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

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