NonMarkovian

Non-Markovian describes stochastic processes whose future evolution depends on more of the past than is captured by the present state. In a Markov process, the conditional distribution of the future given the entire past depends only on the current state. A non-Markovian process thus exhibits memory effects: the history influences future dynamics beyond what the present value contains. The term is commonly written non-Markovian, though some sources omit the hyphen. It is widely used in probability theory, statistics, physics, and applied disciplines to characterize memory-bearing dynamics.

Common examples include fractional Brownian motion and fractional Gaussian noise, which exhibit long-range dependence, and renewal processes with non-exponential interarrival times. In such systems, correlations decay slowly and the future cannot be predicted from the present alone. Some processes may appear memoryless at short times but reveal memory at longer horizons; others have transition probabilities that depend on the history of the process.

Modeling approaches to non-Markovian dynamics include enlarging the state space to make the process Markovian, introducing memory kernels or fractional derivatives, and using semi-Markov or Hawkes-type processes. Practical methods also treat non-Markovian systems as approximated by Markovian models with a finite set of hidden variables. These strategies aim to balance mathematical tractability with faithful representation of history-dependent behavior.

Non-Markovian dynamics arise in physics (viscoelastic materials, glassy dynamics), biology (neural adaptation, temporal integration), and finance (volatility clustering, long memory in asset returns). They pose analytical and computational challenges because standard Markovian tools do not apply directly. Understanding non-Markovian effects is important for accurate prediction, parameter estimation, and the design of interventions in complex systems.