped) continuous-time Langevin dynamics, governed by the Ito stochastic differential equation (SDE)

where the initial condition is generated according to some probability law , and is the standard -dimensional Brownian motion. Let denote the probability law of . Then, under appropriate regularity conditions on , one can establish the following:

- is the unique invariant distribution of (1), i.e., if , then for all .
- converges to in a suitable sense as — in fact, it is often possible to show that there exists a constant that depends only on , such that one has the exponential convergence to equilibrium
for some distance between probability measures on .

In this sense, the Langevin process (1) gives only *approximate samples* from . I would like to discuss an alternative approach that uses diffusion processes to obtain *exact* samples in *finite* time. This approach is based on ideas that appeared in two papers from the 1930s by Erwin Schrödinger in the context of physics, and is now referred to as the *Schrödinger bridge problem*.

Continue reading “Sampling Using Diffusion Processes, from Langevin to Schrödinger”