A time-space harmonic polynomial for a continuous-time process $X={X_t : t ge 0} $ is a two-variable polynomial $ P $ such that $ { P(t,X_t) : t ge 0 } $ is a martingale for the natural filtration of $ X $. Motivated by Lévy's characterisation of Brownian motion and Watanabe's characterisation of the Poisson process, we look for classes of processes with reasonably general path properties in which a characterisation of those members whose laws are determined by a finite number of such polynomials is available. We exhibit two classes of processes, the first containing the Lévy processes, and the second a more general class of additive processes, with this property and describe the respective characterisations.
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机译：连续时间过程$ X = {X_t：t ge 0 } $的时空谐波多项式是二元多项式$ P $，使得$ {P（t，X_t）：t ge 0 } $是$ X $的自然过滤的mar。受Lévy对布朗运动的刻画和Watanabe对泊松过程的刻画的激励，我们寻找具有合理一般路径属性的过程类，在这些过程中，可以表征那些由有限数量的此类多项式确定其定律的成员。我们展示了两类过程，第一类包含Lévy过程，第二类具有更一般的加性过程，具有此特性并描述了各自的特征。
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