  We consider the behavior of the Gaussian concentration bound (GCB) under
stochastic time evolution. More precisely, we consider a Markovian diffusion
process on $\mathbb{R}^d$ and start the process from an initial distribution
$\mu$ that satisfies GCB. We then study the question whether GCB is preserved
under the time-evolution, and if yes, how the constant behaves as a function of
time. In particular, if for the constant we obtain a uniform bound, then we can
also conclude properties of the stationary measure(s) of the diffusion process.
This question, as well as the methodology developed in the paper allows to
prove Gaussian concentration via semigroup interpolation method, for measures
which are not available in explicit form.
  We provide examples of conservation of GCB, loss of GCB in finite time, and
loss of GCB at infinity. We also consider diffusions ``coming down from
infinity'' for which we show that, from any starting measure, at positive
times, GCB holds. Finally we consider a simple class of non-Markovian diffusion
processes with drift of Ornstein-Uhlenbeck type, and general bounded
predictable variance.
