  We construct and analyze a continuum dynamical percolation process which
evolves in a random environment given by a $\gamma$-Liouville measure. The
homogeneous counterpart of this process describes the scaling limit of discrete
dynamical percolation on the rescaled triangular lattice. Our focus here is to
study the same limiting dynamics, but where the speed of microscopic updates is
highly inhomogeneous in space and is driven by the $\gamma$-Liouville measure
associated with a two-dimensional log-correlated field $h$. Roughly speaking,
this continuum percolation process evolves very rapidly where the field $h$ is
high and barely moves where the field $h$ is low. Our main results can be
summarized as follows.
  1. First, we build this inhomogeneous dynamical percolation which we call
$\gamma$-Liouville dynamical percolation (LDP) by taking the scaling limit of
the associated process on the triangular lattice. We work with three different
regimes each requiring different tools: $\gamma\in [0,2-\sqrt{5/2})$,
$\gamma\in [2-\sqrt{5/2}, \sqrt{3/2})$, and $\gamma\in(\sqrt{3/2},2)$.
  2. When $\gamma<\sqrt{3/2}$, we prove that $\gamma$-LDP is mixing in the
Schramm-Smirnov space as $t\to \infty$, quenched in the log-correlated field
$h$. On the contrary, when $\gamma>\sqrt{3/2}$ the process is frozen in time.
The ergodicity result is a crucial piece of the Cardy embedding project of the
second and fourth coauthors, where LDP for $\gamma=\sqrt{1/6}$ is used to study
the scaling limit of a variant of dynamical percolation on uniform
triangulations.
  3. When $\gamma<\sqrt{3/4}$, we obtain quantitative bounds on the mixing of
quad crossing events.
