Kolmogorov continuity criterion in one parameter

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.