A parametrised diffusion operator on the regular domain \(\Omega \) of a p-adic Schottky group is constructed. It is defined as an integral operator on the complex-valued functions on \(\Omega \) which are invariant under the Schottky group \(\Gamma \) , where integration is against the measure defined by an invariant regular differential 1-form \(\omega \) . It is proven that the space of Schottky invariant \(L^2\) -functions on \(\Omega \) outside the zeros of \(\omega \) has an orthonormal basis consisting of \(\Gamma \) -invariant extensions of Kozyrev wavelets which are eigenfunctions of the operator. The eigenvalues are calculated, and it is shown that the heat equation for this operator provides a unique solution for its Cauchy problem with Schottky-invariant continuous initial conditions supported outside the zero set of \(\omega \) , and gives rise to a strong Markov process on the corresponding orbit space for the Schottky group whose paths are càdlàg.
