 Solution

A Poisson process of rate $\lambda >0$ is a counting process with $N_0=0$, independent stationary increments, and $N_t-N_s\sim \mathrm{Poisson}(\lambda (t-s))$ for $0\le s<t$. It has nondecreasing right-continuous integer-valued paths with unit jumps almost surely. The independent increment requirement distinguishes the process from merely having Poisson marginals.