nLab stochastic process

Idea

A stochastic process describes a dynamical system evolving over a linearly ordered set $T$ (“time”), typically taken to be the (positive) integers or real numbers, whose dynamical laws of motion are morphisms in the Kleisli category of the Giry monad (or any other probability monad). By working in the larger category of algebras of that monad, a characterization of a stochastic processes can be modeled in terms of the expected value of measurements on that process.

By a random process physicists usually mean the same, but mathematicians sometimes allow random processes indexed by more general sets, not usually with meaning of time or equipped with a linear order.

The most studied examples include Markov processes (Def. ), Brownian motion, Ornstein-Uhlenbeck processes? and Lévy processes?.

Definitions

We denote by:

• $\mathbb{N}$ the category with countably many objects, $\{0,1,2,...\}$, and no non-identity morphisms,

• $\mathbf{Meas}_G$ the Kleisli category of the Giry monad,

• $\mathbf{Meas}_G^{\mathbb{N}}$ the category whose objects $\mathbf{\Omega}$ are sequences $\Omega_0, \Omega_1,\ldots$ of objects in $\mathbf{Meas}_G$ which are measurable spaces,

• $\mathbf{Meas}_G^{N} \xrightarrow{\mathbf{\Phi}} \mathbf{Meas}_G^{N}$ the endofunctor given by

  (1)$$\mathbf{\Phi}\big(\Omega_\bullet\big)_n \,\coloneqq\, \textstyle{\prod_{k \lt n}} \Omega_k \mathrlap{\,,}$$

  where $\prod_{k \lt n} \Omega_k$ denotes the product measurable space with the smallest $\sigma$-algebra such that all the coordinate-projection functions are measurable.

  So if $\Omega_n$ is thought of as the space of all possible states of a system at time $n$, then $\mathbf{\Phi}(\Omega_\bullet)_n = \prod_{k \lt n}\Omega_k$ is the space of all possible histories of the system before time $n$.

Stochastic processes

The following definition in category-theoretic probability theory is due to Lawvere 1962 §3.2.

Definition

A discrete time stochastic process is an algebra over the endofunctor $\mathbf{\Phi}$ (1).

This means that a discrete time stochastic process is:

1. a sequence $\Omega_\bullet$ of measurable spaces $\Omega_n$ (an object in $\mathbf{Meas}_G^{\mathbb{N}}$),

2. a morphism $Q_\bullet$ in $\mathbf{Meas}_G^{\mathbb{N}}$:

$$Q_\bullet \,\colon\, \mathbf{\Phi}(\Omega_\bullet) \longrightarrow \Omega_\bullet \mathrlap{\,.}$$

Because $\mathbb{N}$ is a discrete category, the morphism $Q_\bullet$ consists of a sequence of component morphisms $Q_n$ (which are Kleisli morphisms)

$$Q_n \,\colon\, \textstyle{\prod_{k \lt n}} \Omega_k \longrightarrow \Omega_n$$

also called dynamic laws (of motion) at time $n$.

Definition

The homomorphisms between stochastic processes are homomorphisms of algebras of endofunctors. This means:

Given a pair $\mathbf{Q}$ and $\mathbf{Q}'$ of such (discrete time) stochastic processes (Def. ) a morphism

$$\mathbf{Q} \longrightarrow \mathbf{Q}'$$

is a sequence $\big(f_n \colon \Omega_n \to \Omega_n^{'}\big)_{n \in \mathbb{N}}$ of morphisms in $\mathbf{Meas}_G$ such that for each $n \in \mathbb{N}$, the following $\mathbf{Meas}_G$-diagram commutes:

With the evident notion of composition, this defines the category of discrete time stochastic processes, $\mathbf{dStoch}$.

Remark

Beware that some authors use the notation $Stoch$ instead to denote the Kleisli category of the Giry monad, traditionally denoted by $\mathbf{Meas}_G$ or $\mathcal{K}(G)$, which can also be interpreted (for modeling) as the category of Markov stochastic processes.

Markov processes

The following definition is as in Lawvere 1962 §3.3.

