Spectral Properties of Stochastic Processes Possessing Finite Propagation Velocity

This article investigates the spectral structure of the evolution operators associated with the statistical description of stochastic processes possessing finite propagation velocity. Generalized Poisson–Kac processes and Lévy walks are explicitly considered as paradigmatic examples of regular and anomalous dynamics. A generic spectral feature of these processes is the lower boundedness of the real part of the eigenvalue spectrum that corresponds to an upper limit of the spectral dispersion curve, physically expressing the relaxation rate of a disturbance as a function of the wave vector. We also analyze Generalized Poisson–Kac processes possessing a continuum of stochastic states parametrized with respect to the velocity. In this case, there is a critical value for the wave vector, above which the point spectrum ceases to exist, and the relaxation dynamics becomes controlled by the essential part of the spectrum. This model can be extended to the quantum case, and in fact, it represents a simple and clear example of a sub-quantum dynamics with hidden variables.


Introduction
The investigation of micro-and nanoscale physics [1], as well as the extension of statistical physical concepts to new phenomelogies involving active matter and living beings [2,3], have stimulated the development of more refined descriptions of stochastic processes, accounting for background, thermal or quantum fluctuations, aimed at deriving their statistical features and long-term, in some cases anomalous [4], scaling properties [5,6].Experimental results on the motion of bacteria, amoebae, insects and other classes of living beings indicate that the classical paradigm of Wiener fluctuations (the mathematical Brownian motion) does not always apply to the erratic kinematics of these entities [7].
Turning attention to a completely different branch of physics, namely field theory, the physical constraints on the texture of the space-time fabric dictate the requirement of finite propagation velocity as an essential condition for relativistic consistent models of fluctuations [8,9].What is remarkable, in this context, is that the Lorentz covariance applies at the opposite ends of the length scale spectrum, i.e., at very short (quantum fluctuations at particle level) [10] and at very large lengthscale (cosmological models) [11].
But even for processes of everyday human experience, such as those related to mass, momentum and heat transport, representing the natural realm of investigation of thermodynamic and transport theories, the quest for overcoming the classical paradoxes resulting from using stochastic equations with infinite propagation speed, unavoidable within the assumption of constitutive equations of Fickian type (i.e., where the flux of the thermodynamic entities is proportional and opposite to their concentration gradients, such as in Fick, Fourier and Newton constitutive equations for mass, heat and momentum "diffusive" fluxes), is not only dictated by aesthetical or epistemological arguments supporting the internal coherence of these macroscopic and statistical theories [12], but stems from the resolution of physical problems such as solute transport through polymeric materials or heat transport in nanodevices [13,14].
A common denominator in all these issues can be found in the understanding of the emergent properties characterizing stochastic models possessing finite propagation velocity and of their distinctive features, either as regards the local regularity of their trajectories or their collective statistical behaviour.In this article we focus on the latter issue.Specifically, we study the spectral structure of the evolution operators that propagate the probability densities of these models.Henceforth, we refer to these operators as Statistical Evolution Operators, SEO for short.This is because the constraint of bounded velocity enforces specific and characteristic spectral features, distinguishing them from their classical and widely used counterparts, namely Wiener and Wiener-driven fluctuations.
GPK processes originate from the need of defining a sufficiently wide class of stochastic models able to interpret at a mesoscopic level the equations of extended thermodynamics for physical processes far from equilibrium [12,23].This represents an extension of the original model proposed by Kac [15] of a stochastic process driven by the parity of a Poisson counting process, which have been subject of intense investigation in the past as a prototype of a non-Markovian stochastic motion driven by bounded, dichotomous and colored noise [24][25][26][27][28].We choose these two classes of processes because they are subjected to a common and complete statistical description in terms of the partial probability densities parametrized with respect to internal variables of the process (for GPK, the velocity direction; for LWs, the transition age).This was shown especially for LWs by the recent work of Fedotov [29] (see also [30][31][32][33]), and by the unifying theory of Extended Poisson-Kac processes developed by Giona and collaborators [34].Our main result is that the real part of the eigenvalues of the associated SEO of these processes is bounded from below.Consequently, any perturbation of arbitrary wavelength cannot relax faster than an exponential function e −µ l t , where µ l is such lower bound.
The article is organized as follows.In Sections 2 and 3, we discuss one-velocity models.These are processes characterized by a single velocity in norm, where only the direction of motion may change.These processes may seem, at a first glance, a simplistic physical model.In point of fact, they represent an important physical situation related to photon propagation in material media.In this scenario, in fact, the transition in the velocity direction of the photon is solely dictated by scattering events [35].
