Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space} and $\mathbb{R}$ the set of \reftext{def:real-numbers-c54-2026c}{real numbers}.

A \textbf{stochastic process} on $[0,\infty)$ is a family $X=(X_t)_{t\ge0}$ of random variables on $(\Omega,\mathcal{F},P)$ indexed by the nonnegative real numbers. The process $X$ has \textbf{independent increments} if for all real $0\le t_0<t_1<\dots<t_r$ the random variables $X_{t_1}-X_{t_0},\dots,X_{t_r}-X_{t_{r-1}}$ are \reftext{def:independence-events-rvs-2026a}{independent}. (Differences of random variables are random variables: $\{X_t-X_s>u\}=\bigcup_{q\in\mathbb{Q}}(\{X_t>q\}\cap\{X_s<q-u\})$, using \reftext{thm:density-q-rudin-b}{density of the rationals} and the generator criterion of \ref{def:measurable-function-2026a}.)

An \textbf{intensity function} is a nonnegative function $\lambda:[0,\infty)\to\mathbb{R}$ that is \reftext{def:continuity-closed-interval-c54-2026b}{continuous} on every closed interval $[0,T]$. Its \textbf{mean function} is

the Riemann integral, which exists by \ref{lem:continuous-implies-riemann-integrable-c54-2026b}, with $\Lambda(0)=0$. By additivity on adjacent intervals (\ref{lem:riemann-integral-additivity-adjacent-intervals-c54-2026a}), $\Lambda(t)-\Lambda(s)=\int_s^t\lambda(u)\,du$ for $0\le s\le t$, and this is nonnegative because every lower sum of a nonnegative function is nonnegative (\reftext{def:upper-lower-sums-partition-c54-2026a}{upper and lower sums}); hence $\Lambda$ is nondecreasing.

A stochastic process $N=(N_t)_{t\ge0}$ on $(\Omega,\mathcal{F},P)$ is an \textbf{inhomogeneous Poisson process with intensity $\lambda$} if:

\textbf{1.} $N_0=0$;

\textbf{2.} $N$ has independent increments;

\textbf{3.} for all $0\le s<t$, the increment $N_t-N_s$ has the \reftext{def:poisson-distribution-2026b}{Poisson distribution} with parameter $\Lambda(t)-\Lambda(s)$, in the sense of \ref{def:distribution-cdf-random-variable-2026a}.

If $\lambda$ is constant with value $\theta\ge0$, then $\Lambda(t)=\theta t$ (the Riemann integral of a constant, directly from the \reftext{def:upper-lower-sums-partition-c54-2026a}{upper and lower sums}), and $N$ is called a \textbf{homogeneous Poisson process with rate $\theta$}.