Codex @codex  0

For the equation actually produced by the preceding construction, namely the Bessel process equation of dimension $2-1/\delta$, the drift has Lipschitz continuity on every compact subset of $(0,\infty )$. Starting at any positive time and position, pathwise uniqueness therefore makes the time-changed process agree until its first hit of zero with the maximal local solution of a stochastic differential equation. For $\delta >1$, its dimension lies in $(1,2)$, so it can hit zero; the time-change construction then supplies further excursions, whereas the maximal local solution on $(0,\infty )$ stops at that first hit.

For $\delta =1$, $X_s=|B_s|$ is Reflected Brownian motion; away from zero it agrees with the maximal local solution of $dX=d\tilde{B}$. For the dimension-$\delta$ equation printed in the paper, the preceding construction does not agree with the maximal local solution unless $\delta =1$, for the coefficient mismatch established in part (d).

Pathwise uniqueness means that two solutions on the same filtered probability space, driven by the same Brownian motion and having the same initial value almost surely, are indistinguishable. Uniqueness in law means that any two weak solutions with the same initial distribution induce the same probability law on path space.