Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $\lambda$ be the Lebesgue measure on $[0,\infty)$
- $\mathcal D:=C_c^\infty([0,\infty))$ and $\mathcal D'$ be the dual space of $\mathcal D$
- $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ and $$\langle W,\phi\rangle:=\int\phi(t)B_t\;{\rm d}\lambda\;\;\;\text{for }\phi\in\mathcal D\tag 1$$
We can show that $W$ is a $\mathcal D'$-valued Gaussian random variable on $(\Omega,\mathcal A,\operatorname P)$, i.e. $$\left(\langle W,\phi_1\rangle,\ldots,\langle W,\phi_n\rangle\right)\text{ is }n\text{-dimensionally normally distributed}\tag 2$$ for all linearly independent $\phi_1,\ldots,\phi_n\in\mathcal D$, with expectation $$\operatorname E[W](\phi):=\operatorname E\left[\langle W,\phi\rangle\right]=0\;\;\;\text{for all }\phi\in\mathcal D\tag 3$$ and covariance $$\rho[W](\phi,\psi):=\operatorname E\left[\langle W,\phi\rangle\langle W,\psi\rangle\right]=\int\int\min(s,t)\phi(s)\psi(t)\;{\rm d}\lambda(s)\;{\rm d}\lambda(t)\;\;\;\text{for all }\phi,\psi\in\mathcal D\;.\tag 4$$ Moreover, the derivative $$\langle W',\phi\rangle:=-\langle W,\phi\rangle\;\;\;\text{for }\phi\in\mathcal D\tag 5$$ is again a $\mathcal D'$-valued Gaussian random variable on $(\Omega,\mathcal A,\operatorname P)$ with expectation $$\operatorname E[W'](\phi)=0\;\;\;\text{for all }\phi\in\mathcal D\tag 6$$ and covariance \begin{equation} \begin{split} \varrho[W'](\phi,\psi)&=\int\int\min(s,t)\phi'(s)\psi'(t)\;{\rm d}\lambda(s)\;{\rm d}\lambda(t)\\ &=\int\int\delta(t-s)\phi(s)\psi(t)\;{\rm d}\lambda(t)\;{\rm d}\lambda(s) \end{split}\tag 7 \end{equation} for all $\phi,\psi\in\mathcal D$. A generalized Gaussian stochastic process with expectation and covariance given by $(5)$ and $(6)$ is called Gaussian white noise. Thus, the generalized derivative $W'$ of the generalized Brownian motion $W$ is a Gaussian white noise.
Now let $(H,\langle\;\cdot\;,\;\cdot\;\rangle_H)$ be a Hilbert space, $Q$ be a linear, bounded, nonnegative and symmetric operator on $H$ with finite trace and $\tilde B$ be a $Q$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$, i.e.
- $\tilde B$ is a $H$-valued stochastic process on $(\Omega,\mathcal A,\operatorname P)$
- $\tilde B_0=0$ almost surely
- $\tilde B$ is almost surely continuous
- $\tilde B$ has independent increments
- $\tilde B_t$ has the Gaussian distribution $\mu_t:=\mathcal N_{0,tQ}$ defined by its characteristic function $$H\ni x\mapsto\int e^{{\rm i}\langle x,y\rangle_H}\;{\rm d}\mu_t(y)=e^{-\frac t2\langle Qx,x\rangle_H}\tag 8$$
Plese note that for a Gaussian probability measure space $\mu$ on $(H,\mathcal B(H))$ $$H\to\mathbb R,\;\;\;x\mapsto\langle x,y\rangle_H\tag 9$$ is normally distributed, for all $y\in H$.
Question: How can we generalize the result that the generalized derivative $W'$ of the generalized Brownian motion $W$ is a Gaussian white noise to the $Q$-Brownian motion $\tilde B$?
Especially: $(1)$ is the "naturally" generalized stochastic process induced by $B$, but what's the generalized stochastic process induced by $\tilde B$, i.e. what's the generalized $Q$-Brownian motion $\tilde W$? And how do we obtain $(2)$-$(7)$?