Persistent Kac-type Processes

• Persistent Kac-type processes are finite-velocity stochastic models characterized by directional persistence and Markovian switching mechanisms.
• They generalize the classical telegraph process into multi-state and continuous velocity frameworks, yielding hyperbolic transport equations and piecewise smooth trajectories.
• These processes exhibit key features such as finite propagation speed, emergent diffusive limits, and the incorporation of nonlinear and memory effects across various applications.

Searching arXiv for recent and foundational papers on persistent Kac-type stochastic processes. Persistent Kac-type stochastic processes are finite-velocity stochastic dynamics in which motion remains directionally correlated over nonzero time intervals and switching between propagation states is governed by Poissonian or more general Markovian mechanisms. In their classical form they arise from Kac’s telegraph process, where a particle moves with constant speed and randomly reverses direction; in modern generalizations they include multi-state Generalized Poisson–Kac processes, continuum-state velocity-jump models, embedded nonlinear drift processes in Feynman–Kac theory, and broader extended Poisson–Kac constructions with internal variables, memory, and self-consistent couplings (Giona et al., 2016). Across these variants, the common structural features are finite propagation speed, piecewise smooth trajectories, hyperbolic transport equations for partial probability waves, and diffusive or parabolic behavior only as an asymptotic Kac limit (Giona et al., 2016).

1. Historical emergence and conceptual scope

Persistent Kac-type processes originate in the one-dimensional Poisson–Kac or telegraph process, where a particle moves with velocity that alternates between two values under Poissonian switching. In one dimension the classical process is

$dx(t) = v(x(t))\,dt + b\,(-1)^{\chi(t)}\,dt,$

with $\chi(t)$ a Poisson process of rate $\lambda$, $v(x)$ a deterministic drift, and $b>0$ a constant speed (Giona et al., 2016). The particle therefore moves with finite speed between flips, and the stochastic velocity retains its sign on a characteristic time scale $1/\lambda$, which is the simplest form of persistence (Giona et al., 2016).

This formulation already distinguishes Kac-type dynamics from Wiener-driven models. In the diffusive/Wiener benchmark,

$dV = f(V,t)\,dt + \sigma\,dW_t,$

propagation is effectively instantaneous and infinitesimal increments are uncorrelated, whereas Kac-type dynamics is a velocity-jump process with finite propagation speed $v$, persistence time $\tau_{\rm pers}=(2\lambda)^{-1}$, and a random telegraph internal state $s(t)\in\{+1,-1\}$ (Ghose, 12 May 2026). The finite-speed character also underlies the contrast with Lévy flights and other unbounded-fluctuation models: extended Poisson–Kac theory was developed explicitly to address the problem of unbounded random fluctuations by constructing a framework with physically realistic finite propagation velocity (Giona et al., 2020).

Modern usage of the term extends well beyond the original two-state telegraph process. Generalized Poisson–Kac processes enlarge the internal stochastic state from two directions to an $\chi(t)$0-state Markov chain with velocity vectors $\chi(t)$1, while continuous-state versions treat a continuum of velocity states and connect directly to kinetic theory and the Boltzmann equation (Giona et al., 2016). Nonlinear and embedded variants incorporate self-consistent drifts, internal state dependence, and history dependence, thereby linking persistent Kac-type dynamics to nonlinear Fokker–Planck–Kac models, stochastic functional differential equations, and generalized Feynman–Kac representations (Giona et al., 2016).

A further conceptual extension appears in nonlinear drift representations of advection–reaction–diffusion equations, where a single stochastic trajectory is driven by an auxiliary random velocity process and nevertheless reproduces deterministic PDE solutions after taking expectations over paths. In that setting, “persistent Kac-type stochastic processes” denote single stochastic evolutions carrying a nontrivial, time-correlated drift structure generated by an auxiliary random velocity process, yet still reproducing deterministic PDE solutions through expectations over paths (Yaacoub et al., 2024).

2. Canonical formulations and primitive statistical description

In the scalar two-state case, one introduces partial densities

$\chi(t)$2

which satisfy the coupled balance equations

$\chi(t)$3

or, equivalently in the Poisson–Kac notation,

$\chi(t)$4

when the speed and switching rate are state dependent (Ghose, 12 May 2026). The total density $\chi(t)$5 and imbalance $\chi(t)$6 then provide a closed hyperbolic description.

The multidimensional finite-state generalization is the Generalized Poisson–Kac process

$\chi(t)$7

where $\chi(t)$8 is a finite-state Markov process with transition rates $\chi(t)$9 and transition probabilities $\lambda$0 (Giona et al., 2016). The primitive statistical description is given by the partial probability density functions

$\lambda$1

which satisfy the coupled hyperbolic system

$\lambda$2

(Giona et al., 2016). The vectors $\lambda$3 satisfy a zero-bias condition, and in the symmetric case this becomes $\lambda$4 (Giona et al., 2016).

