What's the difference between the Wiener measure on $C^0[0,1]$ and the distribution function of a Brownian motion?

Question

Let $(\Omega,\mathcal A, P)$ be a probability space and $B_t$ a standard Brownian motion on that space ($t \in [0,1]$). I am trying to understand what is called the "Wiener measure".

I never had a course on that, but I always thought that what is called the Wiener measure is simply the dynamical Gaussian measure associated with the Brownian motion, that is $\mu_t(A)= P(B_t \in A)=\int_A \frac{1}{\sqrt{2\pi t}} e^{-x^2/2\sqrt{t}} \, dx$ $\forall A \in \mathcal B(\mathbb R)$, so for each $t>0$, $\mu_t$ is a Gaussian measure (the cumulative distribution function/pushforward measure of the Brownian motion).

But I just discovered that it is not exactly the case. The Wiener measure is a measure on $(C^0[0,1], ||.||_\infty)$. So a measure on an infinite dimensional space. How do we even build such measure ? Intuitively, $\mu_t$ is "the projection on one dimension" of the Wiener measure. So the Wiener measure is able to measure much more than just sets in $\mathcal B(\mathbb R)$. It can measure sets in $\mathcal B(C^0[0,1])$. I have many questions on the Wiener measure, call it $W$:

• How is $W(A)$ defined for $A \in \mathcal B(C^0[0,1]) $ ? Does it have a density ? With regard to what measure ?
• Is it enough the give all "one dimensional projection" of a measure on an infinite dimensional space to define this measure ? So in that case, is knowing all $\mu_t$ enough to build $W$ ? How to do that ?
• Is knowing $W$ enough to build all $\mu_t$ ? If so, how to do that ? Same question but for a general measure on an infinite dimensional space, not necessarily $W$.
• What's the connection between $W$ and $B$ besides through the one dimensional projections ? Is $W$ the pushforward measure of a Brownian motion on $\mathbb R ^\infty$ ? Is the latter well defined ?

Answer 1

If $(B_t)_{t\in[0,1]}$ is a standard Brownian motion defined on $(\Omega, \mathcal A,P)$, then $\Psi: \omega\to (B_t(\omega))_{t\in[0,1]}$ is a measurable map from $(\Omega,\mathcal A)$ to $(C^0[0,1],\mathcal B(C^0[0,1]))$. The measure induced on $(C^0[0,1],\mathcal B(C^0[0,1]))$ by this map is the Wiener measure; $$ W(A)=P\left[\Psi^{-1}(A)\right]=P\left[\{\omega\in\Omega: \Psi(\omega)\in A\}\right], $$ for $A\in \mathcal B(C^0[0,1])$

Answer 2

Though this may no longer be of interest to OP over two years later, I noticed this is related to my own recent question, so I'll try to provide some (hopefully useful and correct) insight. See that question and my answer to it for references, in particular David Elworthy's lecture notes which I used extensively here:

The measure of the classical Wiener space is what I'd call "the Wiener measure" and I'd also say that this is essentially also the distribution of Brownian motion, though that will depend on what you consider the underlying $\Omega$ of Brownian motion to be.

Regarding the technical construction of (abstract) Wiener spaces: They consist of a Hilbert space $\mathcal{H}$ (the Cameron–Martin Hilbert space), a Banach space $E$ and a map $i: \mathcal{H} \to E$ which must be continuous, linear, injective with a dense range, and, crucially be $\gamma$-radonifying (or "radonify the canonical Gaussian cylinder set measure $\left\{\gamma_T^\mathcal{H}\right\}_T$").

To explain what the latter condition means, we first need to take a step back: First consider the set of all linear maps from the Hilbert space $\mathcal{H}$ to any other real vector space $F$, $\mathbb{L}(E;F)$, which will be a huge set of maps. Of this set of maps, consider only those maps $T \in \mathbb{L}(\mathcal{H}; F)$ which are surjective and map to a finite-dimensional vector space, and denote the image-space of the map $T$ by $F_T$ (i.e., $F_T = T(\mathcal{H})$ and $\dim{F_T} < \infty$). Note that the "simplest possible" maps $T$ here would be linear functionals where $F_T = \mathbb{R}$, which relates to your question, since this is essentially what you meant by "projection on one dimension". Call the set of all of these surjective finite-dimensional linear maps $\mathcal{A(H)}$. Then a cylinder set measure (CSM) $\{\mu_T\}_{T\in\mathcal{A(H)}}$ is defined to be a family of measures on the collection of all the $F_T$ for $T \in \mathcal{A(H)}$ which are in a sense self-consistent upon projecting from one $F_T$ to another $F_S$.

