Brownian Motion Construction and Continuity

Fix $n\ge 1$ and times $t_1,\dots ,t_n\in I$. By [F1], the vectors $X^{(n)}=(X_{t_1},\dots ,X_{t_n}),Y^{(n)}=(Y_{t_1},\dots ,Y_{t_n})$ are multivariate normal. Their mean vectors agree by the first given identity. Their covariance matrices agree entry by entry by the second identity, even if some times repeat and the common matrix is singular.

Write the common mean vector and covariance matrix as $m$ and $\mathrm{\Sigma }$. By [F2], both vector characteristic functions equal $u⟼\exp (iu\cdot m-u^T\mathrm{\Sigma }u\mathrm{/}2)\mathrm{.}$ The uniqueness clause of [F2] therefore gives . Since the finite time list was arbitrary, all finite-dimensional distributions agree. The empty-coordinate law, if included as a convention, is the unique probability law on the singleton empty tuple. AC is used exactly through [F1]–[F2], not to choose a version of either process.

Symmetry is immediate from $\min (s,t)=\min (t,s)$. If $n=0$, the displayed quadratic form is the empty sum $0$. Now take $n\ge 1$, let $0=r_0<r_1<\cdots <r_k$ be the increasing list of the distinct positive values among $t_1,\dots ,t_n$, and put $A_{\mathrm{\ell }}={\sum }_{i:t_i\ge r_{\mathrm{\ell }}}{a}_i$. The list is finite and uniquely fixed by the given times; no choice function is used.

For every $i,j$, $\min (t_i,t_j)={\sum }_{\mathrm{\ell }=1}^k(r_{\mathrm{\ell }}-r_{\mathrm{\ell }-1}){\mathbf{1}}_{\{t_i\ge r_{\mathrm{\ell }}\}}{\mathbf{1}}_{\{t_j\ge r_{\mathrm{\ell }}\}}\mathrm{.}$ Indeed, if the smaller of $t_i,t_j$ is $0$ both sides vanish, while if it is $r_m$ the right side telescopes to ${\sum }_{\mathrm{\ell }=1}^m(r_{\mathrm{\ell }}-r_{\mathrm{\ell }-1})=r_m$.

Substituting step 2.1 and rearranging only finite sums gives ${\sum }_{i,j=1}^n{a}_i{a}_j\min (t_i,t_j)={\sum }_{\mathrm{\ell }=1}^k(r_{\mathrm{\ell }}-r_{\mathrm{\ell }-1}){\left({\sum }_{i:t_i\ge r_{\mathrm{\ell }}}{a}_i\right)}^2={\sum }_{\mathrm{\ell }=1}^k(r_{\mathrm{\ell }}-r_{\mathrm{\ell }-1})A_{\mathrm{\ell }}^2\mathrm{.}$ Every weight is positive and every square is nonnegative, so the quadratic form is nonnegative. This includes coincident times, zero times, zero coefficients, and the case $k=0$, where the last sum is empty.

For $n=0$, set ${\mu }_{()}$ equal to the unique law on the singleton empty tuple. For $n\ge 1$, [F1] makes $\mathrm{\Sigma }(\mathbf{t})$ an admissible covariance matrix, so [F2] supplies the centered law ${\mu }_{\mathbf{t}}=N_n(0,\mathrm{\Sigma }(\mathbf{t}))\mathrm{.}$ This remains well-defined when a time is zero, times repeat, or the covariance is singular.

Let $\theta :\{1,\dots ,m\}\to \{1,\dots ,n\}$ be any map, let $R:{\mathbb{R}}^n\to {\mathbb{R}}^m$ be the coordinate map $(Rx{)}_j=x_{\theta (j)}$, and take $X\sim {\mu }_{\mathbf{t}}$. For $u\in {\mathbb{R}}^m$, [F3] gives $E{e}^{iu\cdot RX}=E{e}^{i(R^{T}u)\cdot X}=\exp \text{\hspace{0.17em}⁣}(-\frac{1}{2}{u}^{T}R\mathrm{\Sigma }(\mathbf{t})R^{T}u)\mathrm{.}$ The $(j,k)$ entry of $R\mathrm{\Sigma }(\mathbf{t})R^T$ is $\min (t_{\theta (j)},t_{\theta (k)})$, so [F3] identifies the law of $RX$ with ${\mu }_{(t_{\theta (1)},\dots ,t_{\theta (m)})}$. The case $m=0$ is the unique empty-tuple law.

