A Schrödinger bridge is a probability measure on paths that has prescribed initial and final marginal
distributions and is closest in relative entropy
to a given reference path measure. If 
is the reference measure and 
are the required endpoint distributions, the Schrödinger bridge
is the solution of
 |
where 
is the relative entropy of 
with respect to 
.
When the reference measure is the law of a Markov process, the minimizing measure is also the law of a Markov process and, under suitable regularity conditions, has the form
 |
for nonnegative endpoint factors 
and 
. The intermediate marginal
distributions form an entropic interpolation between 
and 
. When the endpoint distributions
are point masses, the construction reduces to a Markov
bridge; for a Brownian reference process it includes the Brownian
bridge as a special case.
The problem was introduced by Schrödinger (1931) in connection with the unlikely evolution of a large collection of independent particles between two observed distributions. It is also closely connected with control theory and with variational problems for transporting probability distributions (Léonard 2014).