Definition

A stochastic process $Q_\bullet$ (Def. ) is called a Markov process if each $Q_n$ for $n \geq 1$ depends only on the current state in that it factors as:

where $\pi_{n-1}$ is the canonical coordinate projection (measurable) function which specifies the deterministic Kleisli morphism (with the same notation).

In the special case where $\Omega_\bullet$ is constant on some $\Omega$, $\forall_n \colon \Omega_n = \Omega$, then a Markov stochastic process on $\Omega_\bullet$ is called a Markov chain.

Remark

When a non-Markov stochastic process can be re-expressed as a Markov stochastic process by formally augmenting its states with a suitable collection of unobservable variables, then the resulting process is called a hidden Markov model. The unobservable variables added to make the process look Markov are said to be latent or hidden variables. For example, pilot wave theories can be viewed as Hidden Markov Models (Barandes 2026)

(In)Divisible processes

The term “(in)divisible stochastic process” is not classical but has been brought up by Barandes 2025 in a suggestion to give an “indivisible stochastic process interpretation of quantum mechanics”. Here is a way to state the definition in line with the above category-theoretic definitions and beyond finite state spaces:

Definition

Given a discrete time stochastic process $Q_\bullet$ (Def. ), its multi-step marginal transition from time $0$ to time $n$, denoted $Q_{0,n} \colon \Omega_0 \to \Omega_n$, is the Kleisli morphism obtained by forming the joint distribution over all intermediate states up to time $n$, and subsequently projecting onto the state space at time $n$.

Explicitly, one constructs a sequence of history-accumulating morphisms $J_n \colon \Omega_0 \to \prod_{k \le n} \Omega_k$ in $\mathbf{Meas}_G$ by induction:

1. Base case ($n=0$): $J_0 \colon \Omega_0 \to \Omega_0$ is the identity morphism in $\mathbf{Meas}_G$ (representing the deterministic assignment $x \mapsto \delta_x$).
2. Inductive step ($n \ge 1$): Given $J_{n-1} \colon \Omega_0 \to \prod_{k \lt n} \Omega_k$, define $J_n$ as the Kleisli composition:
   $$J_n \;\coloneqq\; \langle \mathbf{id}, Q_n \rangle \circ J_{n-1}$$

   where $\langle \mathbf{id}, Q_n \rangle \colon \prod_{k \lt n} \Omega_k \longrightarrow \prod_{k \le n} \Omega_k$ is the morphism given by pairing the deterministic identity on the history with the dynamic law $Q_n$. In the context of the Giry monad, this pairing is canonically constructed via the monadic tensorial strength (or equivalently, via the copy morphism in the framework of Markov categories).

The multi-step marginal transition $Q_{0,n}$ is then defined as the Kleisli composition of $J_n$ with the canonical coordinate projection $\pi_n$:

$$Q_{0,n} \;\coloneqq\; \pi_n \circ J_n$$

where $\pi_n \colon \prod_{k \le n} \Omega_k \to \Omega_n$ is pushed forward as a deterministic Kleisli morphism. This construction is summarized by the following commuting diagram in $\mathbf{Meas}_G$:

Definition

A stochastic process $Q_\bullet$ (Def. ) is called divisible if its multi-step marginal transitions $Q_{0,n} \colon \Omega_0 \to \Omega_n$ (Def. ) satisfy the Chapman-Kolmogorov property:

Namely, for all stages $m \le n$, there exists a Kleisli morphism $\tilde{Q}_{m,n} \colon \Omega_m \to \Omega_n$ such that the following diagram in $\mathbf{Meas}_G$ commutes:

Meaning that $Q_{0,n} = \tilde{Q}_{m,n} \circ Q_{0,m}$.

Example

In discrete time, every Markov process (Def. ) is automatically divisible (Def. ) by defining $\tilde{Q}_{m,n}$ as the composition of its 1-step transitions.

Therefore, indivisibility is inherently a feature of non-Markovian processes (or continuous-time processes where intermediate transition kernels may fail to exist).

