  A self-adjoint operator is constructed on the $L_2$-functions on the
$K$-rational points $X(K)$ of a Mumford curve $X$ defined over a
non-archimedean local field $K$. It generates a Feller semi-group, and the
corresponding heat equation describes a Markov process on $X(K)$. Its spectrum
is non-positive, contains zero and has finitely many limit points which are the
only non-eigenvalues, and correspond to the zeros of a given regular
differential 1-form on $X(K)$. This allows to recover the genus of X from the
spectrum. The hyperelliptic case allows in principle an explicit genus
extraction.
