Existence and scaling of $d$-dimensional Brownian motion

Let $\mathsf{W}$ be Wiener measure on $C=C([0,\mathrm{\infty }),\mathbb{R})$ from [F1]. By [F2], on some probability space there is a countable independent family $(X_{\alpha }{)}_{\alpha \ge 1}$ of $C$-valued random elements, each with law $\mathsf{W}$. Retain its first $d$ members.

For the scaling assertion, put $I=[0,\mathrm{\infty })$ and define $T_c:{\mathbb{R}}^I\to {\mathbb{R}}^I$ by $(T_{c}f)(t)=c^{-1\mathrm{/}2}f(ct)$. This map is cylinder-measurable: the inverse image of a finite-coordinate cylinder supported on $F$ is a finite-coordinate cylinder supported on $cF$, with its base pulled back by coordinatewise scalar multiplication.

Define ${\widehat{B}}_t^{\alpha }={\pi }_t(X_{\alpha })$ and ${\widehat{B}}_t=({\widehat{B}}_t^1,\dots ,{\widehat{B}}_t^d)$ for $1\le \alpha \le d$ and $t\ge 0$. Every $X_{\alpha }(\omega )$ is a continuous path. Its finite evaluation laws are Brownian by [F1], so ${\widehat{B}}^{\alpha }$ is standard one-dimensional Brownian motion, with the whole sample space as a continuity event and with its initial and increment laws supplied by $\mathsf{W}$. The inclusion $J:C\to {\mathbb{R}}^{[0,\mathrm{\infty })}$, $J(f)=(f(t){)}_t$, is measurable because every target cylinder pulls back through finitely many Borel evaluation maps by [F1] and [F5]. Therefore the process random elements $J\circ X_{\alpha }$ are independent by [F4].

Return now to the arbitrary standard $d$-dimensional Brownian motion $B$ in the scaling hypothesis. By [F3], its coordinate-process random elements ${\mathrm{\Phi }}_{\alpha }=(B_t^{\alpha }{)}_t$ are independent and each coordinate is standard one-dimensional Brownian motion. Since $(Y_t^{\alpha }{)}_t=T_c{\mathrm{\Phi }}_{\alpha }$, [F4] and step 1.2 show that the coordinate processes of $Y$ remain independent, while scalar Brownian scaling makes every one of them standard Brownian motion.

Applying the coordinate equivalence [F3] to step 2.1 makes the assembled process $\widehat{B}$ a standard $d$-dimensional Brownian motion. This proves existence and realizes it from independent copies of the constructed scalar path law.

The reverse direction of [F3] applied to step 2.2 shows that $Y_t=c^{-1\mathrm{/}2}{B}_{ct}$ is standard $d$-dimensional Brownian motion. The same common coordinate argument includes continuity at $t=0$; it is not inferred from finite-dimensional laws.

Any two standard $d$-dimensional Brownian motions have the same finite-dimensional laws. Indeed, on a sorted finite time grid their vector increments are mutually independent with the same respective laws $N_d(0,(t_j-t_{j-1})I_d)$ by [F3]. Their joint increment laws therefore agree on measurable rectangles and hence on the finite product sigma-algebra by [F5]; the cumulative-sum map gives equality of the evaluation-vector laws. An arbitrary finite, unordered, or repeated time list is a coordinate projection of the sorted distinct-time vector, and an empty list has the unit law. Applying this to $B$ and $Y$ uses step 3.2.

The path maps ${\mathrm{\Theta }}_B,{\mathrm{\Theta }}_Y:\mathrm{\Omega }\to ({\mathbb{R}}^d{)}^I$ are measurable because every finite-coordinate cylinder has a measurable preimage. Their laws are probabilities by [F5]. Step 4.1 makes those laws agree on all finite-coordinate cylinders, a generating pi-system, and their total masses are both one. Finite-measure uniqueness in [F5] therefore gives $\mathcal{L}({\mathrm{\Theta }}_B)=\mathcal{L}({\mathrm{\Theta }}_Y)$ on the full cylinder sigma-algebra.

The construction in steps 1.1--3.1 and the scaling and law conclusions in steps 1.2--5.1 establish both claims. When $d=1$, the construction and scaling reduce to the scalar results. The empty coordinate case $d=0$ is excluded; empty time lists and repeated times were handled in step 4.1. At $c=1$, $T_c$ is the identity, while $c=0$ is excluded because $c^{-1\mathrm{/}2}$ is undefined and would not preserve Brownian covariance. AC is used through [F1] and [F3] for Brownian and Gaussian laws and through [F2] for the countable product; the finite truncation to $d$, fixed scaling map, and cylinder-law comparison use no further choice. Thus the required process exists and the displayed scaling has the same $d$-dimensional law.