Non-local Dirichlet forms and symmetric jump processes

We consider the non-local symmetric Dirichlet form $(\mathcal {E}, \mathcal {F})$ given by \[ \mathcal {E} (f,f)=\int \limits _{\mathbb {R}^d} \int \limits _{\mathbb {R}^d} (f(y)-f(x))^2 J(x,y) dx dy \] with $\mathcal {F}$ the closure with respect to $\mathcal {E}_1$ of the set of $C^1$ functions on $\mathbb {R}^d$ with compact support, where $\mathcal {E}_1 (f, f):=\mathcal {E} (f, f)+\int _{\mathbb {R}^d} f(x)^2 dx$, 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 sizes depend 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 $(\mathcal {E}, \mathcal {F})$. We prove a parabolic Harnack inequality for non-negative functions that solve the heat equation with respect to $\mathcal {E}$. Finally we construct an example where the corresponding harmonic functions need not be continuous.