  We define a Hesse soliton, that is, a self-similar solution to the Hesse flow
on Hessian manifolds. On information geometry, the $e$-connection and the
$m$-connection are important, which do not coincide with the Levi-Civita one.
Therefore, it is interesting to consider a Hessian manifold with a flat
connection which does not coincide with the Levi-Civita one. We call it a
proper Hessian manifold. In this paper, we show that any compact proper Hesse
soliton is expanding and any non-trivial compact gradient Hesse soliton is
proper. Furthermore, we show that the dual space of a Hesse-Einstein manifold
can be understood as a Hesse soliton.