In Section 4, we extend the analysis to processes characterized by a continuous structure of stochastic states.While the main qualitative result obtained in Sections 2 and 3 is confirmed also in this case, novel and qualitatively interesting features arise, such as the incompleteness of the spectrum, and the fact that the eigenvalue spectrum can be defined solely for particular wavevectors (in the case discussed, these belong to an interval).The consequence of the latter property is that the relaxation decay in density dynamics is controlled not only by the point spectrum but also by the essential component of the spectrum.For a mathematical definition of the point and essential spectra the reader is referred to the classical monograph by Kato [36].Throughout this article we adopt an "operational" interpretation of the essential spectrum as the spectral complement to the point spectrum: an eigenvalue of the linear operator A defined in a functional space F (say the space of square summable functions) belongs to the essential spectrum if the corresponding eigenfunction does not belong to the functional space F.
Given the strong analogy between Poisson-Kac processes and quantum mechanics, provided by the seminal paper by Gaveau et al. [37], we explore an extension of the Gaveau model based on the functional structure of the model developed in Section 4 possessing a continuum of velocity states, focusing on its spectral properties.This provides an interesting and exactly solvable example of a prototypical sub-quantum theory, in the meaning of David Bohm [38,39], the emergent properties of whose theory coincide with the classical Schrödinger equation.

One-velocity GPK processes
Consider the simplest GPK process on the real line possessing two states s = {1, 2}, a uniform transition rate λ 0 and a transition probability matrix A given by where r ∈ [0, 1).The velocities in the two states are equal in absolute value and opposite in sign.We denote them as b This is therefore a one-velocity model.The stochastic model can be expressed as where χ 2 (t; λ 0 , A) is a 2-state finite Poisson process attaining values {1, 2}, i.e., a Markov process characterized by the transition rate λ 0 and by the transition probability matrix A. The parameter r in eq. ( 1) determines a bias in the stochastic motion.The occurrence of two stochastic states implies that the statistical description of the process involves the partial probability densities p(x, t) = (p 1 (x, t), p 2 (x, t)), and the SEO is given by ∂p(x, t)/∂t = L[p(x, t)] where the infinitesimal generator L is [18] and λ = λ 0 /2.Consider the associated eigenvalue problem, L[ψ(x)] = µ ψ(x).From the structure of L defined by eq. ( 3), the eigenfunctions are the imaginary exponentials, ψ(x) = e ik x (ψ 0 1 , ψ 0 2 ).Introducing the dimensionless quantities µ * = µ/ λ, k * = k b 0 / λ, the eigenvalues of L can be expressed by the relation where the square root should be interpreted in its multivalued meaning, and i = √ −1.The eigenvalue structure described by Eq. ( 4) is depicted in figure 1. Specifically: (i) the real part of the eigenvalues is lower-bounded, i.e., Re[µ * ] ≥ −2 (and of course Re[µ * ] ≤ 0).For finite λ 0 this implies that the dynamics of any perturbation of the probability density, δp(x, t), is lower-bounded as ||δp(x, t)|| ≥ Ce −2 t * , where || • || is any suitable norm (for instance the L 2 -norm) and t * = t λ.For this process, therefore, µ l = λ 0 .(ii) For any wavevector k, two spectral branches exist.While in the symmetric case, r = 0, the real parts of the spectral branches collapse into a single one for k * > 1, the effect of a bias, r > 0, is to keep the two branches separate for any k * .This phenomenon can be understood easily in the limit k * → ∞.Assuming k * ≫ 1 in eq. ( 4), we can write , and consequently, µ * = −1 ± i(k * + i r).This implies that the high-wavevector region of the spectrum is well approximated by the two eigenvalue branches µ * = −(1 + r) + i k * and µ * = −(1 − r) − i k * , thus explaining the qualitative features depicted in figure 1.
The latter result finds a direct interpretation in terms of the salient properties of the Green functions for this class of random motion.Consider the dynamics of an ensemble of particles moving according to eq. ( 2) and starting from the origin.Assume symmetric initial conditions, i.e., p 1 (x, 0) = p 2 (x, 0) = δ(x)/2.Figure 2 (a) depicts the evolution of the particle ensemble, expressed in terms of the overall probability density function, p(x, t) = p 1 (x, t) + p 2 (x, t), obtained from stochastic simulations of N p = 5 × 10 7 particles with b 0 = 1, λ 0 = 1/2, so that D = b 2 0 /2λ 0 = 1.A characteristic feature of this class of finite-velocity stochastic motions is that the diagonal entries of the matrix-valued Green function are characterized by the superposition of a continuous, compactly supported component and of an impulsive contribution.Expressed in terms of the overall probability density p(x, t) starting from an impulsive initial condition, this implies that p(x, t) = p c (x, t) + p δ (x, t), where p c (x, t) is the continuous term and p δ (x, t) the impulsive one, namely p δ (x, t) = I + (t) δ(x − b 0 t) + I − (t)δ(x + b 0 t).The functions I ± (t) progressively fade away as t increases [40][41][42].The previous discussion suggests that the decay of the impulsive branches is controlled by the real part of the two asymptotic eigenvalues Re . This phenomenon is depicted in figure 2 (b).
