Fundamental solutions for differential equations associated with the number operator

Abstract:

Let $(H, B)$ be an abstract Wiener space. If $u$ is a twice $H$-differentiable function on $B$ such that $Du(x) \in {B^{\ast }}$ and ${D^2}u(x)$ is of trace class, then we define $\mathfrak {N}u(x) = - \Delta u(x) + (x, Du(x))$, where $\Delta u(x) = {\operatorname {trace} _H} {D^2}u(x)$ is the Laplacian and $( \cdot , \cdot )$ denotes the $B$-${B^{\ast }}$ pairing. The closure $\overline {\mathfrak {N}}$ of $\mathfrak {N}$ is known as the number operator. In this paper, we investigate the existence, uniqueness and regularity of solutions for the following two types of equations: (1) ${u_t} = - \mathfrak {N}u$ (initial value problem) and (2) ${\mathfrak {N}^k}u = f(k \geqslant 1)$. We show that the fundamental solutions of (1) and (2) exist in the sense of measures and we represent their solutions by integrals with respect to these measures.