= Brownian conditioning by a stopped Bessel density
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{title2=$\mathcal L(R^{T_M})=\mathcal L_x(B^{T_M}\mid T_M<T_0)$}
For $0<x<M$, a <three-dimensional Bessel process> stopped at $M$ has the law of <Brownian motion> conditioned to hit $M$ before zero and stopped at $M$. On $(\varepsilon,M)$, the <Girsanov theorem> and the <Itô formula> give terminal density $X_\tau/x$. Its value is $M/x$ at the upper exit and $\varepsilon/x$ at the lower exit. Letting $\varepsilon\downarrow0$ in bounded stopped-path functionals gives density $(M/x)\mathbf1_{\{T_M<T_0\}}$ relative to the corresponding stopped <Wiener measure>. The exit probability is $x/M$ by the <optional stopping theorem>, so the density is normalized. On the unstopped full path space the same density defines the conditioned Brownian law; pushing it forward by the stopping map gives the stopped Bessel law. The already-stopped law is not absolutely continuous relative to unstopped Wiener measure, since eventually constant paths have Wiener probability zero.
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