Off-diagonal short time expansion of the heat kernel on a certain nilpotent Lie group (Q756824)

Let \(L=(1/2)\sum V^ 2_{\alpha}\) be a differential operator of Hörmander type, where \(V_{\alpha}\), \(\alpha =1,...,r\), are \(C^{\infty}\)-vector fields on \({\mathbb{R}}^ d\) satisfying the Hörmander condition. Then the heat equation \(\partial u/\partial t=Lu\) has the fundamental solution p(t,x,y). We consider the short time expansion of p(t,x,y); \[ (0)\quad p(t,x,y)\sim \exp (-d(x,y)^ 2/2t)t^{-N/2}(c_ 0+c_ 1t+...)\quad (t\downarrow 0). \] Here d(x,y) is the control metric of the pair (x,y). It is known that (0) holds for \(N=d\) when the pair (x,y) is out of cut-locus. In this paper, we investigate the above expansion (0) by the method of Wiener functionals in a concrete case of the nilpotent Lie group \(N_{4,2}\) realized by \({\mathbb{R}}^{10}\). In this case, it occurs the situation that the pair (x,y) is in the cut-locus, and in this situation we show that the expansion (0) holds for some constant N which is determined by the following three factors, i.e. the dimension of the totality of minimal horizontal curves connecting x and y, the degeneracy of the corresponding deterministic Malliavin covariance and \(d=10\), the dimension of the whole space.