  Researching elliptic analogues for equalities and formulas is a new trend in
enumerative combinatorics which has followed the previous trend of studying
$q$-analogues. Recently Schlosser proposed a lattice path model in the square
lattice with a family of totally elliptic weight-functions including several
complex parameters and discussed an elliptic extension of the binomial theorem.
In the present paper, we introduce a family of discrete-time excursion
processes on $\mathbb{Z}$ starting from the origin and returning to the origin
in a given time duration $2T$ associated with Schlosser's elliptic
combinatorics. The processes are inhomogeneous both in space and time and hence
expected to provide new models in non-equilibrium statistical mechanics. By
numerical calculation we show that the maximum likelihood trajectories on the
spatio-temporal plane of the elliptic excursion processes and of their reduced
trigonometric versions are not straight lines in general but are nontrivially
curved depending on parameters. We analyze asymptotic probability laws in the
long-term limit $T \to \infty$ for a simplified trigonometric version of
excursion process. Emergence of nontrivial curves of trajectories in a large
scale of space and time from the elementary elliptic weight-functions exhibits
a new aspect of elliptic combinatorics.
