Poisson point process: Difference between revisions

Revision as of 16:47, 8 March 2023

Revision as of 19:46, 20 March 2023

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===Laplace functionals===

For a Poisson point process <math>\textstyle N</math> with intensity measure <math>\textstyle \Lambda</math> on some space <math>X</math>, the [[Laplace functional]] is given by:<ref name="baccelli2009stochastic1"/>

:<math> L_N(f)= \mathbb{E} e^{-\int_{int_X f(x)\mathbb{R, N(\mathrm dx)} = e^d{-\int_{X}(1-e^{-f(x)})\Lambda(\mathrm dx)}, </math>

:<math> L_N(f)=e^{-\lambda\int_{\mathbb{R}^d}(1-e^{-f(x)})\,\mathrm dx}. </math>

One version of [[Campbell's theorem (probability)#Second definition: Poisson point process|Campbell's theorem]] involves the Laplace functional of the Poisson point process.