Beurling–Deny formula

Remarks

The proof starts from the Markov property, which gives $\mathcal{E}(u, v) \le 0$ for nonnegative $u, v \in \mathcal{F} \cap C_c(X)$ with disjoint supports. So $-\mathcal{E}(u, v)$ is a positive functional of such pairs, and a Riesz representation argument turns it into the measure $J$. The killing measure comes from the behaviour of $\mathcal{E}$ on functions equal to $1$ near a compact set, and $\mathcal{E}^{(c)}$ is what remains. Some texts put a factor $2$ or $\tfrac12$ in front of the jump term, which only rescales $J$. For the form $\int_{\R^n} \nabla u \cdot \nabla v \, dx$ on $W^{1,2}(\R^n)$, the form is all local: $J = 0$ and $k = 0$. For the fractional form $\iint (u(x) - u(y))(v(x) - v(y)) \abs{x - y}^{-n - \alpha} \, dx \, dy$ with $0 < \alpha < 2$, only the jump part is present. For a Hunt process attached to $\mathcal{E}$, the three parts describe its continuous motion, its jumps and its killing.

Going deeper

For a student

The formula is the analytic counterpart of the Lévy–Khintchine decomposition of a Lévy process into a diffusion part, a jump part and a killing rate. A regular Dirichlet form is local, meaning that $J = 0$, exactly when its associated Hunt process has continuous paths up to its lifetime. In practice one reads off $J$ and $k$ from a given form to understand the process, or one builds forms from chosen $J$ and $k$ to construct jump processes.