On-diagonal asymptotics for heat kernels of a class of inhomogeneous partial differential operators

Abstract

We consider certain constant-coefficient differential operators on Rd that have positive-definite symbols. Each such operator Λ with symbol P defines a semigroup of operators e−tΛ, t>0, admitting a continuous convolution kernel HPt for which the large-time behavior of HPt(0) cannot be deduced by basic scaling arguments. The simplest example has symbol P(ξ)=(η+ζ2)2+η4, ξ=(η,ζ)∈R2. We devise a method that allows us to determine the large-time behavior of HPt(0) for several classes of examples of this type and we show that these large-time asymptotics are preserved by perturbations of Λ by certain higher-order differential operators. For the P just given, it turns out that HPt(0)∼cPt−5/8 when t tends to infinity. We show how such results are relevant to understand the iterated convolution powers of certain finitely-supported complex functions on Zd. We also discuss how these techniques provide precise small-time asymptotics for HPt(0) in some cases when the operator Λ is not hypoelliptic. The simplest such example Λ has symbol P(ξ)=η2+(η−ξ2)4 and we show that HPt(0)∼cPt−1/2 as t tends to 0 in this case. Our work represents a first basic step towards a good understanding of the semigroups associated with these differential operators. Obtaining meaningful off-diagonal upper bounds for the convolution kernels of these semigroups remains an interesting challenge.