  We study ancient Ricci flows which admit asymptotic solitons in the sense of
Perelman. We prove that the asymptotic solitons must coincide with Bamler's
tangent flows at infinity. Furthermore, we show that Perelman's
$\nu$-functional is uniformly bounded on such ancient solutions; this fact
leads to logarithmic Sobolev inequalities and Sobolev inequalities. Lastly, as
an important tool for the proofs of the above results, we also show that, for a
complete Ricci flow with bounded curvature, the bound of the Nash entropy
depends only on the local geometry around an $H_n$-center of its base point.
