Definition Strong Markov property in Stroock/Varadhan

Let $$(\mathbb{P}_x)_{x \in \mathbb{R}^d}$$ be a continuous Markov process. So the $$\mathbb{P}_x$$ are the laws on the space of continuous function from the positive real half line to $$\mathbb{R}^d$$, subsequently denoted by $$\Omega$$. In the book by Stroock\Varadhan on multidimensional diffusions, they define the strong Markov property as follows:

For every $$x \in \mathbb{R}^d$$ and every finite stopping time $$\tau$$, the family $$(\delta_{\omega} \otimes_{\tau(\omega)}\mathbb{P_{\pi_{\tau(\omega)}(\omega)}})_{\omega \in \Omega}$$ is a regular conditional probability distribution of $$\mathbb{P}_x$$ w.r.t. $$\mathcal{F}_{\tau}$$, where the latter one denotes the usual $$\sigma$$-field of the $$\tau$$-past. $$\pi$$ denotes the canonical projection from the path space to the value at time $$t$$.

For some $$\omega \in \Omega$$ (i.e. a continuous path), a time $$t \geq 0$$ and a probability measure $$\mu$$ on $$\Omega$$, the measure $$\delta_{\omega}\otimes_{t}\mu$$ is defined as follows: On sets $$A \in \mathcal{F}_t$$, it is defined as $$\delta_{\omega}(A)$$ and on $$B \in \sigma(\pi_r, r \geq t)$$, it is given by $$\mu(B)$$. Intuitively, this means that the measure gives mass to paths only if they look like the reference path $$\omega$$ until time $$t$$ and after time $$t$$, the measurement of paths is inherited by the action of $$\mu$$.

If you spell this out, you can conclude $$\mathbb{E}_x[\phi|\mathcal{F}_{\tau}] = \mathbb{E}_{\delta_{\omega}\otimes_{\tau(\omega)}\mathbb{P}_{\pi_{\tau(\omega)}(\omega)}}[\phi] \,\,\,a.s.$$ for all measurable and bd. $$\phi:\Omega \to \mathbb{R}$$. (How) Is this equivalent to the "usual" definition of the strong Markov property, i.e. $$\mathbb{E}_x[\phi\circ \mathcal{v}_{\tau}|\mathcal{F}_{\tau}] = \mathbb{E}_{\pi_{\tau}}[\phi]\,\, a.s.?$$ Intuitively it seems close, but I am not able to make the rigorous connection. I would apprectiate any help!

• I'd be grateful for any input on this! – Marco Mar 1 at 9:53