A bijective $\theta$ permutes coordinates, an injective coordinate selection deletes the unselected coordinates, and a noninjective $\theta$ repeats coordinates. Thus step 2.1 proves all three promised compatibilities, with both orders of any inverse permutation covered. AC is spent only in the existence and characteristic-function uniqueness suppliers [F2]–[F3].

By [F5], the usual real line is complete and has the countable dense subset $\mathbb{Q}$, hence is Polish; its Borel measurable space is therefore standard Borel via the identity presentation. This verifies, rather than merely assumes, the coordinate-space hypothesis of [F2].

For every finite $F\subseteq [0,\mathrm{\infty })$, use [F1] to put on ${\mathbb{R}}^F$ the centered Gaussian law with covariance $(\min (s,t){)}_{s,t\in F}$; step 1.1 and [F2] give a probability $P$ on the cylinder sigma-algebra of ${\mathbb{R}}^{[0,\mathrm{\infty })}$ with exactly those marginals. The empty marginal has mass one on the singleton empty product.

Let $X_t(x)=x(t)$. By [F3], every finite evaluation vector of $X$ has the prescribed centered Gaussian law. Thus [F4] makes $X$ Gaussian, and its one- and two-coordinate marginals give $E[X_t]=0$ and $E[X_s{X}_t]=\min (s,t)$, including $X_0=0$ almost surely because its variance is zero.

The conclusion of [F2] is only a measure on the cylinder sigma-algebra realizing finite-coordinate laws; neither [F2] nor [F3] supplies a common full-measure set on which $t\mapsto X_t(x)$ is continuous. Consequently step 3.1 does not establish the path-continuity clause needed for Brownian motion. AC is used in [F1]–[F2] and the Gaussian interface [F4], including the choices inside arbitrary-index Kolmogorov extension; no stronger regularity is inferred from it.

Assume condition 1 and fix $0=t_0<t_1<\cdots <t_n$. By [F1], the increment vector $\mathrm{\Delta }=(X_{t_j}-X_{t_{j-1}}{)}_{j=1}^n$ is multivariate normal. It is centered, and $\mathrm{Var}({\mathrm{\Delta }}_j)=t_j-t_{j-1}\mathrm{.}$ If $i<j$, expanding the four covariance terms and using $t_{i-1}<t_i\le t_{j-1}<t_j$ gives $\mathrm{Cov}({\mathrm{\Delta }}_i,{\mathrm{\Delta }}_j)=t_i-t_i-t_{i-1}+t_{i-1}=0.$ Thus $\mathrm{\Delta }\sim N_n(0,D)$ for $D=\mathrm{diag}(t_j-t_{j-1})$, by [F3].

Conversely assume condition 2. Given any finite time list, discard repetitions only for the construction and write its distinct positive values as $0=r_0<r_1<\cdots <r_k$. Put ${\mathrm{\Delta }}_{\mathrm{\ell }}=X_{r_{\mathrm{\ell }}}-X_{r_{\mathrm{\ell }-1}}$. Then condition 2 makes these independent with ${\mathrm{\Delta }}_{\mathrm{\ell }}\sim N(0,r_{\mathrm{\ell }}-r_{\mathrm{\ell }-1})$, and telescoping with $X_0=0$ gives $X_t={\sum }_{\mathrm{\ell }:r_{\mathrm{\ell }}\le t}{\mathrm{\Delta }}_{\mathrm{\ell }}$ almost surely at every time in the original list.

By [F2], $N_n(0,D)$ is also the joint law of the vector $(\sqrt{t_j-t_{j-1}}{Z}_j{)}_{j=1}^n$ with independent standard normals $Z_j$. The joint-law identity in step 1.1 transfers every measurable-rectangle probability, so [F7] makes the ${\mathrm{\Delta }}_j$ mutually independent; their one-coordinate laws are $N(0,t_j-t_{j-1})$. For $n=0$ this is the vacuous empty family. Hence condition 2 holds.

