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

We consider certain constant-coefficient differential operators on ${\mathbb{R}}^d$ with positive-definite symbols. Each such operator $\mathrm{\Lambda }$ with symbol $P$ defines a semigroup $e^{-t\mathrm{\Lambda }}$ , $t>0$ , admitting a convolution kernel $H_P^t$ for which the large-time behavior of $H_P^t(0)$ cannot be deduced by basic scaling arguments. The simplest example has symbol $P(\xi )=(\eta +{\zeta }^2{)}^2+{\eta }^4$ , $\xi =(\eta ,\zeta )\in {\mathbb{R}}^2$ . We devise a method to establish large-time asymptotics of $H_P^t(0)$ for several classes of examples of this type and we show that these asymptotics are preserved by perturbations by certain higher-order differential operators. For the $P$ just given, it turns out that $H_P^t(0)\sim c_P{t}^{-5\mathrm{/}8}$ as $t\to \mathrm{\infty }$ . We show how such results are relevant to understand the convolution powers of certain complex functions on ${\mathbb{Z}}^d$ . Our work represents a first basic step towards a good understanding of the semigroups associated with these operators. Obtaining meaningful off-diagonal upper bounds for $H_P^t$ remains an interesting challenge.