Green measure
From Wikipedia, the free encyclopedia
In mathematics — specifically, in stochastic analysis — the Green measure is a measure associated to an Itō diffusion. There is an associated Green formula representing suitably smooth functions in terms of the Green measure and first exit times of the diffusion. The concepts are named after the British mathematician George Green and are generalizations of the classical Green's function and Green formula to the stochastic case using Dynkin's formula.
Let X be an Rn-valued Itō diffusion satisfying an Itō stochastic differential equation of the form
Let Px denote the law of X given the initial condition X0 = x, and let Ex denote expectation with respect to Px. Let LX be the infinitesimal generator of X, i.e.
Let D ⊆ Rn be an open, bounded domain; let τD be the first exit time of X from D:
![{\displaystyle G(x,H)=\mathbf {E} ^{x}\left[\int _{0}^{\tau _{D}}\chi _{H}(X_{s})\,\mathrm {d} s\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4c31b6290036aab7780f2db087113cf571f14d0)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9144f11efbc0cab48bd5a27aa6c3894badeb28e)
![{\displaystyle f(x)=\mathbf {E} ^{x}{\big [}f{\big (}X_{\tau _{D}}{\big )}{\big ]}-\int _{D}L_{X}f(y)\,G(x,\mathrm {d} y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bdab6106cf6376e5a62d691cbd9010cd1a85e53)
![{\displaystyle \mathbf {E} ^{x}{\big [}f{\big (}X_{\tau _{D}}{\big )}{\big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f5f9e8e9023fb7ef600ba577b4198575cff4c7)
![{\displaystyle =f(x)+\mathbf {E} ^{x}\left[\int _{0}^{\tau _{D}}L_{X}f(X_{s})\,\mathrm {d} s\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a9cd5bcc1b5eece6956f01d891d7541ff3c0f0)