Figure 3 shows the real part of the eigenvalue spectrum of these models for two different values of N as a function of the norm k * = |k * | of the wave vector.The real part of the eigenvalue spectrum is lower bounded, specifically Re[µ * ] ≥ −1.For this 2d GPK process, therefore, we obtain the same lower bound of the 1d biased model, µ l = λ 0 .The comparison of the data shown in the two panels suggests the existence of a well defined spectral limit for N → ∞, i.e., in the case the velocities are uniformly distributed on the unit circumference.This limit case, corresponding to the Markov motions analyzed by Kolesnik [17], suggests that for a continuum of stochastic states the spectral properties of the resulting SEO give rise to a much simpler structure of the spectral branches.

Lévy walks
Owing to the analysis developed by Fedotov [29], subsequently elaborated in [30][31][32][33], and extended in a unitary theory of stochastic processes possessing bounded velocity in [34], a complete statistical description of a LW involves a system of partial probability densities, as for GPK processes.The only difference is that in the LW case, we need to parametrize the partial densities also with respect to an internal parameter, the transition age.This is due to the more complex nature of the density function for the transition times that is no longer exponential (as for Poisson-Kac processes).We consider a one-dimensional LW switching between two states, corresponding to the directions of motion, keeping constant the absolute value of the velocity b 0 .The statistical properties of such a LW are fully described by the partial probability densities p ± (x, t; τ).Setting the vector-valued density p(x, t; τ) satisfies the evolution equation [29] where M τ represents the evolution operator for the transition age, I 2 is the 2 × 2 identity matrix, and A x represents the advection operator with σ z = 1 0 0 −1 a Pauli matrix.Both operators act on the two partial probability density waves p ± .Assuming that at each transition, particles invert their direction of motion, eq. ( 8) is equipped with the boundary condition where σ x is the Pauli matrix σ x = 0 1 1 0 .We now consider the spectral structure underlying the SEO, namely we solve the eigenvalue problem, The eigenfunction ψ(x, τ) can be expressed as planar waves with respect to the spatial coordinate, i.e., ψ(x, τ) = e i k x φ(τ), while the dependence on the transition time τ is accounted for by the vector-valued function φ(τ) = (φ + (τ), φ − (τ)).Its two components satisfy the equations and thus, where A ± are two complex-valued constants.
Imposing the boundary conditions (11) on the function φ, we find that the two constants A ± satisfy the relation where dθ is the probability density function for the transition times.Eliminating A ± from the two eqs.(15), we obtain that the characteristic equation for the eigenvalues of L τ,x is given by For k = 0 one recovers the spectral properties of the "renewal mechanism" of the LW.In this case, eq. ( 16) reduces to Equation ( 17) can be interpreted as follows: the spectral structure of the renewal equation of a LW corresponds to the zeroes of the equations T(µ) = ±1 where T(µ) represents the Laplace transform (of argument µ) of the transition-time probability density T(τ).For real µ, only the positive determination of the latter equation makes sense, and it is straightforward to notice that there exists a unique real solution of T(µ) = 1, namely µ = 0, corresponding to the Frobenius (conservation) eigenvalue of the process.
In the rest of this section, we consider three typical LW models and solve for their spectral properties.

Gamma-distributed transition times
For this model, the transition-time density is expressed by it follows that eq. ( 16) reduces to or equivalently, where ϕ α,h = 2 π h/α, and h is an integer, h = 0, 1, . . . .Two cases may occur: (i) if α = P/Q is rational, with P, Q both integers, there are Q distinct values of e i ϕ α,h corresponding to h = 0, 1, . . ., Q; (ii) if α is irrational, there is a countable number of distinct values of e i ϕ α,h , each corresponding to a different spectral branch.The spectral properties of the renewal mechanism (corresponding to k = 0) follow from eq. ( 21) and the eigenvalues are in general complex-valued.In particular, if α is irrational, (µ h + β)/β, h = 0, 1, . . ., fill densely the unit circumference.Next, we consider the case k = 0. Solving the quadratic equation ( 21) one obtains where h e i ω h /2 , it follows that  It is straightforward to show that the real part of the spectrum is lower bounded.Moreover, the explicit calculation of the eigenvalues indicates that In order to give some examples, figures 4 and 5 depict the real and imaginary part of the spectrum in two typical cases: α = 3 (integer), and α = π (irrational).The other parameters are set to β = b 0 = 1.In the irrational case, solely the the spectral branches corresponding to h = 0, . . ., 19, are depicted.

