A theorem of Harris states that a monotone Markov process on a finite partially ordered set has positive correlations at time $t$ (assuming positive correlations at time 0) if and only if each jump of the process is either up or down. A new proof of the sufficiency of the jump condition is presented.
"An Alternate Proof of a Correlation Inequality of Harris." Ann. Probab. 12 (1) 272 - 273, February, 1984. https://doi.org/10.1214/aop/1176993391