  The notion of quasi-Einstein metric in physics is equivalent to the notion of
Ricci soliton in Riemannian spaces. Quasi-Einstein metrics serve also as
solution to the Ricci flow equation. Here, the Riemannian metric is replaced by
a Hessian matrix derived from a Finsler structure and a quasi-Einstein Finsler
metric is defined. In compact case, it is proved that the quasi-Einstein
metrics are solution to the Finslerian Ricci flow and conversely, certain form
of solutions to the Finslerian Ricci flow are quasi-Einstein Finsler metrics.
