Scalar Brownian bridge from x to y

Assume AC. Let $h>0$, $x,y\in \mathbb{R}$, and let $W$ be a standard scalar Brownian motion on $[0,h]$. The scalar Brownian bridge from $x$ to $y$ over duration $h$ is the continuous Gaussian process $B_t^{x,y,h}=x+\frac{t}{h}(y-x)+W_t-\frac{t}{h}{W}_h,0\le t\le h\mathrm{.}$ It starts at $x$ and ends at $y$. Its law does not depend on which Brownian motion is used: Brownian increment laws determine its finite-dimensional laws. For the path-law assertion, give $C([0,h])$ the uniform topology and extend each path $f$ by $E_{h}f(s)=f(s\wedge h)$ for $s\ge 0$. This is a homeomorphism onto the closed subspace of $C([0,\mathrm{\infty }))$ consisting of paths constant after $h$, with inverse restriction to $[0,h]$. Thus Borel sigma-algebra of continuous path space is generated by coordinates makes the Borel sigma-algebra on $C([0,h])$ generated by its coordinates: coordinates of the extension after $h$ equal $f(h)$, and continuity recovers every coordinate, including $h$, from rational times in $[0,h]$. Equality on finite coordinate cylinders now gives equality of path laws by Dynkin's pi-lambda theorem. The endpoint-conditional and restriction-density properties are established separately on the A page.