# Dual of a Reversible Markov Chain [closed]

Let a reversible Markov process $m_{t+1}=m_t P$, where $t$ is time that has a stationary distribution $\pi$. I saw in a paper that the dual system was defined as $x_{t+1}=P x_t$. Can anyone give me some directions in order to understand how this is derived?

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## closed as not a real question by Tsuyoshi Ito, Lev Reyzin♦, Jukka Suomela, David Eppstein, Vijay DJan 16 '13 at 18:48

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That looks like the definition of "dual system". Derived from what? – Jeffε Feb 22 '12 at 23:34

Let $y_t = x_t^T$. Then, $y_{t+1} = y_t P^T$. Thus, $y$ (and therefore $x$) corresponds to the original Markov Chain run backward in time. Hope this gives some motivation towards the concept of dual chains.