 Lazy random walk

A lazy random walk remains at its current position with positive probability and otherwise makes an ordinary random walk step. A common convention replaces transition matrix $P$ by $P_{\mathrm{lazy}}=(I+P)/2$. This preserves every stationary distribution because $\pi P=\pi$ implies $\pi P_{\mathrm{lazy}}=\pi$. It also removes periodic alternation in an irreducible discrete-time chain: the self-loop permits return in one step. A lazy symmetric walk on the integers has increments $-1,0,1$ with probabilities $1/4,1/2,1/4$, so its step variance is $1/2$.