In the Chapter 1.2 of Stochastic Partial Differential Equations: An Introduction by Wei Liu and Michael Röckner, the authors introduce stochastic partial differential equations by considering equations of the form $$\frac{{\rm d}X_t}{{\rm d}t}=F\left(t,X_t,\dot B_t\right)$$ where $\left(\dot B_t\right)_{t\ge 0}$ is a "white noise in time" (whatever that means) with values in a separable Hilbert space $U$. $\left(\dot B_t\right)_{t\ge 0}$ is said to be the "generalized time-derivative of a $U$-valued Brownian motion $(B_t)_{t\ge 0}$.
Question: What exactly do the authors mean? What is a "white noise in time" and why (and in which sense) is it the "generalized time-derivative" of a Brownian motion?
You can skip the following, if you know the answer to these questions. I will present what I've found out so far:
I've searched the terms "white noise" and "distributional derivative of Brownian motion" on the internet and found few and inconsistent definitions.
Definition 1: In the book An Introduction to Computational Stochastic PDEs the authors do the following: Let $(\phi_n)_{n\in\mathbb N}$ be an orthonormal basis of $L^2([0,1])$, e.g. $\phi_n(t):=\sqrt 2\sin(n\pi t)$. Then $$W_t:=\lim_{n\to\infty}\sum_{i=1}^n\phi_i(t)\xi_i\;\;\;\text{for }t\in [0,1]\;,$$ where the $\xi_i$ are independent and standard normally distributed random variables on a probability space $(\Omega,\mathcal A,\operatorname P)$, is a stochastic process on $(\Omega,\mathcal A,\operatorname P)$ with $\operatorname E[W_t]=0$ and $$\operatorname E[W_sW_t]=\delta(s-t)\;\;\;\text{for all }s,t\in [0,1]$$ where $\delta$ denotes the Dirac delta function. They call $(W_t)_{t\in [0,1]}$ white noise.
This definition seems to depend on the explicit choice of the orthnormal basis $(\phi_n)_{n\in\mathbb N}$ and I don't see the connection to a "derivative" of a Brownian motion (needless to say that I don't see how this would generalize to a cylindrical Brownian motion).
However, maybe it has something to do with the following: Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on $(\Omega,\mathcal A,\operatorname P)$. Then the Karhunen–Loève theorem yields $$B_t=\lim_{n\to\infty}\sum_{i=1}^n\sqrt{\zeta_i}\phi_i(t)\xi_i\;\;\;\text{for all }t\in [0,T]$$ in $L^2(\operatorname P)$ and uniformly in $t$, where $(\phi_n)_{n\in\mathbb N}$ is an orthonormal basis of $L^2([0,1])$ and $(\xi_n)_{n\in\mathbb N}$ is a sequence of indepedent standard normally distributed random variables on $(\Omega,\mathcal A,\operatorname P)$. In particular, $$\zeta_i=\frac 4{(2i-1)^2\pi^2}$$ and $$\phi_i(t)=\sqrt 2\sin\frac t{\sqrt{\zeta_i}}\;.$$
The authors state, that we can formally consider the derivative of $B$ as being the process $$\dot B_t=\lim_{n\to\infty}\sum_{i=1}^n\phi_i(t)\xi_i\;.$$ I have no idea why.
Nevertheless, we may notice the following: Let $${\rm D}^{(\Delta t)}_t:=\frac{B_{t+\Delta t}-B_t}{\Delta t}\;\;\;\text{for }t\ge 0$$ for some $\Delta t>0$. Then $\left({\rm D}^{(\Delta t)}_t\right)$ is a stochastic process on $(\Omega,\mathcal A,\operatorname P)$ with $$\operatorname E\left[{\rm D}^{(\Delta t)}_t\right]=0\;\;\;\text{for all }t\ge 0$$ and $$\operatorname{Cov}\left[{\rm D}^{(\Delta t)}_s,{\rm D}^{(\Delta t)}_t\right]=\left.\begin{cases}\displaystyle\frac{\Delta t-|s-t|}{\Delta t^2}&\text{, if }|s-t|\le \Delta t\\0&\text{, if }|s-t|\ge \Delta t\end{cases}\right\}=:\eta^{(\Delta t)}(s-t)\;\;\;\text{for all }s,t\ge 0\;.$$ Since $$\int\eta^{(\Delta t)}(x)\;{\rm d}x=\int_{-\Delta t}^{\Delta t}\eta^{(\Delta t)}(x)\;{\rm d}x=1$$ we obtain $$\eta^{(\Delta t)}(x)\stackrel{\Delta t\to 0}\to\delta(x)\;,$$ but I have no idea how this is related to white noise.
