Constructing the 'idealized white noise' stochastic process

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.

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.

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.

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 $$\dot W$$?

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 $$\dot W$$?

• What about $\otimes_{t \in \mathbb{R}_+} \mathcal{N}(0,1)$, the product of standard normal distributions? Mar 31, 2021 at 11:27
• You mean to say that we use the Kolomogorov extension theorem? Mar 31, 2021 at 11:31
• Seems to be one of the most simple ways to construct white noise. Mar 31, 2021 at 12:19
• What you call 'idealised white noise' is not white noise. It also isn't a stochastic process in the usual sense because it cannot be made measurable in $t$. Mar 31, 2021 at 13:43
• @UserA: "idealized white noise" in the book is to be understood as "heuristic/physicsy white noise". Moreover, they don't give the variance of $\dot{W}_t$ because this is a "$N(0,\infty)$" variable. The issue here is one should not take seriously the $t$-indexed process $\{\dot{W}_t\}_{t\ge 0}$. Mar 31, 2021 at 20:16

1 Answer

$$\newcommand\ep\varepsilon$$

• 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" $$\dot W$$.

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".

• @oferzeitouni : The variance of $\dot W^{(\varepsilon)}_t$ is $1$. Mar 31, 2021 at 15:52
• @UserA If you want your $\dot W$ to be actual white noise (which is rigorously defined in one of the two ways you mention in your question), then this can heuristically be thought of as i.i.d. Gaussians with infinite variance, just like the 'delta function' can be heuristically thought of as a function that vanishes everywhere except at the origin where it is infinite. Real white noise is of course not built by Kolmogorov's extension theorem in the way suggested. (It can as a test function-indexed process though.) ... Apr 2, 2021 at 10:42
• ... On the other hand, you can use Kolmogorov's extension theorem to construct an $\mathbb{R}$-indexed i.i.d. collection of Gaussian random variables with some fixed finite variance (say $1$). This is what most commenters here believe you mean when you talk about 'idealised white noise' and what I mentioned cannot be realised as a jointly measurable stochastic process. The relation between the two is like the relation between the 'delta function' and the indicator function of the set $\{0\}$, i.e. intuitively appealing but actually pretty much nonexistent. Apr 2, 2021 at 10:46
• @IosifPinelis What exactly would you mean by such a statement given that the set of measurable functions isn’t itself measurable with respect to the product sigma-algebra? Apr 2, 2021 at 15:48
• @UserA If by random field you mean any $\mathbb{R}$-indexed family of random variables then yes. Most authors would however implicitly assume some measurability with respect to the underlying space when talking about a 'random field', so I would think that in most cases it wouldn't qualify... Apr 2, 2021 at 20:43