Penney, Richard C.Urban, Roman2023-03-292023-03-292022-01-10Penney, R. C., & Urban, R. (2022). Poisson measures on semi-direct products of infinite-dimensional Hilbert spaces. <i>Electronic Journal of Differential Equations, 2022</i>(04), pp. 1-15.1072-6691https://hdl.handle.net/10877/16508Let G = X ⋊ A where X and A are Hilbert spaces considered as additive groups and the A-action on G is diagonal in some orthonormal basis. We consider a particular second order left-invariant differential operator L on G which is analogous to the Laplacian on ℝn. We prove the existence of "heat kernel" for L and give a probabilistic formula for it. We then prove that X is a "Poisson boundary" in a sense of Furstenberg for L with a (not necessarily) probabilistic measure ν on X called the "Poisson measure" for the operator L.Text15 pages1 file (.pdf)enAttribution 4.0 InternationalPoisson measureGaussian measureHilbert spaceBrownian motionEvolution kernelDiffusion processesArticle