The reflection principle for Brownian motion roughly states that a Brownian motion reflected a stopping time is also a Brownian motion. More precisely, if $W$ is a Brownian motion and $T$ a stopping time then $\hat{W}_t = W_t$ on $t\leq T$ and $\hat{W}_t = 2W_T-W_t$ if $t>T$ is also a Brownian motion.

The proof of the reflection principle as far as I am aware only uses symmetry of the normal distribution, continuity of paths and the strong Markov property of $W$. My inclination is that the reflection would surely hold for a larger class of processes (possibly continuous symmetric stable processes?) than the Brownian motion, but I have yet to find a generalization.

My question is then if there are known generalizations of the reflection principle, or if not is it that continuous symmetric Markov processes are precisely Brownian motions?

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    $\begingroup$ For deterministic times $T$ the reflection principle should hold, for instance, for any symmetric Lévy process and for stochastic integrals of the form $\int_0^t f(s) \, dW_s$ where $f$ is deterministic and $W$ a Brownian motion. $\endgroup$ – saz Mar 8 at 16:05

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