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Markov Process

The general term for any stochastic process with the Markov (memoryless) property, covering discrete or continuous state spaces and discrete or continuous time, of which a Markov chain is the discrete-state special case.

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In Plain English

"Markov process" is the umbrella term, and "Markov chain" is one room inside that house. Strictly, a chain is a Markov process confined to a countable state space (a finite or countable list of possible states); once the state can take any real value, as in a stock price or a particle's position, the same memoryless idea is usually just called a Markov process instead. The two terms are often used loosely and interchangeably in casual conversation, but the distinction matters when precision counts.

Definition

A Markov process is a stochastic process ${X_{t}}$ , indexed by discrete or continuous time $t$ , taking values in an arbitrary (discrete or continuous) state space, that satisfies the Markov property: $P (X_{t + s} \in A ∣ F_{t}) = P (X_{t + s} \in A ∣ X_{t})$ for all $s > 0$ and all measurable sets $A$ , where $F_{t}$ is the history of the process up to time $t$ . This single defining property spans a wide zoo of specific models: discrete-time, countable-state Markov chains; continuous-time Markov chains (e.g. Poisson and birth-death processes); and continuous-state diffusion processes governed by stochastic differential equations (e.g. Brownian motion, the Ornstein-Uhlenbeck process). All are Markov processes; only the first category is, strictly, a Markov chain.

Properties

The distinction between 'chain' and 'process' is mostly about state-space cardinality rather than time: a continuous-time process on a countable state space (like a continuous-time Markov chain, used in queueing theory) is still conventionally called a chain, while any process on an uncountable state space, discrete or continuous time, is called a process.
In everyday statistical writing the two terms are frequently used interchangeably, and this entry's distinction, while textbook-standard, is not universally enforced, when precision matters (e.g. distinguishing diffusion processes from countable-state chains), check which state space the author has in mind.
The Markov property alone says nothing about long-run behavior, whether a Markov process settles into a stationary distribution, drifts off to infinity, or oscillates depends on additional structure (irreducibility, recurrence, boundedness) layered on top of the basic memorylessness assumption.

At a Glance

Relationship to Markov chainchain = Markov process restricted to a countable state space

State spacecan be discrete or continuous (e.g. all real numbers)

Time indexcan be discrete or continuous

Defining featurefuture depends on the past only through the present state, for any state space

Last updated September 8, 2026← Back to the Dictionary