timeinhomogeneous

Timeinhomogeneous, commonly written as time-inhomogeneous, is a term used in probability theory, stochastic processes, and differential equations to describe systems whose dynamics change over calendar time, as opposed to time-homogeneous (or stationary) models where rules depend only on time differences. In a time-inhomogeneous stochastic process, transition probabilities depend on both the start time s and the end time t, i.e., P(X(t) ∈ A | X(s) = x) = K(s,t; x, A). The family of transition kernels K(s,t) replaces a single time-translation-invariant semigroup. Correspondingly, the infinitesimal generator is time-dependent, Q(t), and the Kolmogorov forward and backward equations become time-dependent.

They appear in many contexts: time-varying Poisson processes with rate λ(t); birth-death processes with rates that depend on time; Markov processes with non-stationary environments. In continuous-state models, stochastic differential equations often have coefficients b(x,t) and σ(x,t) that depend explicitly on time, producing time-inhomogeneous dynamics. In deterministic settings, systems of ordinary differential equations with coefficients that change in time are non-autonomous and time-inhomogeneous.

Applications include finance, where interest rates or volatilities vary with calendar time; queueing theory with time-varying arrival rates; physics and biology under seasonal or externally driven conditions. Time-inhomogeneous models often require different analytical or numerical techniques, as simple spectral methods or steady-state arguments may not apply.

See also: non-stationary processes, Markov process, Poisson process, stochastic differential equation, non-autonomous systems.