Poisson measures on semi-direct products of infinite-dimensional Hilbert spaces

Let \(G=X\rtimes A\) 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 \(\mathbb{R}^n\). 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.