The moment problem on the Wiener space

Abstract

Consider an $L^1$-continuous functional $\ell$ on the vector space of polynomials of Brownian motion at given times, suppose $\ell $ commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, $f_1(\vec b),...,f_m(\vec b)$, mapping the Wiener space to $\mathbb{R}$. In the spirit of Schm\"udgen's solution to the finite-dimensional moment problem, we give sufficient conditions under which $\ell$ can be written in the form $\int \cdot d\mu$ for some finite measure $\mu$ on the Wiener space such that $\mu$-almost surely, all the random variables $f_1(\vec b),...,f_m(\vec b)$ are nonnegative.