For coefficients $a_1,\dots ,a_m$ attached to the original time list, step 1.2 rewrites ${\sum }_{j=1}^m{a}_j{X}_{t_j}={\sum }_{\mathrm{\ell }=1}^k{b}_{\mathrm{\ell }}{\mathrm{\Delta }}_{\mathrm{\ell }},b_{\mathrm{\ell }}={\sum }_{j:r_{\mathrm{\ell }}\le t_j}{a}_j\mathrm{.}$ By [F4]–[F5], its characteristic function is ${\prod }_{\mathrm{\ell }=1}^k\exp \text{\hspace{0.17em}⁣}(-\frac{1}{2}{u}^2{b}_{\mathrm{\ell }}^2(r_{\mathrm{\ell }}-r_{\mathrm{\ell }-1}))=\exp \text{\hspace{0.17em}⁣}(-\frac{1}{2}{u}^2{\sum }_{\mathrm{\ell }=1}^k{b}_{\mathrm{\ell }}^2(r_{\mathrm{\ell }}-r_{\mathrm{\ell }-1}))\mathrm{.}$ By [F4] and [F6], this is a centered normal law, including the empty sum and variance zero. Since the list and coefficients were arbitrary, [F1] makes $X$ a centered Gaussian process.

For $0\le s\le t$, condition 2 decomposes $X_t=X_s+(X_t-X_s)$ into independent centered normal variables. Consequently [F8] gives $E[X_s{X}_t]=E[X_s^2]+E[X_s]E[X_t-X_s]=s,$ while both means are zero by step 2.2; symmetry gives $\mathrm{Cov}(X_s,X_t)=\min (s,t)$ for every order of $s,t$. The cases $s=0$, $s=t$, and $t=0$ follow from the same formula and $X_0=0$ almost surely. Thus condition 1 holds. AC is used exactly in [F1]–[F4] and [F6], for the library's normal-law constructions and characteristic-function uniqueness; no path regularity is asserted.

Let $D=\{k{2}^{-n}:k,n\in {\mathbb{N}}_0\}$ be the nonnegative dyadic times. Fix an integer $N\ge 1$ and a rational $\eta$ with $0<\eta <\beta \mathrm{/}\alpha$. For $n\ge 0$, let $E_{N,\eta ,n}$ be the event that some adjacent level-$n$ dyadic pair $k{2}^{-n},(k+1)2^{-n}$ in $[0,N]$ has $d(X_{(k+1)2^{-n}},X_{k{2}^{-n}})>2^{-n\eta }$. These are measurable finite unions.

Each of the at most $N{2}^n$ edges in step 1.1 has, by [F2] and the moment hypothesis, probability at most $C_N{2}^{-n(1+\beta -\alpha \eta )}$. Hence [F3] gives $P(E_{N,\eta ,n})\le N{C}_N{2}^{-n(\beta -\alpha \eta )}\mathrm{.}$ The exponent is positive, so [F4] makes the sum over $n$ finite and [F5] shows that, almost surely, only finitely many $E_{N,\eta ,n}$ occur.

Intersect the probability-one conclusions of step 2.1 over the countable set of integer $N\ge 1$ and rational $\eta \in (0,\beta \mathrm{/}\alpha )$, and call the resulting measurable event $H$. Countability follows from [F8], and its complement is a countable union of null sets and is null by [F3]. Fix $\omega \in H,N,\eta$. The eventual edge bound can be enlarged over its finitely many exceptional levels to a finite $K=K(\omega ,N,\eta )$ satisfying $d(X_{(k+1)2^{-n}}(\omega ),X_{k{2}^{-n}}(\omega ))\le K{2}^{-n\eta }$ for every level-$n$ edge in $[0,N]$.

