Evolution equations for a cubic stochastic process

Abstract: By analogue of [1,2] we define a cubic stochastic process and study evolution (dynamics) of a system $E$ which contains at least three elements.

• The paper rigorously defines and analyzes cubic stochastic processes, extending classical Markov models through three-body interactions.
• It develops both discrete and continuous evolution equations, including delay and advanced integro-differential forms that capture hereditary dynamics.
• Explicit examples and derivations demonstrate nonlinear coupling and pave the way for applications in population genetics and anticipatory systems.

Evolution Equations for a Cubic Stochastic Process

Introduction

The paper provides a formal construction and analysis for cubic stochastic processes (CSP), a class of measure-valued stochastic dynamical systems whose evolution is governed by the interaction of triplets of system elements. Initiated by parallels to classical Markov processes and motivated by evolutionary genetics, the paper rigorously introduces a CSP on a measurable space and derives the associated evolution (Kolmogorov-Chapman-like) equations, their delay and advance variants, and associated integro-differential and differential forms.

Formal Framework for Cubic Stochastic Processes

A CSP is formally defined as a tuple $(E, F, \mathcal{M})$ where $E$ is a set (“state space”), $F$ a $\sigma$-algebra of its subsets, and $\mathcal{M}$ a collection of probability measures on $(E, F)$. The process state at time $t$ is represented by $m_t \in \mathcal{M}$, and the temporal evolution is characterized by transition laws $P(t, x, y, z, t_2, A)$, denoting the probability that the joint interaction of $x, y, z \in E$ at time $t$ leads to realization in $A \in F$ at $t_2$.

This $P$ function is constrained by:

• Time-homogeneity over unit increments,
• Symmetry under permutation of $(x, y, z)$,
• Measure property in $A$,
• Joint measurability in triplet arguments,
• A cubic "Chapman-Kolmogorov" equation (Eq. (2)), defining memoryless recombination across two time-interval interactions.

The framework thus generalizes standard Markovian transitions, replacing pairwise joint laws by cubic (three-body) ones, which is nontrivial both analytically and for possible applications (e.g., nonlinear population genetics).

Discrete and Continuous Representation

For finite $E$ (e.g., $E = \{1, ..., n\}$), the paper defines the evolution of the state vector $x^{(t)} = (x_1^{(t)}, ..., x_n^{(t)})$ by

$$x_l^{(t+1)} = \sum_{i, j, k=1}^n P_{ijk,l} x_i^{(t)} x_j^{(t)} x_k^{(t)},$$

analogous to the quadratic stochastic operators (QSOs) used in genetics, but extended to interactions among three individuals instead of two. For certain transition kernels ($P_{ijk,l}$), explicit recurrence relations and difference equations are given.

In the continuous case, $E$ is a continuum (e.g., $E=\mathbb{R}$), and $P$ becomes a transition kernel over measures, leading naturally to integro-differential equations for the process density $f(s,x,y,z,t,w)$.

Evolution Equations

The paper derives several distinct evolution equations:

• Cubic Chapman-Kolmogorov Equation: An identity relating the transition law over a compound interval to products over subintervals.
• Difference Equations for Discrete Time: Recurrences for population distributions $x^{(t)}$.
• Integro-Differential Equations: Upon taking limit transitions in time, the process equations become delay- and advance-functional integral equations (e.g., Eqs. (10), (11)), and, under certain regularity assumptions, reduce to nonlinear partial differential equations with delayed/advanced arguments.
• Taylor Expansion and Differential Limit: Detailed Taylor expansions (up to third order) for the kernel function $f$ combined with moment integrals lead to explicit diffusion-like terms (Eqs. (13)–(20)), parallels of the Fokker-Planck equation but for cubic interactions.

The system’s dynamics thus involve not only the instantaneous state but also functionals over delayed and advanced states, characterizing hereditary stochasticity with complexity beyond ordinary Markovian or quadratic models.

Notable Mathematical Results

• Delay and Advance Equations: The evolution equations, especially Eqs. (6), (7), (9), and (18), yield systems of functional differential equations containing both delayed and advanced terms with respect to time—in contrast to standard forward-evolution.
• Nonlinear Coupling: State evolution is governed by nonlinear (cubic) convolution of the current or lagged distributions, reflecting nonpairwise (three-body) interactions.
• Derivation of Limiting Differential Forms: Given suitable existence of limits and regularity (e.g., differentiability in time and space), integro-differential forms reduce to partial differential equations, containing drift and diffusion terms parameterized by functional moments of the transition kernel.
• Explicit Constructive Example: The paper presents explicit (exponential-form) families of transition kernels which satisfy all the axiomatic requirements for a CSP, providing constructive ground for theoretical and numerical investigations.

Implications and Future Directions

This formulation generalizes Markov and quadratic processes, enabling dynamic modeling of systems with multi-individual interactions—of particular relevance to population genetics, where phenomena such as triple recombination or gene regulation networks arise naturally. The presence of retarded and advanced arguments in the evolution equations bridges to fields such as hereditary differential equations and anticipatory systems.

On the theoretical side, the structural similarity to the Kolmogorov-Chapman equation suggests lines for developing a full-fledged operator theory, spectral analysis, and the study of ergodicity or stability for CSPs. On the practical side, stochastic models incorporating cubic interactions may better capture real-world biological and social processes where three-way interactions are fundamentally irreducible.

The derived nonlinear PDEs with nonstandard argument structure prompt further study into their analytical properties, well-posedness, potential for pattern formation, and long-time behaviors. Future work could address classification of stationary states, bifurcations, and detailed applications to evolutionary or ecological modeling.

Conclusion

The paper builds a comprehensive mathematical structure for cubic stochastic processes and associated evolution equations. By extending paradigms from Markov and quadratic processes to models driven by triplet interactions, it introduces new classes of nonlinear functionally delayed/advanced evolution equations, and opens avenues for both theoretical exploration and practical application in complex biological and stochastic systems.