Indivisible Stochastic Processes

• Indivisible stochastic processes are non-Markovian evolutions defined by a single-step law that prevents decomposition into sequential updates.
• They exhibit distinct geometric and algebraic structures, such as conical regions in stochastic matrices and determinant sign changes, that delineate Markovian and non-Markovian regimes.
• Their study bridges classical probability and quantum dynamics, shedding light on memory effects, information backflow, and the emergence of quantum mechanical structures.

An indivisible stochastic process is a non-Markovian stochastic evolution in which future system probabilities are determined linearly from initial data via a single-step "indivisible" law, but where intermediate states generally do not permit a decomposition into sequential stochastic (Markovian) updates. Indivisible processes form the minimal, atomistic constituents of non-Markovian time evolution in both classical stochastic and quantum dynamical systems. Their paper is closely linked with the stochastic-quantum correspondence, foundational questions in non-Markovianity, and the structure of infinitely divisible laws.

1. Foundational Concepts: Divisibility and Indivisibility

A stochastic process on a discrete configuration space $C = \{1, ..., N\}$ is described at time $t$ by the probability vector $p(t) \in \mathbb{R}^N$, with $p_i(t)\geq0$, $\sum_i p_i(t)=1$. The evolution is linear if

$p(t) = T(t,s)\,p(s)$

where $T(t,s)$ is a column-stochastic matrix: $T_{ij}\ge0$, $\sum_i T_{ij}=1$. The process is called divisible if, for all $s \leq u \leq t$, $T(t,s) = T(t,u)T(u,s)$ with $T(t,u),T(u,s)$ also stochastic. If no such decomposition exists, the process is indivisible (Pimenta, 13 May 2025, Barandes, 27 Jul 2025).

In classical terms, divisible processes obey the Chapman-Kolmogorov equation—a key trait of Markovianity. Indivisible processes violate this property, so the evolution between times $s$ and $t$ cannot be factored via stochastic matrices at an intermediate time $u$. In the quantum case, the analogous concept concerns complete positivity divisibility (CP-divisibility) and generalizes to the structure of quantum channels (Utagi et al., 2020).

2. Geometric, Operational, and Algebraic Structure

For $N=2$, the space of column-stochastic matrices forms a convex polytope, coordinatized by their diagonal entries $(p, q)$, $T_{11}=p$, $T_{22}=q$, and off-diagonals $T_{12}=1-q$, $T_{21}=1-p$. Divisibility and indivisibility are organized by conical regions in this plane (Pimenta, 13 May 2025):

• The determinant $\det T = p+q-1$ partitions the space: $\det T >0$ (near identity), $\det T = 0$ (degeneracy), $\det T < 0$ (near swap).
• For a target $T^\ast = (p, q)$, the set of possible $T(u,s)$ yielding $T^\ast$ by $T(t,s)=T(t,u)T(u,s)$ is constrained to two "opposite" cones rooted at $(1,1)$ (identity) and the swap matrix, via explicit linear inequalities (see Pimenta, eqs. 14-18).
• In a basis $X=p-q$, $T=1-(p+q)$, the $T$-axis parameterizes information erasure. Divisible processes flow to $T=0$, indivisible processes move "outside the cone", including tachyonic trajectories violating $|\Delta X| \leq |\Delta T|$.

This construction generalizes to $N>2$, though the geometric structure becomes more intricate (Pimenta, 13 May 2025). For continuous-time maps $t \mapsto T(t)$, only one cone is accessible from identity continuously: indivisibility arises upon crossing between cones, and piecewise-divisible but globally indivisible evolutions are possible.

The table below summarizes key geometric features for $N=2$:

Region of $[0,1]^2$	$\det T$	Dynamical Character
$p+q>1$	$>0$ (interior)	Divisible; Markovian flow
$p+q=1$	$0$ (degenerate)	Full information erasure
$p+q<1$	$<0$ (swap side)	Indivisible ("tachyonic")

Indivisible processes thus encode departures from Markovian decomposability in both classical and quantum systems, with their atomicity reflecting irreducible memory.

3. Stochastic-Quantum Correspondence

The stochastic-quantum correspondence (Barandes, 27 Jul 2025) establishes that any indivisible stochastic law $T_{ij}(t,0)$ (a stochastic matrix) admits a representation as squared moduli of a complex matrix $U(t,0)$: $T_{ij}(t,0) = |U_{ij}(t,0)|^2$ with induced density-matrix evolution

$\rho(t) = U(t,0)\,\rho(0)\,U^\dagger(t,0)$

where initial data $\rho(0)$ is diagonal. Markovianity (divisible structure) forces $U(t,0)$ to satisfy composition laws, whereas in the indivisible (non-Markovian) case, no such factorization exists.

Notably, the emergence of the Hilbert space, wave functions, and gauge structure (e.g., Foldy-Wouthuysen and Schur-Hadamard gauges), arises as secondary constructions from the underlying indivisible stochastic process. Divisibility relates to complete positivity and semigroup structure, while indivisibility encodes quantum memory phenomena, such as information backflow and nontrivial dynamical symmetry.

