Markov processes, time–space harmonic functions and polynomials

We consider stochastic processes (Mt)t≥0(Mt)t≥0 for which the class VV of time–space harmonic functions is rich enough to yield the Markov property for the process. In particular, we prove that denseness for all t≥0t≥0 of Vt≔{f(t,⋅)}∣f∈V}Vt≔{f(t,⋅)}∣f∈V} in Lp(μt)Lp(μt) for any p≥1p≥1, where μtμt denotes the law of MtMt, is sufficient to guarantee the Markov property. We use this to improve upon a result of [Sengupta, Arindam, 2000. Time–space harmonic polynomials martingales for continuous-time processes and an extension. Journal of Theoretical Probability 13 (4), 951–976] concerning pp-harmonizability, describe two new methods for constructing time–space harmonic polynomials and apply them to get some interesting examples.