Superdiffusive Lévy walk
For LWs defined by a transition-time probability density possessing a long-term powerlaw scaling, such as those associated with the distribution function is the Mittag-Leffler function, 0 < ν < 1, τ 0 > 0, and Γ(z) the Gamma function [43,44], the Laplace transform of T(τ) is given by and eq.( 16) provides To solve this equation, we set where ρ ≥ Re[µ].Eq. ( 27) can be thus rewritten as which can be further reduced to This implies the relation which proves the boundedness of the real part of the spectrum.

Power-law distributed transition times
We consider as a third example the classical case of a LW ( 8)-( 11) defined by the the transition-time probability density T(τ) = ξ/(1 + τ) ξ+1 , where ξ > 0. Its Laplace transform is given by where Ei ξ+1 (z) is the Exponential Integral function of order ξ + 1, In this case, it is not possible to derive in closed-form the structure of the eigenvalue branches.Nevertheless, we can still obtain spectral bounds using asymptotic analysis.Let us prove ad absurdum that there exists a lower bound for the real part of the eigenvalue spectrum.Let us suppose the opposite, namely that there are eigenvalues with arbitrarily large real part, that implies arbitrarily large values of |µ|.In this case, we can use eq.( 32), and the asymptotic expansion for the Exponential Integral function Ei ν (z which implies For large Re[µ], one obtains from eq. ( 16) By hypothesis, the real part of the eigenvalues can attain arbitrarily large values so that the higher-order terms in eq. ( 36) can be neglected, reducing it to the equation The solutions of this equation are It follows from eq. ( 38) that the real part of the spectrum is lower-bounded by −ξ/b 0 , contradicting the original hypothesis.The proof is completed.
From the properties of Laplace transforms, namely lim Re[µ]→∞ T(µ) = 0, and applying the same analysis developed above, we can show that all the one-dimensional LWs, whose spectrum fulfils eq. ( 16), are characterized by a lower bound −µ l , with µ l > 0 for Re[µ].

GPK process with a continuum of states
A common feature of one-velocity models is the decomposition of the Green function into a continuous and an impulsive part as discussed in [40,42] and shown pictorially in figure 2.
A question naturally arises, whether it would be possible to define a stochastic process with finite propagation velocity, for which the corresponding matrix-valued Green function does not admit an impulsive contribution but solely the continuous part, similarly to what happen for the solution of the parabolic diffusion equation.While the answer to this question is negative (for reasons that are addressed below), it is indeed possible to construct processes such that, if the overall initial density is impulsive, say p(x, 0) = δ(x), then for any t > 0 the overall density function p(x, t) is smooth and compactly supported, i.e., it does not not possess any impulsive term.This construction is interesting for further addressing the spectral properties of these models, as the sudden annihilation of the impulsive initial conditions necessarily implies a spectral counterpart in the high-wavevector limit.Examples of these models have been discussed in [45].
Processes possessing this property implies a continuous set of stochastic states possessing velocities defined in some bounded and continuous set (in the present case an interval), and smooth transitions amongst them.In this case, the velocity itself can be used to parametrize the stochastic states.A typical example is given by the stochastic dynamics where Ξ(t; λ, K) is a Poisson field defined in [48], and attaining values in [−1, 1], i.e., a stochastic process such that the probabilities densities P(β, t) where where λ(β) is a smooth positive function expressing the transition rate from the stochastic state β, and K(β, β ′ ) a smooth stochastic kernel, with K(β, β ′ ) ≥ 0, and for all β ′ ∈ [−1, 1].The latter accounts for the transition probabilities from state β ′ to all the other states.Henceforth, we consider the simplest case of uniform λ(β) and K(β, β ′ ), i.e., The statistical description of the stochastic motion defined by eq. ( 39) involves the family of partial probability densities p(x, t; β), continuously parametrized with respect to β ∈ [−1, 1], and satisfying the equation In this model, the overall probability density P(x, t) for the particle position x at time t, and the associated flux J(x, t) are given by Using the method outlined in [18,47,48], it follows that the Kac limit of eq. ( 43), i.e., the limit for unbounded propagation velocity and transition rate, b 0 , λ 0 → ∞, keeping fixed the nominal diffusivity D nom = b 2 0 /2 λ 0 is given by the diffusion equation for P(x, t), with a value of the effective diffusivity D equal to D = 2 D nom /3.Moreover, for any finite value of b 0 and λ 0 , the long-term solutions of eq. ( 43) approach those of the parabolic equation (45).