If $p,q\in D\cap [0,N]$ and $2^{-(m+1)}<\mathrm{\mid }p-q\mathrm{\mid }\le 2^{-m}$, compare each point with its level-$m$ dyadic floor. Successive binary floors differ by at most one edge of level $j+1$, so step 3.1 and [F4] bound each tail by $K{\sum }_{j\ge m}{2}^{-(j+1)\eta }=K{2}^{-(m+1)\eta }\mathrm{/}(1-2^{-\eta })$; the two level-$m$ floors differ by at most one level-$m$ edge. Therefore $d(X_p(\omega ),X_q(\omega ))\le K\text{\hspace{0.17em}⁣}(1+\frac{2^{1-\eta }}{1-2^{-\eta }})2^{-m\eta }\le K^{\mathrm{\prime }}\mathrm{\mid }p-q{\mathrm{\mid }}^{\eta }\mathrm{.}$ Equality $p=q$ is trivial. Thus the sample values on the dense dyadic set are locally Hölder.

For $t\ge 0$, let $q_j(t)$ be the largest level-$j$ dyadic not exceeding $t$; then $0\le t-q_j(t)<2^{-j}\to 0$ by [F9]. On $H$, step 4.1 makes $(X_{q_j(t)}(\omega ){)}_j$ Cauchy on any integer compact containing $t$, so [F1] gives a unique limit. Define $Y_t(\omega )$ to be this limit on $H$ and $X_0(\omega )$ on $H^c$. Equivalently, the measurable maps $Z_j^t=X_{q_j(t)}$ on $H$ and $Z_j^t=X_0$ on $H^c$ converge pointwise to $Y_t$.

Each $Y_t$ is a Borel random element. Indeed, for a nonempty closed $F\subseteq S$, continuity of $x\mapsto d(x,F)$ and step 5.1 give $\{Y_t\in F\}={\bigcap }_{r\ge 1}{\bigcup }_{J\ge 1}{\bigcap }_{j\ge J}\{d(Z_j^t,F)<1\mathrm{/}r\};$ the formula is also correct in the limiting direction because $F$ is closed, while the empty closed set has empty inverse image. Thus inverse images of closed, hence Borel, sets are measurable. No point of $S$ was selected: the already given $X_0(\omega )$ supplies the value on $H^c$.

Letting dyadic $p,q$ tend to arbitrary $s,t\in [0,N]$ in step 4.1 shows on $H$ that $d(Y_s,Y_t)\le K^{\mathrm{\prime }}\mathrm{\mid }s-t{\mathrm{\mid }}^{\eta }$. Hence every path of $Y$ on $H$ is continuous and is locally $\eta$-Hölder for every rational $\eta <\beta \mathrm{/}\alpha$. For arbitrary $0<\gamma <\beta \mathrm{/}\alpha$, [F8] supplies one rational $\eta$ strictly between them; on $[0,N]$, the $\eta$-bound implies the $\gamma$-bound after multiplying its constant by $\max (1,N^{\eta -\gamma })$. This gives all exponents simultaneously on the single event $H$.

Fix $t$ and an integer $N>t$. The moment hypothesis and [F2] give $P(d(X_{q_j(t)},X_t)>\epsilon )\le C_N{\epsilon }^{-\alpha }\mathrm{\mid }{q}_j(t)-t{\mathrm{\mid }}^{1+\beta }\to 0$. Since $Z_j^t=X_{q_j(t)}$ off only the null event $H^c$, the same convergence holds with $Z_j^t$. On the other hand, $Z_j^t\to Y_t$ pointwise, so [F6] applied to ${\mathbf{1}}_{\{d(Z_j^t,Y_t)>\epsilon \}}$ makes the corresponding probabilities tend to zero. The triangle inequality now gives $P(d(X_t,Y_t)>2\epsilon )\le P(d(X_t,Z_j^t)>\epsilon )+P(d(Z_j^t,Y_t)>\epsilon )\to 0.$ The left side is independent of $j$, hence is zero; intersecting over $\epsilon =1\mathrm{/}r$ gives $X_t=Y_t$ almost surely. Thus [F7] makes $Y$ a modification of $X$, and step 6.2 supplies all promised path regularity. The constants $(C_T)$ were supplied as data, and every other construction was canonical or countably intersected, so no choice axiom is used.

