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>=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 |W_m|<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) maps to 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 infinity=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<=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<=u}, lie in F_u. Rational coordinates and [F4] finish this claim. By [F3], r(W) is independent of 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,...,E_{n-1}) and that the preceding E_j are iid with law nu=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 intersect {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.