Compactness of semigroups generated by symmetric non-local Dirichlet forms with unbounded coefficients

TL;DR

This paper provides sharp criteria for the compactness of semigroups generated by symmetric non-local Dirichlet forms with unbounded coefficients, focusing on the influence of the weighted function's growth near small jumps.

Contribution

It establishes precise conditions for the compactness of associated semigroups, especially highlighting the role of the weighted function's growth for small jumps.

Findings

Semigroup is compact if and only if p > 2 for the specified weight growth.

Compactness depends heavily on the growth of W(x,y) for |x-y|< 1.

Results hold even with degenerate or singular jumping kernels.

Abstract

Let $(\E,\F)$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^2(\R^d;\d x)$ defined by $$\E(f,g)=\iint_{\R^d\times \R^d} (f(y)-f(x))(g(x)-g(y)){W(x,y)}\, J(x,\d y)\,\d x, \quad f,g\in \F,$$ where $J(x,\d y)$ is regarded as the jumping kernel for a pure-jump symmetric L\'evy-type process with bounded coefficients, and $W(x,y)$ is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup $(P_t)_{t\ge0}$ on $L^2(\R^d;\d x)$. In particular, we prove that if $J(x,\d y)=|x-y|^{-d-\alpha}\,\d y$ with $\alpha\in (0,2)$, and $$W(x,y)= \begin{cases} (1+|x|)^p+(1+|y|)^p, \ & |x-y|< 1 \\ (1+|x|)^q+(1+|y|)^q, \ & |x-y|\geq 1 \end{cases}$$ with $p\in [0,\infty)$ and $q\in [0,\alpha)$, then $(P_t)_{t\ge0}$ is compact, if and only if $p>2$. This indicates that the compactness of $(\E,\F)$ heavily depends on the growth of the weighted function $W(x,y)$ only for $|x-y|<1$. Our approach is based on establishing the essential super Poincar\'e inequality for $(\E,\F)$. Our general results work even if the jumping kernel $J(x,\d y)$ is degenerate or is singular with respect to the Lebesgue measure.