These partial waves are the primitive variables of the theory. The overall density and stochastic flux,

$\lambda$5

are derived quantities rather than the fundamental statistical objects (Giona et al., 2016). This distinction becomes decisive in dissipation theory: consistent $\lambda$6-norms and entropy functions depend on the full collection $\lambda$7, whereas corresponding functionals built only from $\lambda$8 need not be monotone in time (Giona et al., 2016).

Continuous-state generalizations replace the finite internal state by a continuum $\lambda$9, leading to continuous Generalized Poisson–Kac processes

$v(x)$0

with primitive densities $v(x)$1 satisfying an integro-hyperbolic equation whose transport term is structurally the one-dimensional linearized Boltzmann equation (Giona et al., 2016). In that setting the stochastic velocity state becomes the kinetic velocity variable itself.

3. Persistence, finite propagation speed, and hyperbolic transport

The defining property of persistence is that the stochastic velocity remains constant over random intervals, so trajectories are continuous and piecewise smooth rather than fractal. In the one-dimensional telegraph process the internal-state correlation is

$v(x)$2

and the persistence time is $v(x)$3 (Ghose, 12 May 2026). More generally, all GPK processes are persistent Kac-type processes because they generalize Kac’s dichotomic velocity model to arbitrarily many velocity states in higher dimensions while preserving finite propagation speed (Giona et al., 2016).

The associated transport equations are hyperbolic. In the scalar case, eliminating the imbalance variable yields the Telegrapher equation

$v(x)$4

with finite signal cone $v(x)$5 away from the diffusive limit (Ghose, 12 May 2026). In the one-dimensional Poisson–Kac model this same structure appears in Cattaneo form,

$v(x)$6

which yields the telegraph equation for $v(x)$7 (Giona et al., 2016).

In the generalized finite-state case, transport is hyperbolic because each partial density $v(x)$8 is advected at finite velocity $v(x)$9, while inter-state switching produces relaxation and recombination among the partial waves (Giona et al., 2016). This is the sense in which GPK dynamics produces “undulatory transport”: finite-speed, wave-like propagation at mesoscopic scales, with Fickian diffusion only after coarse-graining (Giona et al., 2016).

The Kac limit is the asymptotic regime in which the persistent finite-velocity process converges to Brownian motion. In the scalar case, taking $b>0$0 with

$b>0$1

fixed yields the diffusion equation as the $b>0$2 term becomes negligible (Giona et al., 2016). In continuous-velocity and multidimensional GPK settings the same principle holds: Brownian diffusion is an emergent, long-time, high-rate limit of a more detailed finite-speed dynamics (Giona et al., 2016). This suggests a hierarchy in which persistent Kac-type dynamics is microscopic, hyperbolic transport is mesoscopic, and diffusive parabolic behavior is asymptotic.

Extended Poisson–Kac theory broadens persistence still further by allowing generalized transition rates and additional internal variables. In that framework the state variables and transitional parameters may overlap, as in Lévy walks represented by an age variable $b>0$3 and a state-dependent rate $b>0$4; the resulting dynamics remains finite-speed but can exhibit normal diffusion, superdiffusion, subdiffusion, “Brownian yet non Gaussian” diffusion, and senescence effects (Giona et al., 2020).

4. Nonlinearity, memory, and generalized representations

Nonlinearity enters persistent Kac-type theory in two distinct ways. In one class, the stochastic velocity vectors $b>0$5, transition rates $b>0$6, or transition probabilities $b>0$7 depend on the state variable, producing persistent motion in an inhomogeneous environment (Giona et al., 2016). In another class, the microdynamic equations depend on the statistical state itself, leading to nonlinear Fokker–Planck–Kac models in which the switching rate or drift depends on the partial densities (Giona et al., 2016).

A prototype is the nonlinear Poisson–Kac process

$b>0$8

with rate

$b>0$9

which makes the evolution equations for $1/\lambda$0 nonlinear through $1/\lambda$1 (Giona et al., 2016). Another example is the Shimizu–Yamada-type nonlinear Fokker–Planck–Kac model, where the drift depends on the first moment of the distribution and the dynamics yields nonlinear traveling solitons with finite asymptotic width (Giona et al., 2016). These examples show that self-consistent feedback and persistence can combine to produce anomalous diffusion, non-Gaussian traveling structures, and nonlinear transport behavior not present in purely diffusive models.