4. Infinitely Divisible Processes and Integral Representations

Processes with stationary independent increments—paradigmatic examples of infinitely divisible (i.d., sum-i.d., max-i.d.) laws—are described by broader classes of stochastic evolution equations (Li, 2018, Kabluchko et al., 2012). The path-integral slicing method yields convolution integral equations for the evolution of event-count distributions or random fields: $$f_n(t) = \frac{1}{t}\sum_{k=0}^{n} \int_0^t f_k(\tau)\,f_{n-k}(t-\tau)\,d\tau$$ or, in Laplace transform,

$$-\frac{d}{dp}\hat{f}_n(p) = \sum_{k=0}^n \hat{f}_k(p)\,\hat{f}_{n-k}(p)$$

The explicit solution requires specification of a small-$t$ generator, e.g., the Poisson or more general infinitely divisible law. The resulting framework encompasses Poisson, compound Poisson, negative binomial, gamma, and stable processes.

Sum-i.d. and max-i.d. processes admit canonical representations as integrals over Poisson point processes:

• For sum-i.d.: stochastic (compensated) integrals over functions $f_t$ with respect to a Poisson process, yielding additive structure.
• For max-i.d.: extremal integrals (suprema) over Poisson points, yielding maximum-type structure.

Minimality and uniqueness of these representations are guaranteed on Borel spaces, and stationary processes correspond to flow representations by measure-preserving maps (Kabluchko et al., 2012).

5. Indivisible Dynamics in Quantum and Classical Non-Markovianity

Indivisible processes in the quantum setting correspond to non-CP-divisible quantum channels—where intermediate propagators cease to be completely positive (or even positive and trace-preserving for P-indivisibility) (Utagi et al., 2020). The time-local generator in such evolutions has at least one negative decay rate: $$\mathcal{L}(t)[\rho] = -i[H(t),\rho] + \sum_\alpha \gamma_\alpha(t) (L_\alpha(t)\rho L_\alpha(t)^\dagger - \frac12\{L_\alpha^\dagger(t)L_\alpha(t),\rho\})$$ with some $\gamma_\alpha(t)<0$. Operationally, indivisibility coincides with the presence of information backflow, as seen in the dynamics of the channel’s Holevo quantity.

In the classical context, non-Markovianity can even persist in divisible evolutions: e.g., quantum semi-Markov processes can be CP-divisible but exhibit non-zero semigroup deviation (SSS-measure), quantifying memory retained from non-exponential waiting-time statistics.

The table below compares various levels of divisibility:

Class	Intermediate Map Property	Associated Generator Conditions	Memory/Backflow
Semigroup Markov	CPTP, composition law	Constant non-negative rates	None
CP-divisible	CPTP (for all $t,s$)	Time-local Lindblad; $\gamma_\alpha(t)\ge0$	Classical memory
Indivisible	Nonpositive/Non-CP at $t,s$	Some $\gamma_\alpha(t)<0$	Information backflow

6. Construction, Classification, and Generalizations

The theory for sum- and max-i.d. processes (including union-i.d. random sets) is cast in terms of minimal spectral representations, measure-preserving flows, and ergodic decompositions (conservative vs. dissipative flows, periodic/null recurrent structure) (Kabluchko et al., 2012). Specific constructions include:

• Mixed moving maxima, Poisson line-process fields, Penrose-type fields, and matrix analogues (e.g., Laguerre Unitary Process) (Ipsen, 2019).
• Flow representations $f_t = f_0 \circ T_t$ under a measurable, measure-preserving $\{T_t\}$, unique up to measure isomorphism.
• Ergodic decompositions via Hopf/Krengel theorems: conservative/dissipative characterization via integrability criteria of the spectral functions.

Extensions beyond stationary independent increments lead to nonstationary processes, processes with dependent increments, and more generally to processes governed by Volterra-type integral equations.

7. Practical and Conceptual Implications

Indivisible stochastic processes serve as elementary constituents in stochastic and quantum evolution, supplying the atomic blocks for the description of irreducible memory and non-Markovian effects. Their theory informs:

• The foundational understanding and simulation of genuinely non-Markovian systems in both the classical and quantum regimes.
• The dynamical emergence of quantum mechanical structures—Hilbert spaces, wave functions, dynamical gauge symmetries—directly from stochastic underpinnings, with relevance to modifications and reconceptualizations of quantum theory (Barandes, 27 Jul 2025).
• The classification and design of extremal fields and random sets in spatial statistics, via Poisson-driven or flow-based constructions.

Their geometric characterization (cones, tachyonic regions, information erasure axes) provides a transparent framework for determining divisibility in both classical and quantum channels, with operational consequences for information processing, quantum memory, and coarse-grained modeling (Pimenta, 13 May 2025).

In summary, indivisible stochastic processes constitute the foundational building blocks underlying non-Markovian dynamics, linking the deepest structures in classical probability, quantum dynamics, and statistical field theory.