Consider the solutions of eq. ( 43) starting from a spatially impulsive initial distribution P(x, 0) = δ(x).This condition does not specify completely the initial state of the system, as the initial preparation with respect to the internal parametrization β should be also defined, i.e., the whole structure of the partial densities p(x, 0; β) = p 0 (β) δ(x), β ∈ [−1 : 1] at t = 0 should be specified.Two situation can occur: (i) if p 0 (β) = δ(β − β * ) admits an impulsive component at any β = β * , then P(x, t) for t > 0 is characterized by an impulsive contribution centered at x = β * t, propagating with velocity β * superimposed to a continuous distribution deriving from the recombination mechanism amongst the partial waves.Conversely, (ii) if p 0 (β) is smooth, i.e., no impulsive initial term is present, then P(x, t) for t > 0 is also a smooth function of x for any t > 0. The phenomena outlined above are depicted in figure 6 panels (a) and (b).Data have been obtained from stochastic simulations unsing an ensemble of N p = 5 × 10 7 walkers initially placed at x = 0.In the case of the data depicted in panel (a) the initial distribution with respect to the internal state variable is impulsive and centered at β * = 1/2.Conversely, the initial condition for the data depicted in panel (b) is uniform over This example clearly indicates that even in the presence of a continuum of internal states, any initial preparation that is impulsive with respect to the internal-state parametrization β determines an overall density P(x, t) containing an impulsive contribution with respect to x, propagating at constant speed.It is interesting to analyze how this phenomenology can be interpreted in the light of the spectral properties of the associated evolution operator.

Spectral properties
Consider the spectral properties of the infinitesimal generator of the density dynamics ( 43), The eigenvalue equation for L β , admits the eigenfunctions of the form Introducing the dimensionless quantities µ * = µ/λ 0 , k * = k b 0 /λ 0 , the substitution of eq. ( 48) into eqs.( 46)-( 47) provides Consequently, integrating over β and assuming 1 −1 ψ k (β) dβ = 0, the characteristic equation for the eigenvalues follows Setting µ * = r + i ω, and expliciting the integral in eq. ( 50) one obtains the equation Imposing the imaginary part to be null one obtains ω = 0, i.e., the eigenvalues are purely real, µ * = r.Enforcing this property back in eq. ( 51), a simpler equation for the real part r follows, The solution of eq. ( 52) can be easily obtained by introducing the auxiliary variable z = k * /(1 + r), so that eq. ( 52) becomes (1 + r)z = arctan(z).Varying z ∈ [0, ∞), one gets the continuous branch of eigenvalues Given z, the corresponding wavenumber k * follows from the definition of z, namely k * = (1 + r) z.Before addressing the properties of this spectral branch, let us show that there is no other eigenvalue.To show this we have to consider the only remaining condition, namely that In this case, the eigenvalue problem reduces to (µ * + 1 + i k * β) ψ k (β) = 0, which admits no solution for a µ * independent of β.Consequently, the spectrum reduces to the spectral branch defined by eqs.( 52)-( 53).This branch is defined for |k * | < k c = π/2, as depicted in figure 7.In this interval of wavevectors, the associated eigenfunctions defined by eq. ( 49) are complex-valued and can be expressed as where A, is a complex-valued constant and φ k (β) is a real-valued function defined as where the normalization constant Expression (55) with φ k (β) given by eq. ( 56) follows from eq. ( 49), i.e., ψ k (β) = B/[(1 + µ * ) + i k * β], setting B = 1 + i, modulo an arbitrary multiplicative constant.Figure 8 depicts the shape of φ k (β) at k * = 1.5, close to the break-up point k c ≃ 1.57.From the functional form of the eigenfunctions ( 55)-( 56), the dynamical mechanism associated with the spectral break-up becomes clear.As the eigenvalue µ * approaches −1 from below, the associated eigenfunction develops a singularity at β = 0, ψ k (β) → 1/β, and consequently the critical point k c corresponds to the bifurcation point where ψ k (β) ceases to be summable.
The occurrence of a single eigenfunction, restricted solely to the interval [−k c , k c ] is manifestly incomplete in order to represent a generic (square-summable) function of the internal variable β in the interval [−1, 1].It is therefore interesting to analyze the relaxation properties of the operator L β , restricted to a planar wave mode p(x, t; β) = e i k x ζ(β, t), i.e., to consider the dynamics ∂ζ(β, t) where ζ(β, t) is complex valued, and the time variable t has been made nondimensional normalizing it by λ 0 .It follows from eq. ( 57) that  56), symbols (•) the profile obtained from the relaxation of eq. ( 57), integrated numerically.
where Re[•] is the real part of the argument, and ζ(β, t) the complex conjugate of ζ(β, t).
