Let $\lambda$ be an intensity function with mean function $\Lambda$, and let $N=(N_t)_{t\ge0}$ be an \reftext{def:inhomogeneous-poisson-process-2026b}{inhomogeneous Poisson process} with intensity $\lambda$ on a \reftext{def:probability-space-random-variable-2026a}{probability space} $(\Omega,\mathcal{F},P)$. Here $\mathbb{N}$ denotes the set of \reftext{def:natural-numbers-2026a}{natural numbers}, $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ the set of nonnegative integers, and $\mathbb{R}$ the set of \reftext{def:real-numbers-c54-2026c}{real numbers}. For $k\in\mathbb{N}_0$ define

Then the following hold.

\textbf{Claim 1 (explicit form).} For every $k\in\mathbb{N}_0$ and $t\ge0$,

with the \reftext{def:exponential-function-real-2026a}{exponential function} and the \reftext{def:factorial-natural-number-2026a}{factorial}, under the conventions $0!=1$ and $\Lambda(t)^{0}=1$ recorded in \ref{def:poisson-distribution-2026b}.

\textbf{Claim 2 (forward equations).} Each $p_k$ has a \reftext{def:derivative-interior-point-c54-2026b}{derivative} at every $t>0$, and a one-sided derivative at $t=0$ given by the same limit restricted to positive increments; with these derivatives the \textbf{Kolmogorov forward equations} hold for all $t\ge0$:

with the initial values $p_0(0)=1$ and $p_k(0)=0$ for $k\ge1$.

\textbf{Claim 3 (uniqueness).} If $(q_k)_{k\in\mathbb{N}_0}$ is any family of functions $q_k:[0,\infty)\to\mathbb{R}$, differentiable in the same sense, satisfying the same system of equations and the same initial values, then $q_k=p_k$ for every $k\in\mathbb{N}_0$. In particular the functions $t\mapsto P(N_t=k)$ form the unique solution of this system of ordinary differential equations.