Theta-induced diffusion on Tate elliptic curves over nonarchimedean local fields

A diffusion operator on the K-rational points of a Tate elliptic curve Eq is constructed, where K is a nonarchimedean local field, as well as an operator on the Berkovich analytification Eanq of Eq. 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 nonarchimedean curvature forms on Eanq. The spectra of these self-adjoint bounded operators on the Hilbert spaces of L2-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 Eq is retrieved from the spectrum.