Let $Z$ be the coordinate map on the canonical $N(0,1)$ probability space. By [F1]--[F3], for each integer $r\ge 0$, $E\mathrm{\mid }Z{\mathrm{\mid }}^{2r}={\int }_{\mathbb{R}}{x}^{2r}\varphi (x)dx,$ with either side initially allowed to be infinite.

For an integer $R\ge 1$ and $r\ge 1$, [F5] on $[-R,R]$, applied to $u(x)=x^{2r-1}$ and $v(x)=\varphi (x)$, is legitimate by [F6] and gives ${\int }_{-R}^R{x}^{2r}\varphi (x)dx=-2R^{2r-1}\varphi (R)+(2r-1){\int }_{-R}^R{x}^{2r-2}\varphi (x)dx\mathrm{.}$ The sign and factor two come from the odd power at the two endpoints and the evenness of $\varphi$.

Taking $k=r+1$ in [F7] shows $0\le R^{2r-1}\varphi (R)\le \frac{2^{r+1}(r+1)!}{\sqrt{2\pi }{R}^3}⟶0.$ The compact bridge in [F5] identifies every Riemann integral in step 1.2 with the corresponding Lebesgue integral. The truncated nonnegative integrands then increase to their whole-line counterparts, so [F4], step 1.1, and $\int \varphi =1$ from [F1] yield recursively $c_0=1,c_r=(2r-1)c_{r-1}\mathrm{.}$ Thus $c_m=(2m-1)!!$ and every $c_m$ is finite.

Put $h=\mathrm{\mid }t-s\mathrm{\mid }$. By [F1], the law $N(0,h)$ is that of $\sqrt{h}Z$; applying [F2] to the nonnegative function $x\mapsto \mathrm{\mid }x{\mathrm{\mid }}^{2m}$ and using step 2.1 gives $E\mathrm{\mid }{X}_t-X_s{\mathrm{\mid }}^{2m}=E\mathrm{\mid }\sqrt{h}Z{\mathrm{\mid }}^{2m}=h^m{c}_m\mathrm{.}$ This includes $h=0$, when both sides vanish and the law is the Dirac mass at zero.

If $m\ge 2$, set $\alpha =2m$ and $\beta =m-1>0$. Then $m=1+\beta$, so step 3.1 reads $E\mathrm{\mid }{X}_t-X_s{\mathrm{\mid }}^{\alpha }=c_m\mathrm{\mid }t-s{\mathrm{\mid }}^{1+\beta }\mathrm{.}$ The constant $C_T=c_m$ is finite and independent of $T$. AC is used through [F1], which supplies the normal-law probability measure, and through the countable-choice hypothesis of the compact bridge used in step 2.1; all truncations and the recurrence are canonical.

By [F1], on the canonical coordinate probability space there is a centered Gaussian process $X$ with covariance $\min (s,t)$. In particular $X_0=0$ almost surely, and [F4] gives $X_t-X_s\sim N(0,\mathrm{\mid }t-s\mathrm{\mid })$ for all $s,t\ge 0$.

By [F2] with $m=2$, $E\mathrm{\mid }{X}_t-X_s{\mathrm{\mid }}^4=3\mathrm{\mid }t-s{\mathrm{\mid }}^2\mathrm{.}$ By [F6], the real target is complete and separable. Thus [F3], with $\alpha =4$, $\beta =1$, and the supplied constants $C_T=3$, gives a modification $B$ of $X$ and one probability-one event on which every path of $B$ is continuous.

Fix a finite time list $t_1,\dots ,t_n$. Since $B$ is a modification, each event $\{B_{t_j}\mathrm{\ne }{X}_{t_j}\}$ is null; [F7] makes their finite union null. Hence the two evaluation vectors agree almost surely and have the same law. This includes the empty list, for which both laws are the unit mass on the empty tuple. Consequently all finite-dimensional laws of $B$ equal those of $X$, so $B$ is centered Gaussian with covariance $\min (s,t)$ and $B_0=0$ almost surely.