Integrating over β, the dynamics of the where As |Z(t)| 2 ≥ 0, it follows from eq. ( 59) that Eq. ( 60) indicates that all the planar-wave modes cannot decay to zero faster than e −γ * t , with γ * = 1 representing the maximum relaxation rate.This result is valid for any k * , below and above the critical value k c .Consequently, in the statistical evolution of the process, the point spectrum plays a marginal role, especially for high wavevectors, as the dynamics is controlled by the essential spectral component.From eq. ( 60) it follows that also the essential spectrum is lower bounded, and this fact is expressed by the property γ * = 1.The typical evolution of a highwavevector excitation, above the critical value k c , is depicted in figure 9, showing the progressive development of a complex structure in β, not converging to any proper eigenfunction.By expanding ζ(β, t) in truncated Fourier series with respect to β, ζ(β, t) = ∑ N n=−N ζ n (t) e i n πβ , keeping N sufficiently large, the eigenvalue spectrum can be calculated by solving the eigenvalue problem The direct numerical calculation of the spectrum, obtained numerically by setting N = 5000, provides for k * < k c , only a single eigenvalue in the point spectrum, as already found, while for all the values of k * , the essential spectrum is made by essential eigenvalues µ * ess possessing real part identically equal to −1, and imaginary part distributed in an uniform way within the interval [−k * , k * ].This phenomenon is depicted in figure 10.
The overall dynamics of the process involves both the spatial evolution (i.e. the dynamics with respect to the variable x), parametrized by the wavevector k (upon a Fourier transform), and the internal dynamics with respect to the variable β, describing the recombination process amongst the partial density waves.Below the critical threshold k c , the unique point spectrum eigenvalue controls the long-term exponential relaxation of the solutions of eq. (  57).However, the initial/intermediate decay is sensitive to the relaxation exponent γ * pertaining to the essential spectrum.This initial/intermediate behavior becomes more evident starting from initial conditions ζ(β, 0) possessing high-frequency components with respect to the internal variable β.This phenomenon can be clearly observed from the data in figure 11 showing the decay of the , only the long-term exponential relaxation can be observed (line a), the exponent of which is the single point-spectrum exponent r ≃ 0.356.As ν increases, ν > 10 1 , a crossover between the initial/itermediate exponential scaling depending on the essential component of the spectrum ||ζ||(t) ∼ e −γ * t , and the long-term relaxation ||ζ||(t) ∼ e −rt occurs.As a final observation, let us consider the decay of the impulsive component of the overall density function for an initial preparation of the system with particles possessing one and the same initial velocity, i.e., p(x, t; β) = δ(x) δ(β − β * ).This situation corresponds qualitatively to the evolution of the density profiles depicted in figure 2 panel (a) for the simpler one-velocity model. in this case the overall density P(x, t) at time t is the superposition of a continuous distribution P c (x, t) and of an impulsive component centered at x = b 0 β * t, P(x, t; The impulsive component I β * (t) translating in space at velocity β * , corresponds to the fraction of particles that remain in the initial state β * , i.e. not experiencing any transition up to time t.Because of its physical meaning, it is clear that the relaxation of I β * (t) should not depend on the spatial component of the dynamics of p(x, t; β), i.e., should be independent of the wavevector k.Not only this, from the above observation it follows that it is a property exclusively of the Markov recombination mechanism of the partial density waves, described by the internal state dynamics eq. ( 40), and associated with the internal state operator L The spectrum of L is completely degenerate: it possess only the single eigenvalue µ = λ 0 with countable multiplicity, as any function π(β) possessing zero mean in [−1, 1] is an eigen- function associated to it.Specifically, a basis for this eigenspace is given by the sinusoidal functions π ν (β) = sin(ν π β), where ν = 1, 2, ....This eigenproperty of the recombination mechanism, when embedded into the spatio-temporal evolution of the partial density waves p(x, t, β) defined by eq. ( 43) results incompatible with the translation operator −b 0 β ∇ in order to determine any proper eigenstructure, and thus contribute to the essential part of the spectrum of the operator L β associated with the spatial-temporal evolution of the partial density waves.
