---
tag: E7063PCV9P
kind: definition
name: Green's function of a Markov process
msc: [60J45, 60J35, 31C15]
defines: XSQX8X2YQ2
uses: [595AZJAZNA, 46436KE7NT, 22735FGJV6, M36SD56MWQ, 0G48PYM96Q, PR11BZQQPG]
sources:
  - url: https://en.wikipedia.org/wiki/Green%27s_function
---

## Definition

Let $(E, \mathcal{E})$ be a measurable space with a σ-finite [[measure|PR11BZQQPG]] $m$.
Let $(P_t)$ be a sub-Markov transition semigroup on $E$ with [[potential kernel|595AZJAZNA]] $U$.
A *Green's function* (or potential density) of $(P_t)$ with respect to $m$ is a measurable function $u \colon E \times E \to [0, \infty]$ such that
$$U(x, A) = \int_A u(x, y) \, m(dy) \qquad \text{for every } x \in E \text{ and } A \in \mathcal{E}.$$

## Remarks

For each $x$ the function $u(x, \cdot)$ is determined only $m$-almost everywhere.
One usually fixes a version for which $x \mapsto u(x, y)$ is [[excessive|46436KE7NT]] for each $y$; under standard duality hypotheses such a version exists and is unique.
Example: for [[Brownian motion|22735FGJV6]] in $\R^3$ with Lebesgue measure, $u(x, y) = 1/(2\pi\abs{x - y})$, because this is the density of $U(x, \cdot)$.
For Brownian motion killed on leaving a planar domain $D$, the Green's function is $u = g_D/\pi$.
Here $g_D$ is the classical [[Green's function|M36SD56MWQ]] with $\Delta g_D(\cdot, y) = -2\pi\delta_y$, and the factor comes from the generator $\frac12 \Delta$.
For a [[Markov chain|0G48PYM96Q]] on a countable set with counting measure, $u(x, y) = \sum_{n \ge 0} P^n(x, y)$ is the expected number of visits to $y$ starting from $x$.
Non-example: Brownian motion in $\R^2$ on the whole plane has no Green's function with finite values, since $U(x, A) = \infty$ for every nonempty open $A$.