  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.
