The class of continuous functions $f(t, x)$ for which $f(t, X(t))$ are Markov processes is explicitly determined, where $X(t)$ is a Brownian motion on the real line. This extends a result by Walsh.
"Time-dependent Functions of Brownian Motion that are Markovian." Ann. Probab. 7 (3) 515 - 525, June, 1979. https://doi.org/10.1214/aop/1176995051