Constructing the 'idealized white noise' stochastic process

Question

There are some authors, namely H. Holden, B. Øksendal, and J. Ubøe T. Zhang in their book Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach, that define idealized 1-dimensional white noise as a "stochastic process" $\{\dot W_t\}_{t\geq 0}$ with the following properties:

1. It is a mean zero Gaussian process.
2. If $t_1\neq t_2$ then $\dot W_{t_1}$ and $\dot W_{t_2}$ are independent.
3. It is a stationary process.

As the commenters observed and mentioned this already, I have found that author B. Øksendal of the book Stochastic Differential Equations mentions that :

If we require that $\mathbb{E}\big[\dot W_t^2\big]=1$, then the function $(\omega,t)\mapsto \dot W_t(\omega)$ cannot even be measurable, with respect to the $\sigma$-algebra $\mathcal{B}\times \mathcal{F}$, where $\mathcal{B}$ is the Borel $\sigma$-algebra on $[0,\infty]$.

However, in several references I have read so far, I do not see an explicit construction of this process with index set $T=\mathbb{R}^+$. Rather, white noise is defined in the following two ways.

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White noise indexed by a $\sigma$-algebra of a $\sigma$-finite measure space.

Let $(M,\mathcal{F},\sigma)$ be a $\sigma$-finite measure space and let $T$ be the collection of all sets in $\mathcal{F}$ of finite measure. White noise $\{W_A\}_{A\in T}$ is then defined as mean zero Gaussian process such that the following hold.

• For each $A\in T$, $W_A$ is a normal random variable with zero mean and variance $\sigma(A)$.
• If $A, B\in T$ and $A\cap B=\emptyset$ then $W_A$ and $W_B$ are independent.
• For all $A,B\in T$ we have that $W(A\cup B)=W(A)+W(B)-W(A\cap B)$.

We use some version of the Bochner-Minlos theorem to get the existence of such a process. It has the property that for any $A,B\in T$ that $\text{Cov}(W_A,W_B)=\sigma(A\cap B)$. Note that if $M=\mathbb{R}$, $\mathcal{F}=\mathcal{B}$ is the Borel $\sigma$-algebra and $\sigma=\lambda$ is the Lebesgue measure then the process $$\{B_t\}_{t\geq 0}:=\left\{W\left((0,t]\right)\right\}_{t\geq 0}$$ is just regular Brownian motion.

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White noise indexed by Shwartz functions.

Let $T:=\mathcal{S}(\mathbb{R})$ be the Shwartz space rapidly decreasing smooth functions. Let $\Omega:=\mathcal{S'}(\mathbb{R})$ be the space of tempered distributions equipped with the cylinder algebra $\mathscr{C}$. Another version of the Bochner-Minlos theorem states that there is a probability measure $\mathbb{P}$ on $(\Omega,\mathscr{C})$ such that $$\int_\Omega e^{i\langle \omega,\varphi\rangle} d\mathbb{P}(\omega)=\exp\left(-\frac{1}{2}\|\varphi\|_{L^2(\mathbb{R})}\right),\;\;\;\text{ for all }\varphi\in T.$$ White noise is then defined as $$W(t,\omega):=\langle \omega, \varphi \rangle,\;\;\;\text{ for all }(t,\omega)\in T\times \Omega. $$ Note that for $\omega\in\Omega$ and for any $\varphi\in L^2(\mathbb{R})$ one can define $\langle \omega,\varphi\rangle$ by using a sequence of functions $\{\varphi_n\}$ in $T$ that converge to $\varphi$. This allows to define the following: $$B_t(\omega):=\langle \omega,\mathbf{1}_{(0,t]}\rangle,\;\;\;\text{ for all }t\in\mathbb{R}^+. $$ Then $\{B_t\}$ is just one dimensional Brownian motion.

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The Question:

As you can see, one can use white noise (defined in any of the above ways) to construct Brownian motion.

• Can we use the Shwartz space indexed or $\sigma$-algebra indexed white noise $W$, to otain the 'idealized white noise' $\{\dot W_t\}$ indexed by time?
• Is there another way to construct $\dot W$?

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My idea: (can skip this part)

We can use the second construction of white noise above to define a process called the smoothed white noise process. It is defined as follows: For $\varphi\in L^2(\mathbb{R})$ and any $t\in \mathbb{R}$ let $$\varphi_t(x):=\varphi(t-x),\;\;\;\text{ for }x\in \mathbb{R}.$$ Then define $$W_\varphi(t,\omega):=W(\varphi_t,\omega)=\langle \omega,\varphi_t\rangle,\;\;\;\text{ for all }(t,\omega)\in T\times\Omega. $$

$W_\varphi$ is then a zero mean stationary Gaussian process with constant variance equal to $\|\varphi\|_{L^2}$. It has the properties that for any $t_1\neq t_2$, if $\varphi_{t_1}$ and $\varphi_{t_2}$ have disjoint supports then $W(t_1,\cdot)$ and $W(t_2,\cdot)$ are independent.

So I was thinking, if we take a sequence $\{\varphi_n\}$ in $C_c^\infty(\mathbb{R})$ with shrinking supports, then would the sequence of smoothed white noises $\{W_{\varphi_n}\}$ converge (in a appropriate sense) to an idealized white noise $\dot W$?

Answer 1

$\newcommand\ep\varepsilon$

• Can we use white noise $W$ to construct idealized white noise $\dot W$?

First, use your favorite white noise $W$ to construct a standard Brownian motion $B$, as you did in your post. Then, by the Kolmogorov extension theorem, the finite-dimensional distributions of the process $\dot W^{(\ep)}$ defined by the formula $$\dot W^{(\ep)}_t:=\frac{B_{t+\ep}-B_t}{\sqrt\ep}\tag{1}$$ for real $t\ge0$ converge (as $\ep\downarrow0$) to the finite-dimensional distributions of what you refer to as an idealized white noise $\dot W$; here we used the fact that, for any $k$-tuple $(t_1,\dots,t_k)$ of nonnegative real numbers and all small enough $\ep>0$, the intervals $[t_1,t_1+\ep],\dots,[t_k,t_k+\ep]$ will be pairwise disjoint.

• Is there another way to construct $\dot W$?

Yes, as suggested in comments by Dieter Kadelka, you can use the Kolmogorov extension theorem directly to construct an "idealized white noise" $\dot W$.

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Comments:

1. I think the notation $\dot W$ for your "idealized white noise" is not appropriate, because the denominator in (1) is, not $\ep$, but $\sqrt\ep$.

2. It is unclear to me why you need this "idealized white noise".