It follows from the above discussion, that the impulsive part of the dynamics I β * (t) entering eq. ( 62) should decay exponentially as corresponding to the decay defined by the essential spectral component.This phenomenon is depicted in figure 12.The numerical data have been obtained from the stochastic simulation of an ensemble of N p = 10 8 particles evolving in space according to eq. ( 43), initially placed at x = 0 and all possessing the same velocity b 0 β * .Therefore, this preparation of the system corresponds to the initial condition for the partial density waves expressed by p(x, 0; β) = δ(x) δ(β − β * ).By tracking the evolution of the particle system, the intensity of the impulsive peak I β * (t) can be determined.Data in figure 12 refer to different values of b 0 , keeping fixed to ratio b 2 0 /2λ = 1, so that the decay of the impulsive peak depends on the velocity b 0 as

Quantum mechanical extension
There is an intertwined network of mutual analogies between the theory of Brownian motion (and more generally stochastic dynamics) and quantum mechanics that has led to fruitful physical insigths and useful mathematical representations, borrowing ideas and techniques from one field to the other [49][50][51][52].Specifically, there is a strong connection between stochastic processes possesing finite propagation velocity and quantum mechanics, whenever the relativistic constraint on the upper bound for velocities in the space-time propagation of physical phenomena is taken into account.This was already observed by Feynman [53] in his development of the path-integral formulation of quantum mechanics, as regards the path of a relativistic free particle (see Fig. 2.4 in [53]).Hovewer, Gaveau et al. [37] provided the derivation of the strict analogy between the 1 + 1 Dirac equation (1 spatial dimension + 1 temporal dimension) and Poisson-Kac processes in the presence of imaginary transition rates, i.e., where c is the velocity of light in vacuo, i = √ −1 and (ψ + , ψ − ) are the two components of the vector-valued wave function.The spatial probability density is |ψ + | 2 + |ψ − | 2 , consistent with the Dirac theory [54].As for the probabilistic theory of Poisson-Kac processes, eq. ( 64) converges towards the Schrödinger equation in the limit of c, λ 0 → ∞, keeping fixed the ratio c 2 /2λ 0 (Kac limit).This model has been analyzed and generalized by several authors [55][56][57][58][59].
In this Section we extend the stochastic model considered in the previous paragraphs to a quantum mechanical setting following the approach by Gaveau et al. [37] discussed above, by considering a continuous velocity instead of a single velocity c, as in (64).The reason for this extension, stems from the observation that the complete description of the dynamics of the stochastic process (43) requires the introduction of the addition variable β parametrizing the partial density functions p(x, t; β).In a quantum extension, this variable can be viewed as a "hidden variable" associated with a sub-quantum level of description of the system, from which classical quantum theory can be viewed as an emergent property.In other words, this quantum extension is aimed at providing an archetype for possible subquantum descriptions of reality, in the spirit of the work by David Bohm [38,39], showing how from this "hidden" level, classical quantum theory can be emergently derived.As we are primarily interested in highlighting these two aspects, the simple case of a free particle is considered in a one-dimensional spatial setting, in agreement with the stochastic model developed in the previous Sections.
The quantum mechanical counterpart of eq. ( 43) is given by where ψ(x, t; β) is the wavefunction at the subquantum level, parametrized with respect to the additional variable β ∈ [−1, 1].The constants b 0 and λ 0 are determined below.Let Ψ(x, t) be the classical (Schrödinger) wavefunction, and introduce the flux From eq. ( 65) it follows that J ψ (x, t) fulfils the equation Let us consider the Kac limit of this model, letting b 0 , λ 0 → ∞, keeping constant the nominal diffusivity b 2 0 /2λ 0 = D nom .From eq. ( 65), in the Kac limit, i.e. ψ(x, t; β) in the Kac limit does not depend on β, while J ψ (x, t) attains the expression where eq. ( 69) has been used.From eq. ( 70) inserted in eq. ( 67) it follows that if D nom is such that , where D h = h/2 m is the quantum diffusivity for a massive particle of mass m, then Ψ(x, t) satisfies the Schrödinger equation Moreover, it is easy to show that eq. ( 65) is a proper quantum mechanical equation, in the meaning that a Born-rule can be defined from it.In point of fact, elementary calculations provides ∂ ∂t and therefore 1 −1 |ψ(x, t; β)| 2 dβ is a conserved quantity corresponding to the quantum probability density function with respect to the position x.Observe that, in the Kac limit, ψ(x, t; β) ∼ Ψ(x, t), and therefore in agreement with the Schrödinger's nonrelativistic quantum theory.