Practically speaking, this is the extension of the concept of "one-dimensional projections" to an arbitrary but finite number of projections, which comes in handy because we can have a natural definition of Gaussian measures for a finite number of dimensions, namely $$\gamma^{n}(A) = (2\pi)^{-n/2} \int_A e^{-\langle x, x \rangle/2}\ \mathrm{d}^n x$$ for $A \in \mathbb{R}^n$, where $\langle\cdot,\cdot\rangle$ is the regular $n$-dimensional Euclidean inner product. We extend this idea to define a canonical Gaussian CSM $\left\{\gamma^\mathcal{H}_T\right\}_{T \in \mathcal{A(H)}}$ on an arbitrary Hilbert space $\mathcal{H}$ with inner product $(\cdot, \cdot)_\mathcal{H}$ and norm $\|\cdot\|_\mathcal{H}$ as $$\gamma^\mathcal{H}_T(A) = (2\pi)^{-(\dim{F_T})/2} \int_A \exp\left(-\frac{1}{2} {\left\|T\rvert^{-1}_{(\ker{T})^\perp} x \right\|_\mathcal{H}}^2\right)\ \mathrm{d}^{\dim{F_T}}x.$$ As an example, if $T$ is a linear functional which is the dual of some basis vector $e_k \in \mathcal{H}$, then $T\rvert^{-1}_{(\ker{T})^\perp}$ is simply $e_k$ itself and $\gamma_T^\mathcal{H}$ is the standard 1D Gaussian measure that OP mentioned; but, of course, this extends to arbitrary "projections" $T$ onto finite-dimensional spaces $F_T$.

Finally, we define a radonifying map $i \in \mathbb{L}(\mathcal{H}; E)$ as a map whose pushforward CSM $\left\{i_\ast(\mu_\bullet)_S \mid S \in \mathcal{A}(E)\right\})$ is induced by a true measure $\mu$ on $E$ in the sense that $\mu_S = S_\ast(\mu)$ for every $S \in \mathcal{A}(E)$. Effectively, $i$ takes a space $\mathcal{H}$ with a cylinder set measure, which is defined on all finite-dimensional "projections" of $\mathcal{H}$, and maps onto another space $E$ in such a way that it consistently turns the CSM into a true measure on $E$. If the CSM is the canonical Gaussian CSM, then $i$ is called $\gamma$-radonifying and the resulting triple of map $i$, Hilbert space $\mathcal{H}$ and Banach space $E$ is called an abstract Wiener space, and the measure $\mu$ which we have constructed as the radonification of the canonical Gaussian CSM is precisely the Wiener measure.

Though this construction is quite technical, I believe it elucidates some of the aspects mentioned in the question. The classical Wiener space is an abstract Wiener space with $E = \{f(\tau) \in C[0,T] \mid f(0) = 0\}$ and $\mathcal{H} = L^{2,1}_0[0,T]$ (in one dimension), where the latter is the space of functions fulfilling $f(0) = 0$ and $\dot{f} \in L^2[0,T]$, and where $i: \mathcal{H} \hookrightarrow E$ is the inclusion map.

The fun part is now that as a direct consequence of the aforementioned properties, the Wiener measure has the property that (the values $\in \mathbb{R}$ of) any linear functional on $E$ that is induced by an element $h$ of $\mathcal{H}^\ast \cong \mathcal{H}$ is distributed as $\mathcal{N}(0, {\|h\|_\mathcal{H}}^2)$. Furthermore, if we introduce the adjoint $j: E^\ast \to \mathcal{H}^\ast \cong \mathcal{H}$ of the map $i$, then every functional $\ell \in E^\ast$ is distributed as $\mathcal{N}(0, {\|j(\ell)\|_\mathcal{H}}^2)$. This is how we recover the "one-dimensional projections" from the more general Wiener measure.

