We study ancient Ricci flows which admit asymptotic solitons in the sense of Perelman [33, Proposition 11.2]. We prove that the asymptotic solitons must coincide with Bamler’s tangent flows at infinity [5]. 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.
