Shifted moments of Gaussian measures in Hilbert spaces

Abstract

If H is a Hilbert space, then a measure $\mu$ is completely described by the function $$y→∈t_H||x+y||?pd\mu(x),\quad y∈H,$$ provided that $p≠2,4,6,\ldots$.We prove that for Gaussian measures on a Hilbert space H this is valid for all $p>0$ with $p≠ 2$.