Scalar Brownian bridge barrier-crossing probability

First assume $x,y>c$, put $a=x-c>0$, $z=x-y$, and let $W^z$ denote the canonical bridge from $0$ to $z$ over duration $h$. The bridge event in the Statement has the law of $\{{\sup }_{s\le h}{W}_s^z\ge a\}$: subtracting the bridge path from $x$ negates its centered Gaussian residual and has terminal value $z$. Since $z=a-(y-c)<a$, the restriction and endpoint-disintegration statement [F2] applies to this endpoint.

Write $q(z)=\mathbb{P}\{{\sup }_{s\le h}{W}_s^z\ge a\}$. From [F1], for $u<v<a$, $\mathbb{P}\{{\sup }_{s\le h}{W}_s\ge a,u<W_h\le v\}={\int }_u^v{p}_h(2a-r)dr\mathrm{.}$ Indeed, subtract the reflection identities at $v$ and $u$ and change variables in the Gaussian endpoint integral [F3]. The full endpoint disintegration [F2] then identifies $q(r)p_h(r)=p_h(2a-r)$ for Lebesgue-almost every $r<a$.

The function $q$ is continuous on $(-\mathrm{\infty },a)$. Under the explicit bridge representation, changing endpoint from $r$ to $r^{\mathrm{\prime }}$ changes the path uniformly by at most $\mathrm{\mid }r-r^{\mathrm{\prime }}\mathrm{\mid }$. Thus the crossing indicators converge almost surely as $r^{\mathrm{\prime }}\to r$ unless the bridge maximum equals $a$. For fixed $r<a$, if that maximum equals $a$, it is attained before $h$; hence for some $t_k=h(1-1\mathrm{/}k)<h$ the bridge maximum on $[0,t_k]$ equals $a$. By the restriction density in [F2], this event has probability zero because the Brownian maximum on $[0,t_k]$ has a continuous distribution by the reflection identity [F1] and the continuous Gaussian density [F3]. The countable union over $k\ge 2$ is null. Dominated convergence (Dominated convergence) applies with the integrable constant majorant $1$ and proves continuity of $q$.

Both $q(r)p_h(r)$ and $p_h(2a-r)$ are continuous on $r<a$. Step 1.2 shows they agree almost everywhere there; if their difference were nonzero at one point, continuity would make it nonzero on an interval of positive Lebesgue measure, a contradiction. Hence $q(z)=\frac{p_h(2a-z)}{p_h(z)}$ for the specified endpoint $z<a$. Since $2a-z=(x-c)+(y-c)$ and $z=(x-c)-(y-c)$, the Gaussian density ratio is $\exp (-\frac{((x-c)+(y-c){)}^2-((x-c)-(y-c){)}^2}{2h})=\exp (-\frac{2(x-c)(y-c)}{h})\mathrm{.}$

If $x=c$ or $y=c$, the bridge begins or ends at the barrier, so the event in the Statement is certain and the right side is $e^0=1$. These boundary cases complete the claim for $x,y\ge c$. The strict case used $z<a$, which follows exactly from $y>c$; no formula is asserted when an endpoint lies below the barrier. AC is used only through [A1].