Apply [F4] to $B$: it has the independent $N(0,t_j-t_{j-1})$ increments on every finite increasing list. Together with $B_0=0$ from step 3.1 and the common continuity event from step 2.1, [F5] says that $B$ is standard Brownian motion. AC is used through [F1], [F2], [F4], and [F5] for normal-law construction and the arbitrary-index extension; the continuity construction [F3] adds no choice.

For each integer $m\ge 2$, [F1]--[F3] give a continuous modification $Y^{(m)}$ of $B$ and a probability-one event $H_m$ on which its paths are locally Hölder for every exponent below $(m-1)\mathrm{/}(2m)$. Use [F7] to select one such pair for each $m$.

Let $C$ be the probability-one continuity event for $B$ from [F1]. For each $m$ and nonnegative rational $q$, modification gives $P(B_q=Y_q^{(m)})=1$. By countability and subadditivity in [F4], the intersection $H=C\cap {\bigcap }_{m\ge 2}{H}_m\cap {\bigcap }_{m\ge 2}{\bigcap }_{q\in {\mathbb{Q}}_{\ge 0}}\{B_q=Y_q^{(m)}\}$ has probability one.

Fix $\omega \in H$ and $m\ge 2$. On every interval $[0,N]$, the two real functions $t\mapsto B_t(\omega )$ and $t\mapsto Y_t^{(m)}(\omega )$ are continuous and agree on the dense rational subset. By [F4]--[F5] they agree everywhere on $[0,N]$, hence on $[0,\mathrm{\infty })$. Therefore $B(\omega )$ inherits all local Hölder exponents below $(m-1)\mathrm{/}(2m)$.

Given $0<\gamma <1\mathrm{/}2$, [F6] supplies an integer $m\ge 2$ so large that $1\mathrm{/}(2m)<1\mathrm{/}2-\gamma$, equivalently $\gamma <(m-1)\mathrm{/}(2m)$. Step 3.1 then gives the displayed Hölder bound on every $[0,T]$, with its constant allowed to depend on $\omega ,T,\gamma$. The same event $H$ works for all uncountably many $\gamma$, because only the countable integer family was intersected. AC is used exactly at step 1.1 and through the normal-law content of [F1]--[F2].

By [F3] and [F6], each maximum in the definition exists and is finite. Every summand lies in $[0,2^{-n}]$, so [F4] proves that the series converges to a finite value in $[0,1]$.

Let $K\subseteq [0,\mathrm{\infty })$ be compact and let $\epsilon >0$. If $K$ is empty, its basic neighborhood from [F1] is all of $C$. Otherwise the identity function attains a finite maximum on $K$ by [F3], and [F5] gives an integer $N\ge 1$ with $K\subseteq [0,N]$. Put $a=\min (1\mathrm{/}2,\epsilon )$. If $d_{\mathrm{u}\mathrm{o}\mathrm{c}}(f,g)<2^{-N}a$, then the $N$th term gives $1\wedge {\max }_{t\le N}\mathrm{\mid }f(t)-g(t)\mathrm{\mid }<a,$ hence $\mathrm{\mid }f-g\mathrm{\mid }<\epsilon$ throughout $K$. Thus every compact-convergence basic neighborhood contains a $d_{\mathrm{u}\mathrm{o}\mathrm{c}}$-ball.

Conversely, given a $d_{\mathrm{u}\mathrm{o}\mathrm{c}}$-ball of radius $\epsilon >0$, use [F4] to choose $N$ with $2^{-N}<\epsilon \mathrm{/}2$. If ${\max }_{t\le N}\mathrm{\mid }f(t)-g(t)\mathrm{\mid }<\epsilon \mathrm{/}2$, then the first $N$ terms sum to less than $\epsilon \mathrm{/}2$, while [F4] makes the remaining tail at most $2^{-N}<\epsilon \mathrm{/}2$. Hence $d_{\mathrm{u}\mathrm{o}\mathrm{c}}(f,g)<\epsilon$. The neighborhood controlling the compact interval $[0,N]$ therefore lies inside the metric ball.