Green measure

The Green measure

Intuitively, the Green measure of a Borel set H (with respect to a point x and domain D) is the expected length of time that X, having started at x, stays in H before it leaves the domain D. That is, the Green measure of X with respect to D at x, denoted G(x, ⋅), is defined for Borel sets H ⊆ Rⁿ by

${\displaystyle G(x,H)={\mathbf{E}}^x\left[{\int }_0^{{\tau }_D}{\chi }_H(X_s)\mathrm{d}s\right],}$

or for bounded, continuous functions f : D → R by

${\displaystyle {\int }_{D}f(y)G(x,\mathrm{d}y)={\mathbf{E}}^x\left[{\int }_0^{{\tau }_D}f(X_s)\mathrm{d}s\right]}$

The name "Green measure" comes from the fact that if X is Brownian motion, then

${\displaystyle G(x,H)={\int }_{H}G(x,y)\mathrm{d}y,}$

where G(x, y) is Green's function for the operator L_X (which, in the case of Brownian motion, is ⁠1/2⁠Δ, where Δ is the Laplace operator) on the domain D.

The Green formula

Suppose that E^x[τ_D] < +∞ for all x ∈ D, and let f : Rⁿ → R be of smoothness class C² with compact support. Then

${\displaystyle f(x)={\mathbf{E}}^x[f(X_{{\tau }_D})]-{\int }_D{L}_{X}f(y)G(x,\mathrm{d}y).}$

In particular, for C² functions f with support compactly embedded in D,

${\displaystyle f(x)=-{\int }_D{L}_{X}f(y)G(x,\mathrm{d}y).}$

The proof of Green's formula is an easy application of Dynkin's formula and the definition of the Green measure:

${\displaystyle {\mathbf{E}}^x[f(X_{{\tau }_D})]}$

${\displaystyle =f(x)+{\mathbf{E}}^x\left[{\int }_0^{{\tau }_D}{L}_{X}f(X_s)\mathrm{d}s\right]}$

${\displaystyle =f(x)+{\int }_D{L}_{X}f(y)G(x,\mathrm{d}y).}$