# Question concerning a proof on Stroock and Varadhan 1971

In the proof of theorem 2.3 of the article diffusion processes with boundary conditions (1971) one reads:

where $Q_{s,x}$ is the unique solution to the martingale problem for $a,b$ starting from $x$ at time $s$ (which is a probability measure on $\Omega, \mathcal{F}^s = \sigma(X(u), u \geq s)$

The point I don't understand is the following $Q_{s,x}$ is defined for events that occur after time $s$. So in this sense, $Q_{\tau'\wedge T, x(\tau'\wedge T)}$ is defined only for events after $\tau'\wedge T$. On the other hand the $r.c.p.d$ (regular conditional probability distribution) for $P^T$ given $\mathcal{M}^{t_0}_{\tau'\wedge T}=\mathcal{F}^{t_0}_{\tau'\wedge T}= \sigma(X(u\wedge \tau'\wedge T), u \geq t_0)$ is a probability measure defined on $\mathcal{F}^{t_0}$.

Lemma 3.6 of [8 -diffusion processes with continuous coefficients I] is the following

I suspect that $P^T$ of th 2.3 is $P'$ of lemma 3.6, but in that case $P^T$ should equal $P$ on $\mathcal{M}^{t_0}_{\tau'\wedge T}$ and the $r.c.p.d$ of $P^T$ given $\mathcal{M}^{t_0}_{\tau'\wedge T}$ agrees with $Q_{\tau'\wedge T, x(\tau'\wedge T)}$ on $\mathcal{M}^{\tau'\wedge T}$. Is that what is meant in those lines?