Memory effects arise naturally when the internal dynamics depends on history. In the stochastic functional differential equation framework, a process $1/\lambda$2 depends on its segment

$1/\lambda$3

and the joint state $1/\lambda$4 is Markovian in path space (Belloni, 2016). This formalism is used to analyze persistent motion with delayed response, notably bacterial run-and-tumble dynamics in which the run ends when an internal SFDE-driven variable hits a threshold (Belloni, 2016). This yields a persistent Kac-type process with internal memory: finite-speed runs, random switching of direction, and history-dependent tumble timing (Belloni, 2016).

A different and more recent extension concerns nonlinear Feynman–Kac theory. For the advection–reaction–diffusion equation

$1/\lambda$5

the drift field may itself be given by a probabilistic velocity sub-model,

$1/\lambda$6

possibly coupled to $1/\lambda$7 through Poisson–Nernst–Planck, Keller–Segel, or Navier–Stokes relations (Yaacoub et al., 2024). The key result is an embedded representation in which a single process

$1/\lambda$8

is driven by an actual realization of the random velocity process rather than the mean drift, yet expectations over paths still reproduce the deterministic PDE solution (Yaacoub et al., 2024). In this setting, a persistent Kac-type process is a particle trajectory whose drift is given by an embedded stochastic velocity $1/\lambda$9 with finite correlation time or nontrivial structure, such that the expectation over trajectories reproduces the deterministic solution of a drift–diffusion PDE with velocity coupling (Yaacoub et al., 2024).

5. Entropy, irreversibility, and Markovian structure in extended state space

For persistent Kac-type processes, irreversibility is formulated at the level of the primitive partial waves rather than the overall density. In the one-dimensional Poisson–Kac process, the energy-dissipation functional

$dV = f(V,t)\,dt + \sigma\,dW_t,$0

satisfies

$dV = f(V,t)\,dt + \sigma\,dW_t,$1

whereas the corresponding functional built only from the total density $dV = f(V,t)\,dt + \sigma\,dW_t,$2 does not satisfy monotonicity requirements at finite $dV = f(V,t)\,dt + \sigma\,dW_t,$3 (Giona et al., 2016). The same principle persists in the GPK case: for transitionally symmetric dynamics the natural dissipation functional

$dV = f(V,t)\,dt + \sigma\,dW_t,$4

is monotonically nonincreasing, but scalar functionals of $dV = f(V,t)\,dt + \sigma\,dW_t,$5 alone may oscillate (Giona et al., 2016).

The entropy theory is analogous. In the one-dimensional Poisson–Kac process the Boltzmann–Shannon entropy

$dV = f(V,t)\,dt + \sigma\,dW_t,$6

satisfies

$dV = f(V,t)\,dt + \sigma\,dW_t,$7

while the scalar entropy

$dV = f(V,t)\,dt + \sigma\,dW_t,$8

is not monotone in time at finite switching rate (Giona et al., 2016). For transitionally symmetric GPK processes the entropy

$dV = f(V,t)\,dt + \sigma\,dW_t,$9

obeys the same H-theorem-type monotonicity (Giona et al., 2016). This establishes that physically consistent energy dissipation and entropy functions depend on the full system of primitive statistical variables $v$0, not on the overall density alone (Giona et al., 2016).

This feature is closely related to Markovianity. Persistent Kac-type processes are typically Markovian only in an enlarged state space: $v$1 for the telegraph process, $v$2 for GPK, $v$3 for age-structured Lévy-walk embeddings, or $v$4 for SFDE-driven persistent motion (Giona et al., 2016). After projecting onto position alone, the process generally becomes non-Markovian. This is why reduced descriptions can display memory, contextual temporal structure, or oscillatory correlations even when the full lifted process remains Markovian (Ghose, 12 May 2026).

A related consequence appears in discrete-time persistence theory for Gaussian stationary processes. There, the persistence probability

$v$5

plays a role analogous to continuous-time Kac/Rice zero-crossing theory, with large-time decay

$v$6

and a discrete persistence constant $v$7 (Nyberg et al., 2018). Although this discrete-time framework is not itself a velocity-jump dynamics, it illuminates how persistence questions generalize when only coarse temporal observations are available.

6. Applications, analogies, and current extensions

Persistent Kac-type processes now appear across kinetic theory, transport theory, computational modeling, and broader process-based physics.

In kinetic theory, continuous-state GPK processes yield a stochastic derivation of the nonlinear Boltzmann equation. With stochastic velocity $v$8, density $v$9, and a kernel bilinear in the distribution,

$\tau_{\rm pers}=(2\lambda)^{-1}$0

the evolution equation becomes exactly the nonlinear Boltzmann equation after choosing collision kernels that enforce conservation of momentum and kinetic energy (Giona et al., 2016). This is described as furnishing a positive answer to Kac’s program in kinetic theory (Giona et al., 2016).