Examples

Example

A 3-stage Markov stochastic process $\mathbf{Q}$ with measurements $\Omega_i \xrightarrow{f_i} \mathbb{R}$, modeled within the category of algebras of the Giry monad $(G, \eta, \mu)$ so that the process $\mathbf{Q}$ can be characterized in terms of the expected values of the measurements, is given by

The standard Borel space $\mathbb{R}_{\infty}$ is the one point compactification of the real line. (There is no $G$-algebra $G(\mathbb{R}) \rightarrow \mathbb{R}$; so to model any process with measurement $\Omega \xrightarrow{f} \mathbb{R}$ we first need to embedd $\mathbb{R}$ into $\mathbb{R}_{\infty}$.) The operator $\mathbb{E}_{\bullet}(id_{\mathbb{R}_{\infty}})$ is defined at each $P \in G(\mathbb{R}_{\infty})$ by $\mathbb{E}_{P}(id_{\mathbb{R}_{\infty}}) =\int id_{\mathbb{R}_{\infty}} \, dP$.

Note that measurements $\Omega \xrightarrow{f} X$ can be taken over any object $X$ which lies in the category of algebras of the monad $G$. In the case where $X$ is a standard Borel space the $G$-algebra $G(X) \xrightarrow{\mathbb{E}_{\bullet}(id_X)} X$, which is a morphism in the category of algebras, is defined as: $\mathbb{E}_{P}(id_X)$ is the unique element in $X$ such that for all affine measurable maps $X \xrightarrow{m} \mathbb{R}_{\infty}$, the property

$$m(\mathbb{E}_P(id_X)) = \int_X m \, dP$$

holds. (Every algebra $X$ necessarily possesses a convex space structure. The case $X=\mathbb{R}_{\infty}$ is just a special case which trivially satisfies the above property since the affine maps $\mathbb{R}_{\infty} \xrightarrow{m} \mathbb{R}_{\infty}$ are of the form $m(r) =\lambda r + t$ (scale + translate) and hence $\mathbb{E}_{\bullet}(id_{\mathbb{R}_{\infty}})$ simplifies to the standard expectation operator. A derivation of the category of algebras for standard Borel space is given on the Giry monad page.)

Example

Here are two elementary examples of an indivisible stochastic process due to Barandes 2025. Taking the finite space $\Omega = \{1,2\}=\Omega_n$ for all $n$, and defining the dynamic law $\Omega \xrightarrow{\tilde{Q}_n} \Omega$ by

$$\tilde{Q}_n = \begin{pmatrix} e^{-\frac{t_n^2}{\tau^2}} & 1-e^{-\frac{t_n^2}{\tau^2}} \\ 1-e^{-\frac{t_n^2}{\tau^2}} & e^{-\frac{t_n^2}{\tau^2}} \end{pmatrix}$$

where $t_n$ is the time at stage $n$, and $\tau$ is a constant. (For a finite space $\Omega$ a dynamic law $\Omega \rightarrow \Omega$ can be represented by a matrix.)

Similarly, on the same space $\Omega$, we can define

$$\tilde{Q}_n = \begin{pmatrix} \cos^2(\omega t_n) & \sin^2(\omega t_n) \\ \sin^2(\omega t_n) & \cos^2(\omega t_n) \end{pmatrix}$$

where $\omega$ is a constant.

References

General:

• Wikipedia: Stochastic process

Discussion in the context of category-theoretic approaches to probability theory:

• William Lawvere: The Category of Probabilistic Mappings With Applications to Stochastic Processes, Statistics, and Pattern Recognition (1962), including abstract and commentary added by Lawvere in 2020, Lawvere Archive (2025) [pdf]

• Xiao-Qing Meng: Categories of convex sets and of metric spaces with applications to stochastic programming and related areas, PhD thesis, New York (1988) [djvu, pdf]

With an eye towards quantum noise;

• Stéphane Attal: Stochastic Processes, Lecture 4 in: Lectures on Quantum Noises [pdf, webpage]

On some kind of supersymmetry in stochastic PDEs:

• Igor V. Ovchinnikov: Ubiquitous order known as chaos [arXiv:2503.17157]

In the context of a proposed “indivisible stochastic process interpretation of quantum mechanics”:

• Jacob Barandes, Quantum Systems as Indivisible Stochastic Processes [arXiv:2507.21192]

• Jacob Barandes, Pilot-Wave Theories as Hidden Markov Models [arXiv:2602.10569]