Asymptotic Behaviors at Infinity of Ancient Solutions to the Parabolic Monge-Ampère Equation

This paper aims at providing asymptotic behaviors at infinity of solutions to the parabolic Monge-Amp & egrave;re equations -u(t)detD(x)(2)u = f in R_(n+1) for n >= 1, where -u(t) has a positive upper and lower bound and f tends to 1 at infinity. We obtain results that are quite different from the elliptic case, and also improve the existing works of the parabolic case. There are counterexamples to illustrate that the asymptotic behavior is optimal.