Gaussian bounds for degenerate parabolic equations

Let A be a real symmetric, degenerate elliptic matrix whose degeneracy is controlled by a weight w   in the A2A2 or QC   class. We show that there is a heat kernel Wt(x,y)Wt(x,y) associated to the parabolic equation wut=divA∇uwut=divA∇u, and WtWt satisfies classic Gaussian bounds:|Wt(x,y)|⩽C1tn/2exp(−C2|x−y|2t). We then use this bound to derive a number of other properties of the kernel.