  We prove an integral-representation result for limits of non-local quadratic
forms on $H^1_0(\Omega)$, with $\Omega$ a bounded open subset of $\mathbb R^d$,
extending the representation on $C^\infty_c(\Omega)$ given by the Beurling-Deny
formula in the theory of Dirichlet forms. We give a counterexample showing that
a corresponding representation may not hold if we consider analogous
functionals in $W^{1,p}_0(\Omega)$, with $p\neq 2$ and $1<p\le d$.
