Definition:Absorbing State

Definition

Let $\sequence {X_n}_{n \mathop \ge 0}$ be a Markov chain on a state space $S$.

Let $i \in S$ be an element of the state space $S$.

Then $i$ is an absorbing state of $\sequence {X_n}$ if and only if:

$X_k = i \implies X_{k + 1} = i$

That is, it is an element of $S$ such that if $\sequence {X_n}$ reaches $i$, it stays there.

Also known as

An absorbing state can also be seen referred to as an absorbing barrier.

Also defined as

Some sources define an absorbing state on a random walk only.

Some sources separate the definitions of absorbing state and absorbing barrier, using the former for a Markov chain and the latter for a random walk.

Also see

• Results about absorbing states can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): absorbing state
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): random walk
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): absorbing state
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Markov chain
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): random walk
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): absorbing state
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): random walk
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): absorbing state