---
tag: EK58S2872F
kind: statement
role: theorem
name: Beurling–Deny formula
msc: [31C25, 60J46]
uses: [MH20Q2N1AH, HZNEQQBCW4, 0FE2SB8WVJ]
about: [MH20Q2N1AH]
sources:
  - url: https://en.wikipedia.org/wiki/Dirichlet_form
---

## Statement

Let $X$ be a [[locally compact|0FE2SB8WVJ]] separable metric space and let $m$ be a positive [[Radon measure|HZNEQQBCW4]] on $X$ with full support.
Let $(\mathcal{E}, \mathcal{F})$ be a regular [[Dirichlet form|MH20Q2N1AH]] on $L^2(X, m)$, and let $d = \set{(x, x) : x \in X}$ be the diagonal of $X \times X$.
Then there are
1. a symmetric form $\mathcal{E}^{(c)}$ on $\mathcal{F} \cap C_c(X)$ with the strong local property: $\mathcal{E}^{(c)}(u, v) = 0$ whenever $v$ is constant on a neighbourhood of the support of $u$;
2. a symmetric positive Radon measure $J$ on $X \times X \setminus d$, the jumping measure;
3. a positive Radon measure $k$ on $X$, the killing measure;

such that for all $u, v \in \mathcal{F} \cap C_c(X)$
$$\mathcal{E}(u, v) = \mathcal{E}^{(c)}(u, v) + \iint_{X \times X \setminus d} \big(u(x) - u(y)\big)\big(v(x) - v(y)\big) \, J(dx, dy) + \int_X u(x) v(x) \, k(dx).$$
The form $\mathcal{E}^{(c)}$ and the measures $J$ and $k$ are uniquely determined by $\mathcal{E}$.

## 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.

## Explanations

### L1 human

A Dirichlet form measures the energy of a function, and this formula splits that energy into three kinds.
One kind comes from smooth local variation, as with the gradient of a function.
One kind comes from differences between values at distant points, which correspond to jumps of a random particle.
The last kind comes from the function's size alone, which corresponds to the particle being killed.

### L2 human

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.