Non-local Dirichlet Forms and Symmetric Jump Processes

We consider the symmetric non-local Dirichlet form $(E, F)$ given by $$E (f,f)=\int_{R^d} \int_{R^d} (f(y)-f(x))^2 J(x,y) dx dy$$ with $F$ the closure of the set of $C^1$ functions on $R^d$ with compact support with respect to $E_1$, where $E_1 (f, f):=E (f, f)+\int_{R^d} f(x)^2 dx$, and where the jump kernel $J$ satisfies $$\kappa_1|y-x|^{-d-\alpha} \leq J(x,y) \leq \kappa_2|y-x|^{-d-\beta}$$ for $0<\alpha< \beta <2, |x-y|<1$. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to $(E, F)$. We prove a parabolic Harnack inequality for nonnegative functions that solve the heat equation with respect to $E$. Finally we construct an example where the corresponding harmonic functions need not be continuous.