Let
- $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
- $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$
- $U$ be an infinite-dimensional separable $\mathbb R$-Hilbert space
- $Q$ be a nonnegative and self-adjoint nuclear operator on $U$
- $U_0:=Q^{1/2}U$ be equipped with $$\langle u_0,v_0\rangle_{U_0}:=\langle Q^{-1/2}u_0,Q^{-1/2}v_0\rangle_U\;\;\;\text{for }u_0,v_0\in U_0\;,$$ where $Q^{-1/2}$ denotes the pseudo inverse of $Q$
- $W$ be a $Q$-Wiener process on $(\Omega,\mathcal A,\operatorname P)$ with respect to $\mathcal F$
Let $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ and $$I:=\mathbb N\cap[0,\operatorname{rank}Q]\;.$$ Assume
- $(e_n)_{n\in I}$ is an orthonormal basis of $(\ker Q)^\perp$ with $$Qe_n=\lambda_ne_n\;\;\;\text{for all }n\in I\tag1$$ for some nonincreasing $(\lambda_n)_{n\in\mathbb N}\subseteq[0,\infty)$ with $$\lambda_n>0\Leftrightarrow n\in I\;\;\;\text{for all }n\in\mathbb N\tag2$$
- $(e_n)_{n\in\mathbb N\setminus I}$ is an orthonormal basis of $\ker Q$
Note that $$Q^{1/2}e_n=\sqrt{\lambda_n}e_n\;\;\;\text{for all }n\in\mathbb N\tag3$$ and that $(\sqrt{\lambda_n}e_n)_{n\in I}$ is an orthonormal basis of $U_0$.
How can we show that $W_t\in U_0$ for all $t\ge0$ almost surely?
