On anomalous asymptotics of heat kernels on groups of polynomial growth
Nick Dungey, Elst, ter, A.F.M., Derek W. Robinson · ANU Open Research (Australian National University) · 2000
Let K denote the kernel of the continuous semigroup S generated by d' H = (_1)m/2 LA~ i=l where A l , ... ,Ad' are a generating set of right-invariant fields acting on L 2 (G) with G a solvable Lie group of polynomial growth and m an even positive integer.If G is connected, simply connected, and has an abelian nilshadow we establish thatfor all 9 E G and all t ~1, where N a is a subgroup of the abelian nilradical, G(m) denotes an m-th order Gaussian over G and G(2) the second-order Gaussian over N a .The group N a is determined by the choice of the generating set and in general is non-zero.Analogous estimates are derived for various derivatives of the kernel.Further, through the use of homogenization theory, we establish asymptotic estimates for Sand K.These estimates imply that the above kernel bounds give the correct asymptotic behaviour of K, e.g., if m ~4 and N a # {O} then K decreases faster than G(m) as t ~00.