Heat kernel on smooth metric measure spaces with nonnegative curvature

We derive a local Gaussian upper bound for the -heat kernel on complete smooth metric measure space with nonnegative Bakry-A parts per thousand mery Ricci curvature. As applications, we obtain a sharp -Liouville theorem for -subharmonic functions and an -uniqueness property for nonnegative solutions of the -heat equation, assuming is of at most quadratic growth. In particular, any -integrable -subharmonic function on gradient shrinking and steady Ricci solitons must be constant. We also provide explicit -heat kernel for Gaussian solitons.