  Let $E$ be a locally compact separable metric space and $m$ be a positive
Radon measure on it. Given a nonnegative function $k$ defined on $E\times E$
off the diagonal whose anti-symmetric part is assumed to be less singular than
the symmetric part, we construct an associated regular lower bounded
semi-Dirichlet form $\eta$ on $L^2(E;m)$ producing a Hunt process $X^0$ on $E$
whose jump behaviours are governed by $k$. For an arbitrary open subset
$D\subset E$, we also construct a Hunt process $X^{D,0}$ on $D$ in an analogous
manner. When $D$ is relatively compact, we show that $X^{D,0}$ is censored in
the sense that it admits no killing inside $D$ and killed only when the path
approaches to the boundary. When $E$ is a $d$-dimensional Euclidean space and
$m$ is the Lebesgue measure, a typical example of $X^0$ is the stable-like
process that will be also identified with the solution of a martingale problem
up to an $\eta$-polar set of starting points. Approachability to the boundary
$\partial D$ in finite time of its censored process $X^{D,0}$ on a bounded open
subset $D$ will be examined in terms of the polarity of $\partial D$ for the
symmetric stable processes with indices that bound the variable exponent
$\alpha(x)$.
