timehomogeneous

Timehomogeneous, also written as time-homogeneous, describes a stochastic process whose probabilistic behavior does not change when shifted in time. Formally, for any times t ≥ 0 and h ≥ 0 and any set A, the distribution of X_{t+h} conditional on X_t = x depends only on h, not on t. Equivalently, the transition probabilities P_t(x, A) satisfy P_{t+s}(x, A) = ∫ P_t(y, A) P_s(x, dy), and the family {P_t} forms a semigroup.

In discrete time, a time-homogeneous Markov chain has a fixed transition matrix P that is independent of the current time, so the distribution of X_{n+k} given X_n = x depends only on k. In continuous time, a time-homogeneous Markov process has a time-independent generator, and its transition semigroup satisfies P_{t+s} = P_t P_s.

Common examples include the Poisson process with a constant rate λ, Brownian motion, and geometric Brownian motion, all of which maintain a time-invariant transition structure. A simple random walk with fixed step distribution is also time-homogeneous.

Time homogeneity is distinct from, though related to, stationarity. A process can be time-homogeneous without having a stationary distribution, and can have stationary increments without being time-homogeneous in a broader sense. Conversely, time inhomogeneity arises when transition probabilities or generators depend explicitly on the absolute time.

Timehomogeneous models are often preferred for mathematical tractability, enabling semigroup methods, explicit Kolmogorov equations, and simpler calibration in applications across finance, queueing, physics, and beyond.