Successive Brownian exit segments are independent copies

The normalized process X has measurable coordinates, all its paths are continuous, and $X_0=0$ everywhere. It retains every finite-dimensional Brownian law, since its coordinates were changed on one measurable null event. By [F4] it is a $C$-valued random element, supported on the closed subspace $C_0$ of paths starting at zero. Work with the usual augmentation of this process, as defined in [F2], when applying strong Markov. This does not transfer a stopping-time claim to the original B filtration.

For $w\in C_0$ define $\sigma (w)=\inf \{t\ge 0:\mathrm{\mid }w(t)\mathrm{\mid }=1\}$. This is measurable: for $t\ge 0$, continuity gives $\{\sigma \le t\}=\{{\sup }_{0\le s\le t}\mathrm{\mid }w(s)\mathrm{\mid }\ge 1\}$, and the supremum is over the countable set of rationals in $[0,t]$ together with t. Continuity, the intermediate value property and attainment of the maximum on a compact interval prove this equality. It also shows $\sigma$ is strictly positive on $C_0$. For a continuous Brownian W, [F2] makes $\sigma (W)$ a stopping time. If $\sigma (W)>m$, then $\mathrm{\mid }{W}_m\mathrm{\mid }<1$; hence [F1] gives $P(\sigma >m)\le P(\mathrm{\mid }{W}_m\mathrm{\mid }<1)\le 2\mathrm{/}\sqrt{2\pi m}$. Continuity from above in [F5] proves $\sigma$ finite almost surely. At a finite $\sigma$, continuity gives $W_{\sigma }\in \{-1,1\}$.

Define the segment map $e(w)=(\sigma (w),(w(t\wedge \sigma (w)){)}_{t\ge 0})$ into $[0,\mathrm{\infty }]\times C$, and the remainder $r(w)(t)=w(\sigma (w)+t)-w(\sigma (w))$ if $\sigma$ is finite, and the zero path otherwise. Both are Borel maps. Indeed evaluation $(w,s)\mapsto w(s)$ continuously for finite s: if $w_j$ converges uniformly on compact sets and $s_j$ tends to s, bound by the uniform error on one common compact interval plus continuity of w there. Compose with measurable $\sigma$ for each fixed coordinate, using $t\wedge \mathrm{\infty }=t$ and the zero convention for r; then [F4] proves path-valued measurability. The duration $\sigma$ is measurable by step 2.1.

For a continuous Brownian W and its $\sigma$, $e(W)$ is measurable for the stopped $\sigma$-algebra. Sigma itself is measurable there by its defining tests. For a fixed t and u, on $\{\sigma \le u\}$ the value $W_{t\wedge \sigma }$ equals $W_{t\wedge \sigma \wedge u}$, which is measurable at time u: approximate $t\wedge \sigma \wedge u$ from below by a finite grid in $[0,u]$, use adaptation and continuity. Thus the Borel inverse images of $W_{t\wedge \sigma }$, intersected with $\{\sigma \le u\}$, lie in ${\mathcal{F}}_u$. Rational coordinates and [F4] finish this claim. By [F3], $r(W)$ is independent of ${\mathcal{F}}_{\sigma }$ and has Brownian finite-dimensional laws. It is $C_0$-valued by construction; [F4] and the pi-lambda theorem identify its $C_0$ law with that of X and promote coordinate independence to independence of every Borel path event. Therefore $e(W)$ and $r(W)$ are independent, and $r(W)$ again has the law of X.

Set $W^0=X$, $W^{n+1}=r(W^n)$, ${\sigma }_n=\sigma (W^n)$, $E_n=e(W^n)$, and $S_n={\sum }_{j<n}{\sigma }_j$. These are measurable by step 3.1. Inductively each $W^n$ has the law of X by step 4.1, so each ${\sigma }_n$ is finite and strictly positive almost surely by step 2.1. Their countable intersection is a measurable full event by [F5]. On it, finite induction gives $W_t^n=X_{S_n+t}-X_{S_n}$ simultaneously for all t, and hence the formulas for $S_n$ and $E_n$ in the statement. Outside this event the just-defined measurable $E_n$ provide the promised convention; $S_n$ are extended sums, so subsequent times after infinity stay infinite.

Prove by induction that $W^n$ is independent of $H_n=\sigma (E_0,\dots ,E_{n-1})$ and that the preceding $E_j$ are iid with law $\nu =\mathrm{law}(e(X))$. For n=0 this is vacuous. If it holds at n, the pair $(e(W^n),r(W^n))$ is independent of $H_n$, since it is a measurable function of $W^n$. Its components are independent by step 4.1. Thus for $A\in H_n$ and Borel segment and path sets D,L, $P(A\cap \{E_n\in D\}\cap \{W^{n+1}\in L\})=P(A)\nu (D)P(X\in L)\mathrm{.}$ For fixed L, the pi-lambda theorem extends this identity from intersections $A\cap \{E_n\in D\}$ to all of $H_{n+1}$. It follows that $W^{n+1}$ is independent of $H_{n+1}$; taking L to be the whole path space also proves $E_n$ independent of $H_n$ with law nu. Induction proves mutual independence of every finite family of segments, hence the asserted iid sequence. It does not assert independence of the nested entire future processes $W^n$.

Negation preserves the law of X on $C_0$: centered independent normal increments are unchanged jointly under sign reversal by the even normal density and reflection substitution in [F6], so finite-dimensional laws agree, and [F4] and pi-lambda give equality of path laws. The map $d(w)=w(\sigma (w))$ on finite $\sigma$, zero otherwise, is Borel by the evaluation argument in step 3.1. Also $\sigma (-w)=\sigma (w)$ and $d(-w)=-d(w)$. Step 2.1 makes its value a sign almost surely, so its two probabilities are equal and sum to one. Each displacement $d(W^n)$ is a function of $E_n$: for finite duration it is the stopped path evaluated at that duration. Step 6.1 therefore makes these displacements independent fair signs, including the first $d(W^0)=X_{S_1}$. Telescoping gives $X_{S_n}={\sum }_{j<n}d(W^j)$ on the full event in step 5.1, proving the embedded-walk claim.

At time zero X starts at zero and each duration is strictly positive; at exit the value is exactly one of the two endpoints. Infinite durations lie in a measurable null event and have explicit segment/remainder conventions. The empty history in the induction is the trivial $\sigma$-algebra. The claims concern exit segments and signs; no inference about moments of a two-sided exit time is made from a one-sided hitting-time density. Full AC is inherited through [F5]; normalization uses one supplied full event, not a choice of paths.