An account of known upper and lower bounds for standard diagonal Markov transition functions and standard $p$-functions taking on a given value at time $t$ is presented, and some new results are derived.
"The Maximal Oscillation Problem for Regenerative Phenomena." Ann. Probab. 1 (3) 405 - 416, June, 1973. https://doi.org/10.1214/aop/1176996935