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

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 Lévy-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 $(P_t{)}_{t\ge 0}$ on $L^2(\R^d;\d x)$. In particular, we prove that if $J(x,\d y)=|x-y|^{-d-α}\,\d y$ with $\alpha \in (0,2)$, and $$W(x,y)=\{\begin{array}{ll}(1+|x|{)}^p+(1+|y|{)}^p,& |x-y|<1\\ (1+|x|{)}^q+(1+|y|{)}^q,& |x-y|\ge 1\end{array}$$ with $p\in [0,\infty )$ and $q\in [0,\alpha )$, then $(P_t{)}_{t\ge 0}$ 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 Poincaré 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.

22 pages

Cited by in corpus (1)

• Transience of symmetric non-local Dirichlet forms