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 (defined below), Brownian motion, Ornstein-Uhlenbeck processes? and Lévy processes?. The most elementary stochastic process (to define) is an indivisible stochastic process which Barandes 2025 has used to give an indivisible stochastic process interpretation of quantum mechanics.

Definitions

We define discrete time stochastic processes following Lawvere 1962. Let $N$ be the category with countably many objects, $\{0,1,2,...\}$, and no non-identity morphisms, and let $\mathbf{Meas}_G$ denote the Kleisli category of the Giry monad. Let $\mathbf{Meas}_G^{N}$ denote the category whose objects $\mathbf{\Omega}$ are sequences $\Omega_0, \Omega_1,\ldots$ of objects in $\mathbf{Meas}_G$ which are measurable spaces. We define an endofunctor $\mathbf{Meas}_G^{N} \xrightarrow{\mathbf{\Phi}} \mathbf{Meas}_G^{N}$ by $\mathbf{\Phi}(\mathbf{\Omega})= \{ \prod_{k \lt n} \Omega_k \}_{k \in N}$ 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. If $\Omega_n$ is thought of as the space of all possible states of a system at time $n$, then $\mathbf{\Phi}(\mathbf{\Omega})_n=\prod_{k \lt n}\Omega_k$ is the space of all possible histories of the system before time $n$.

We define a discrete time stochastic process $\mathbf{Q}$ to be a natural transformation

$$\mathbf{\Phi} \xrightarrow{\mathbf{Q}} \mathbf{id}$$

where $\mathbf{id}$ is the identity functor on $\mathbf{Meas}_G^N$. Given any two (discrete time) stochastic processes $\mathbf{Q}$ and $\mathbf{Q}'$ we define a sequence of morphisms in $\mathbf{Meas}_G$, $\Omega_n \xrightarrow{f_n} \Omega_n^{'}$ to be a morphism from $\mathbf{Q}$ to $\mathbf{Q}'$ when, for each $n \in N$, the $\mathbf{\Meas}_G$-diagram

commutes. Since there is an obvious notion of composition for such maps, all stochastic processes and all such maps of such determine the category of discrete time stochastic processes $\mathbf{dStoch}$. (Warning: Some authors have used the notation $\mathbf{Stoch}$ (Stoch) to mean 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.)

In the preceding diagram the morphisms $Q_n$, which are Kleisli morphisms $\prod_{k \lt n} \Omega_k \xrightarrow{Q_n} \Omega_n$ are called dynamic laws (of motion), and stochastic processes are classified by properties of the dynamic laws.

A Markov dynamic law $Q_n$ is a dynamic law depending only on the current state, i.e., $Q_n$ factors as where $\pi_{n-1}$ is the canonical coordinate projection (measurable) function which specifies the deterministic Kleisli morphism (with the same notation). Thus a Markov dynamic law only depends upon the current state and not its history.

If $\mathbf{Q}$ is a stochastic process such that the dynamic law at every stage is a Markov dynamic law we say the stochastic process $\mathbf{Q}$ is a Markov stochastic process. In the special case where $\mathbf{\Omega}$ defines a constant sequence, $\Omega_n = \Omega$ for all $n$, then a Markov stochastic process on $\mathbf{\Omega}$ is called a Markov chain.

An even more elementary way of defining a stochastic process is by taking $\Omega_n = \Omega$, so each $\Omega_n$ is a copy of $\Omega$, and for all $n$ defining the dynamic law $Q_n$ such that it factorizes as a composite of the projection map $\pi_0$ and a Kleisli morphism $\tilde{Q}_n$, For such dynamic laws the time at stage $0$ is thought of as the conditioning event time - subsequent stages are ‘’conditioned on $\Omega_0$’’.

We say a stochastic process $\mathbf{Q}$ is divisible if and only if for all stages $m$ and $n$, with $m \le n$ there exists a Kleisli morphism $\tilde{Q}_{m,n}$ such that the triangle on the right hand side of the Kleisli-diagram commutes, i.e., $\tilde{Q}_n = \tilde{Q}_{m,n} \circ \tilde{Q}_m$. (We have abused notation by using the notation $\pi_0$ to denote the two projection maps onto the $0^{th}$ coordinate despite the domains being distinct when $m \lt n$.)

An indivisible stochastic process is a stochastic process which is not divisible. Such processes are employed in the indivisible stochastic process interpretation of quantum mechanics.

An indivisible stochastic process is a very general type of stochastic process which can exhibit wild behavior because, for any $\epsilon \gt 0$, the two dynamics laws $\Gamma_t$ and $\Gamma_{t+\epsilon}$ need not have any relationship between them. For example, even if $\Omega$ was a standard Borel space and for every $U \in \Sigma_{\Omega}$ all the measurable functions $\Omega_0 \xrightarrow{\Gamma_t(U | \bullet)} [0,1]$ and $\Omega_0 \xrightarrow{\Gamma_{t+\epsilon}(U | \bullet)} [0,1]$ were continuous functions the two probability measures obtained by composition with an initial probability measure on $\Omega_0$, $\mathbf{1} \xrightarrow{P_0} \Omega_0 \xrightarrow{\Gamma_t} \Omega_t$ and $\mathbf{1} \xrightarrow{P_0} \Omega_0 \xrightarrow{\Gamma_{t+\epsilon}} \Omega_{t+\epsilon}$, can vary significantly. Such a process is in stark contrast to a Markov process on a standard Borel space where continuous Markov kernels yield a continuously varying probability measure on the measurable space $\Omega$.

Hidden Markov Models

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

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 Barendes 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

• Wikipedia stochastic process

• 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 (djvu)

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

• Jacob Barandes, Pilot-Wave Theories as Hidden Markov Models [

  [arXiv:2602.10569] (https://arxiv.org/abs/2602.10569)]

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]