= Heat-kernel density change for a Brownian bridge
{title2=$M_t=p(1-t,B_t,y)/p(1,0,y)$}
The <Brownian transition semigroup> identity proves that this positive density process has mean one for $t<1$. The <backward heat equation> and <Itô's formula> give $dM_t=M_t(y-B_t)(1-t)^{-1}dB_t$. On every fixed horizon $T<1$, <Girsanov theorem> under $dQ=M_TdP$ makes $B_t-\int_0^t(y-B_s)(1-s)^{-1}ds$ <Brownian motion>. Thus the original coordinate acquires the bridge <drift coefficient> toward $y$ at time one, without defining an equivalent endpoint density at the singular time.
Back to article page
