  A diffusion operator on the $K$-rational points of a Tate elliptic curve
$E_q$ is constructed, where $K$ is a non-archimedean local field, as well as an
operator on the Berkovich-analytification $E_q^{an}$ of $E_q$. These are
integral operators for measures coming from a regular $1$-form, and kernel
functions constructed via theta functions. The second operator can be described
via certain non-archimedan curvature forms on $E_q^{an}$. The spectra of these
self-adjoint bounded operators on the Hilbert spaces of $L^2$-functions are
identical and found to consist of finitely many eigenvalues. A study of the
corresponding heat equations yields a positive answer to the Cauchy problem,
and induced Markov processes on the curve. Finally, some geometric information
about the $K$-rational points of $E_q$ is retrieved from the spectrum.