In transport and hydrodynamics, GPK dynamics leads naturally to hyperbolic balance laws for mass, momentum, and magnetic fields. Partial mass and momentum densities $\tau_{\rm pers}=(2\lambda)^{-1}$1 satisfy continuity and momentum equations containing a stochastic acceleration term and a nontrivial inertial term $\tau_{\rm pers}=(2\lambda)^{-1}$2, which vanish or simplify only in the Kac limit (Giona et al., 2016). This suggests an extended thermodynamic theory with extra internal variables embodying memory of the Kac-type microstructure (Giona et al., 2016).

In confined geometries and engineering transport, embedded persistent Kac-type processes are used to represent drift–diffusion in porous photoanodes. In the stationary Poisson–Nernst–Planck-type model,

$\tau_{\rm pers}=(2\lambda)^{-1}$3

the embedded continuous branching stochastic process simulates a small number of main paths with Maruyama discretization, and the method is reported to be insensitive to geometric refinement: computation times do not increase with the number of triangles in the mesh representing the porous domain, only the geometry loading costs change (Yaacoub et al., 2024). The same paper notes analogies with path sampling in computer graphics and transport in complex geometries (Yaacoub et al., 2024).

In machine learning, the telegrapher’s equation has been used as the basis of a flow-based generative model. In one dimension,

$\tau_{\rm pers}=(2\lambda)^{-1}$4

and the corresponding Kac process

$\tau_{\rm pers}=(2\lambda)^{-1}$5

provides a telegraph analogue of the diffusion/Brownian correspondence (Duong et al., 25 Jun 2025). The resulting Kac flow gives a probability flow that is globally Lipschitz in the Wasserstein distance and whose velocity field is globally bounded in $\tau_{\rm pers}=(2\lambda)^{-1}$6, in contrast to reverse diffusion flows where the kinetic energy can blow up near the target distribution (Duong et al., 25 Jun 2025). This suggests that persistent finite-speed transport can also be advantageous as a generative prior.

In neuroscience, finite-velocity Kac-type processes have been proposed as a non-diffusive alternative to cable-equation or Wiener models for single-neuron dynamics. The core claim is that diffusive models produce purely monotonic temporal correlations, whereas persistent telegrapher-type transport can generate damped oscillatory correlations of the form

$\tau_{\rm pers}=(2\lambda)^{-1}$7

which may violate Leggett–Garg inequalities without invoking microscopic quantum coherence (Ghose, 12 May 2026). This suggests that finite-speed persistent stochastic transport can generate effective non-Markovianity and contextual temporal structure in reduced observables (Ghose, 12 May 2026).

A more speculative extension appears in process-theoretic electrodynamics, where persistent Kac-type stochastic processes are proposed as the underlying dynamical substrate from which Dirac and Maxwell equations emerge after analytic continuation of Telegrapher-type dynamics (Ghose, 23 May 2026). In that framework, mass is interpreted as a persistence scale and charge as a stochastic coupling scale, while particles and fields are treated as emergent collective modes of coupled persistent stochastic dynamics (Ghose, 23 May 2026). This suggests a broader methodological trend: persistent Kac-type processes are increasingly used not only as transport models but as candidate microdynamic foundations for wave-like or relativistic effective theories.

A common misconception is that because trajectories are random, persistent Kac-type processes cannot recover deterministic streamlines or PDE transport. The embedded nonlinear Feynman–Kac result directly counters this: in a pure-advection half-space example with random persistent velocities $\tau_{\rm pers}=(2\lambda)^{-1}$8, the distribution of first hitting points concentrates around the deterministic streamline endpoint as the time step $\tau_{\rm pers}=(2\lambda)^{-1}$9, with variance proportional to $s(t)\in\{+1,-1\}$0 and vanishing in the limit (Yaacoub et al., 2024). Another misconception is that the relevant thermodynamic observables should depend only on the total density; the entropy and $s(t)\in\{+1,-1\}$1-dissipation theory for GPK shows this fails at finite propagation speed, where the primitive partial waves remain essential (Giona et al., 2016).

Persistent Kac-type stochastic processes therefore form a unified class of finite-velocity stochastic models characterized by internal propagation states, persistent trajectories, and hyperbolic statistical equations. Their theory now encompasses classical telegraph motion, generalized and continuous Poisson–Kac processes, nonlinear and path-dependent velocity-jump systems, embedded Feynman–Kac representations, and extended constructions capable of describing anomalous transport, kinetic equations, structured dissipation, and nontrivial memory effects across multiple disciplines (Giona et al., 2020).