# How can we prove that the derivative of a generalized Hilbert space valued Brownian motion is a Gaussian white noise?

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 $(2)$-$(7)$?

• I could imagine that we need to consider $$\langle\tilde W,\phi\rangle:=\int\langle\tilde B_t,\phi\rangle_H\;{\rm d}\lambda\;\;\;\text{for }\phi\in C_c^\infty([0,\infty);H)$$ or $$\langle\tilde W,\phi\rangle:=\int\tilde B_t\phi\;{\rm d}\lambda\;\;\;\text{for }\phi\in C_c^\infty([0,\infty))$$ where the last integral is a Bochner integral, but I really don't know it and I can't find any suitable reference. Either they consider a Gaussian white noise induced by a real-valued BM or they consider a "Gaussian white noise induced by a $Q$-valued BM", but don't define what they mean by that. – 0xbadf00d Feb 1 '16 at 12:56
• I'm not too familiar with the Bochner integral, but if it admits some form of integration by parts, my bet would be on the second definition. One of the key properties of real-valued white noise is that it acts on test functions by stochastic integration (by stochastic integration by parts). I would go with the definition that generalizes this property (unless there's some other reason not to). In the first definition, you won't get far trying to integrate by parts because you can't do much with $\langle \tilde{B}_t, \phi' \rangle_H$. Of course, it all depends on what properties you want. – Ben CW Feb 10 '16 at 6:02