  Enriching Brownian motion with regenerations from a fixed regeneration
distribution $\mu$ at a particular regeneration rate $\kappa$ results in a
Markov process that has a target distribution $\pi$ as its invariant
distribution. For the purpose of Monte Carlo inference, implementing such a
scheme requires firstly selection of regeneration distribution $\mu$, and
secondly computation of a specific constant $C$. Both of these tasks can be
very difficult in practice for good performance. We introduce a method for
adapting the regeneration distribution, by adding point masses to it. This
allows the process to be simulated with as few regenerations as possible and
obviates the need to find said constant $C$. Moreover, the choice of fixed
$\mu$ is replaced with the choice of the initial regeneration distribution,
which is considerably less difficult. We establish convergence of this
resulting self-reinforcing process and explore its effectiveness at sampling
from a number of target distributions. The examples show that adapting the
regeneration distribution guards against poor choices of fixed regeneration
distribution and can reduce the error of Monte Carlo estimates of expectations
of interest, especially when $\pi$ is skewed.