Definition 2: In Stochastic Differential Equations with Applications to Physics and Engineering, Modeling, Simulation, and Optimization of Integrated Circuits and Generalized Functions – Vol 4: Applications of Harmonic Analysis they take a real-valued Brownian motion $(B_t)_{t\ge 0}$ on $(\Omega,\mathcal A,\operatorname P)$ and define $$\langle W,\phi\rangle:=\int\phi(t)B_t\;{\rm d}\lambda\;\;\;\text{for }\phi\in\mathcal D:=C_c^\infty([0,\infty))\;.$$ Let $\mathcal D'$ be the dual space of $\mathcal D$. We can show that $W$ is a $\mathcal D'$-valued Gaussian random variable on $(\Omega,\mathcal A,\operatorname P)$, i.e. $$\left(\langle W,\phi_1\rangle,\ldots,\langle W,\phi_n\rangle\right)\text{ is }n\text{-dimensionally normally distributed}$$ for all linearly independent $\phi_1,\ldots,\phi_n\in\mathcal D$, with expectation $$\operatorname E[W](\phi):=\operatorname E\left[\langle W,\phi\rangle\right]=0\;\;\;\text{for all }\phi\in\mathcal D$$ and covariance $$\rho[W](\phi,\psi):=\operatorname E\left[\langle W,\phi\rangle\langle W,\psi\rangle\right]=\int\int\min(s,t)\phi(s)\psi(t)\;{\rm d}\lambda(s)\;{\rm d}\lambda(t)\;\;\;\text{for all }\phi,\psi\in\mathcal D\;.$$ Moreover, the derivative $$\langle W',\phi\rangle:=-\langle W,\phi\rangle\;\;\;\text{for }\phi\in\mathcal D\tag 1$$ is again a $\mathcal D'$-valued Gaussian random variable on $(\Omega,\mathcal A,\operatorname P)$ with expectation $$\operatorname E[W'](\phi)=0\;\;\;\text{for all }\phi\in\mathcal D\tag 2$$ 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}
\end{equation}
for all $\phi,\psi\in\mathcal D$. Now they call a generalized Gaussian stochastic process with expectation and covariance given by $(1)$ and $(2)$ a Gaussian white noise. Thus, the generalized derivative $W'$ of the generalized Brownian motion $W$ is a Gaussian white noise.
Again, I don't know how I need to generalize this to the case of a cylindrical Brownian motion. Moreover, this definition seems to be less naturally to me and I don't think that this is the notion Liu and Röckner had in mind.
Definition 3: In some lecture notes, I've seen the following the definition: Let $W$ be a centered Gaussian process, indexed by test functions $\phi\in C^\infty([0,\infty]\times\mathbb R^d)$ whose covariance is given by $$\operatorname E\left[W_\phi W_\psi\right]=\int_0^\infty{\rm d}t\int_{\mathbb R^d}{\rm d}x\int_{\mathbb R^d}{\rm d}y\phi(t,x)\psi(t,x)\delta(x-y)\tag 3$$ or $$\operatorname E\left[W_\phi W_\psi\right]=\int_0^\infty{\rm d}t\int_{\mathbb R^d}{\rm d}x\phi(t,x)\psi(t,x)\tag 4\;.$$ Then $W$ is called "white noise in time and colored noise in space" in the case $(3)$ and "white noise, both in time and space" in the case $(4)$. They simply state that $\delta$ is some "reasonable" kernel which might blow up to inifinity at $0$.
I suppose this is related to Definition 2. Again, I don't know how I need to generalize this to the case of a cylindrical Brownian moton.