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To answer your questions as far as I can:

So a measure on an infinite dimensional space. How do we even build such measure? Intuitively, $\mu_t$ is "the projection on one dimension" of the Wiener measure.

I hope what I wrote above answered those questions sufficiently. $\mu_t$ here is the pushforward of the Wiener measure by the evaluation functional $E^\ast \ni \delta_t : B \mapsto B(t)$, which is evaluated as I stated in my last paragraph (see also my own linked question in the first sentence).

So the Wiener measure is able to measure much more than just sets in $\mathcal{B}(\mathbb{R})$. It can measure sets in $\mathcal{B}(C^0[0,1])$.

Yes. The Wiener measure actually can't measure sets in $\mathcal{B}(\mathbb{R})$, only its pushforward measures can.

How is $W(A)$ defined for $A\in \mathcal{B}(C^0[0,1])$?

I don't think there is a way to define an explicit function for $W(A)$; technically, if such a function existed, its input would be infinite-dimensional, so any rule that you could write down about how to map from the infinite-dimensional input to $\mathbb{R}$ may also be impossible to actually write down (reminiscent of the axiom of choice, but I don't know if that actually plays a role here).

Does it have a density? With regard to what measure?

You could define a density given some other measure. However, it is known that there is no infinite-dimensional Lebesgue measure, which is the measure w.r.t. which densities are usually given, so there can also be no density w.r.t. a Lebesgue measure. You can, however, give a density of one Gaussian measure (including Wiener measures) w.r.t. another, translated Gaussian measure, which is the statement of the Cameron–Martin theorem. This probably won't help in identifying/understanding the nature of the Gaussian measure itself, though.

Is it enough the give all "one dimensional projection" of a measure on an infinite dimensional space to define this measure? So in that case, is knowing all $\mu_t$ enough to build $W$? How to do that?

Technically, I don't know the answer to this, but I strongly believe it is no. The one-dimensional projections are a subset of the $T \in \mathcal{A(H)}$ and I would think that using only the associated subset of the cylinder set measure is not strict enough to lead to the same measure on $E$ (i.e., I think there might be a number of different measure candidates then, one of which is the correct Wiener measure, but this is speculation). However, if we expand your idea to include all finite-dimensional projections, then we essentially get the idea that leads to the correct construction of the Wiener measure.

Is knowing $$ enough to build all $\mu_t$? If so, how to do that? Same question but for a general measure on an infinite dimensional space, not necessarily $W$.

Yes, see above. For the more general case, the structure theorem tells us that any Gaussian measure is actually a Wiener measure as constructed above, and in the non-Gaussian case, the same cylinder set measure construction can be used to evaluate the measure of linear functionals on the (topological) dual space.

What's the connection between $W$ and $B$ besides through the one dimensional projections?

This is slightly a matter of notation or interpretation, but I'd say $B : [0,T] \times \Omega \to \mathbb{R}$ is essentially a sample from the set $E = C^0[0,T]$, which is measured by the Wiener measure $W$ or $\mu$. $B_t: \Omega \to \mathbb{R}$ is equivalent to the value of all $f \in E$ at time $t$, which is distributed according to pushfoward by the evaluation functional $E^\ast \ni \delta_t : B \mapsto B(t)$ that we mentioned above, and for arbitrary "multi-time evaluations", you can define a tuple of such evaluations functionals.

Is $W$ the pushforward measure of a Brownian motion on $\mathbb{R}^\infty$? Is the latter well defined?

And finally, as I mentioned at the beginning, the Wiener measure is in a sense the distribution of Brownian motion itself. The difference, as I understand, is that Brownian motion is generally taken to be a family of random variables $\{B_t\}_{t\in[0,T]}$ without any claim to "knowing" the distribution of the function-valued object $B$ itself; whereas the Wiener measure is actually the distribution of this function-valued object itself. However, though the Wiener measure is therefore a rather beautiful object itself, it seems that the more commonplace "time-chopped" stochastic integral formulation may be easier to work with or more versatile than the abstract Wiener measure.