Definition 4: This definition is very sloppy in its notation: Let Let $(W_t)_t$ be a centered Gaussian process with covariance $\operatorname E[W_sW_t]=\delta(s-t)$ where $\delta$ denotes the Dirac delta function. Then, in a [lecture note] I've found (Example 3.56), they state that $$B_t:=\int_0^tW_s\;{\rm d}B_s\tag 5\;\;\;\text{for }t\ge 0$$ is a real-valued Brownian motion. I haven't verified that result. Is it correct? Whatever the case is, if this is the reason, why white noise is considered to be the derivative of a Brownian motion, we should be able that every Brownian motion as a representation of the form $(5)$. Can this be shown?
The same questions as above remain.
Definition 5: Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ and define $$\langle W,\varphi\rangle:=\int_0^\infty\varphi(s)\;{\rm d}B_s\;\;\;\text{for }\phi\in\mathcal D:=C_c^\infty((0,\infty))\;.$$ Then $$\langle W',\varphi\rangle:=\int_0^\infty\varphi'(s)\;{\rm d}B_s\;\;\;\text{for }\phi\in\mathcal D$$ is considered to be the generalized derivative of the generalized Brownian motion $W$.
The same questions as above remain.
Conclusion: I've found different notions of "white noise" and "generalized derivative" of a Brownian motion, but I don't know in which sense they are consistent and which of them Liu and Röckner meant. So, I would be very happy if someone could give a rigorous definition of these terms in the case of a cylindrical Brownian motion or at least in the case of a Hilbert space valued Brownian motion.
Best Answer
I've taken a look at the book and this chapter is introductory and somewhat informal, so I imagine the authors are more specific about what they mean by a white noise in space or time and what they mean by the S(P)DE in your question in later chapters. Nevertheless, I have addressed aspects of your question below.
A discussion of Definitions 2, 3 and 5 are contained in an answer of mine to a similar question here. Everything in that answer is real-valued (which hopefully doesn't make too much of a difference) and indexed by a single real variable (or more precisely a test function of a single real variable); this can make a significant difference depending on what you want to know.
Definition 2
The random distribution that acts on $\phi$ via $(W, \phi) = \int \phi(t) B_t dt$ is just the Brownian motion $B$ (i.e. we can identify the function $B$ with the distribution $W$).
Your definition of $W'$ is then how I define white noise (denoted $X$) in the answer linked to above: white noise $X$ is defined as the random distribution that acts on a test function $\phi$ by $(X, \phi) = -\int_0^\infty B(t) f'(t) dt$. In the parlance of the book you cite, this is a white noise in time (time is the only variable in that answer). However, you can generalize this definition to white noise in space and time (see the discussion of Definition 3 below).
Definition 3
Here $W$ is your white noise (not $W'$ as in Definition 2).
To link this to definition 2, set $d = 0$ (so there is no spatial component to the domain of $\phi$). With $X$ defined as above, $(X_\phi := (X, \phi) : \phi \in C^\infty([0, \infty))$ is a centered Gaussian process with covariance $E(W_\phi W_\psi) = (\phi, \psi)_{L^2}$ (by the Ito isometry). The definition you have stated is a generalization to the case where the process is indexed by space and time (more precisely by test functions of space and time).
Definition 5
Your definition of $W$ is the same (by stochastic integration by parts) as the definition of $X$ above. Thus, $W$ here is once again white noise ($W'$ is then the distributional derivative of white noise).
Definition 1
In this definition, while the realization of the process you get in this way depends on the choice of basis, its (probability) distribution is independent of basis. You can think of a white noise as any process with this distribution.
This definition must be understood in the sense of distributions (now referring to Schwartz distributions) as white-noise is not defined pointwise (so $W_t$ is meaningless). A more precise definition is that $W$ acts on a test function $\phi$ by $W_\phi := (W, \phi) = \sum_{i=1}^\infty \xi_i (\phi, \phi_i)$. Now you can check that $W_\phi$ has mean $0$ and that \begin{equation} E(W_\phi W_\psi) = E\sum_{i=1}^\infty \xi_i \xi_j (\phi, \phi_i) (\psi, \phi_j) = \sum_{i=1}^\infty (\phi, \phi_i) (\psi, \phi_j) = (\phi, \psi)_{L^2}. \end{equation} Thus, the only thing to check to see that $W$ has the same distribution as the processes above is that it is Gaussian.