For the sake of completeness, it is useful to observed that eq. ( 65) refers to a single free particle, but its extension to include the action of a potential or its generalization to a many-body problem is straightforward.If U(x) is a potential acting on a massive particle, the associated quantum model generalizing eq. ( 65) is expressed by (73) Using the same approach outlined above, it is easy to see that eq.(65) converges in the Kac limit to the nonrelativistic Schrödinger equation in the presence of the potential U(x).In a similar way, the extension to a many-body problem is formally simple.To begin with, consider a single spatial dimension.Let x = (x 1 , ..., x n ) be the coordinate vector of the particle system, where x k is the position of the k-th particle, k = 1, . . ., n, and β = (β 1 , . . ., β n ), λ = (λ 1 , . . ., λ n ), so that b 0 β k is the velocity of the k-th particle with β k ∈ [−1, 1], and λ k is its transition rate.The extension of eq. ( 65) to an n-body problem on the real line in the presence of the interaction potential U(x) reads, where In higher dimensions, considering only translational motion, let us define X = (x 1 , . . ., x n ), where x k = (x k,1 , . . ., x k,d ) is the position vector of the k-th particle in a d-dimensional space, and β = (β 1 , . . ., β n ), where the velocity of the k-th particle is b 0 β k = b 0 (β k,1 , . . ., β k,d ), and assuming isotropic transition rates, eq.(74) simply becomes ∂ψ(X, t; where . The inclusion of n-body interactions expressed by the potential U(X) is perfectly admissible in a non-relativistic framework, while it would require care and attention in a relativistic perspective.But this discussion is manifestly outside the scope of the present work.

Spectral properties and dispersion relations
and let ε be an eigenvalue, L q [φ(x, β)] = ε φ(x, β).The eigenfunctions are of the form φ(x, β) = e i k x ψ 0 (β).Introducing, as before, the normalized quantities, ε * = ε/λ 0 , k * = k b 0 /λ 0 , it follows that that the "internal component" of the eigenfunction φ 0 (β) associated with the normalized eigenvalue ε * is given by where A is a constant, leading to the eigenvalue equation that provides, upon quadraturae and algebraic manupulations, the expression for the eigenvalue In the quantum case, for any wavevector k there exists an eigenvalue in the point spectrum of the operator L q , which is not the case for its probabilistic counterpart L β as thoroughly analyzed in Sections 4-4.1.
Let us consider the scaling properties of the eigenvalue spectrum, i.e., the dispersion relation connecting the particle energy E = −h ε to the wave vector k.
The Kac limit (b 0 , λ 0 → ∞, b 2 0 /2 λ 0 = constant), corresponds to small values of k * .Expanding the exponential entering eq. ( 79) in a Taylor series to the leading order, in this case e −2 x = 1 − 2x + 2x 2 − 4x 3 /3, and developing the necessary algebra, one obtains from eq. ( 79) Neglecting the O((k * ) 3 ) term, it implies for the eigenvalue ε, and consequently the energy is given by consistently with the classical quantum mechanical result.Consider the quantity −ε/D h, that in the classical quantum limit equals k 2 , and set σ = For large values |k|, −ε/D h = σ|k * |, and therefore −ε/D h admits the crossover behavior This phenomenon is depicted in figure 13.The linear scaling with |k| in the high wavevector region is the signature in the quantum model of the bounded propagation velocity characterizing the generalized Schrödinger operator L q for finite values of b 0 and λ 0 .

Concluding remarks
In this article we have analyzed the spectral (eigenvalue) properties of several classes of stochastic processes possessing finite propagation velocity, showing the occurrence of a lower bound for the real part of the eigenvalue spectrum.As regard this spectral property, it is immaterial whether the long-term diffusive behavior is regular (linear Einsteinian scaling of the mean square displacement) or anomalous (superdiffusive).This result marks a fundamental and significant difference with respect to the parabolic diffusion model, for which short wavelength perturbations decay at a rate proportional to the squared norm of the wavevector.
There are several general implications of this result.(i) In the approach to the foundations of the thermodynamics of irreversible processes [60]: The results obtained confirm the analysis developed in [61] indicating that the Markov operators associated with SEOs of stochastic processes characterized by a finite propagation velocity are invertible and form a group of transformations parametrized with respect to time t.Conversely, in the case of parabolic diffusion models the weaker semigroup property for t ≥ 0 holds.(ii) In classical field and transport theory: The results indicate that the hyperbolic field equations (e.g. for mass, heat and momentum transfer) that can be developed starting from microscopic equations of motions expresses in the form of stochastic equations, characterized by a bounded velocity of propagation [48], may have completely different stability properties than their parabolic counterparts.
The occurrence of a lower bound for the real part of the eigenspectrum of operators describing the statistical evolution of stochastic processes possessing finite propagation velocity occurs also for continuum models with a spectrum of particle velocities.In this article we have considered a prototypical example on the real line, in which the relaxational properties at high wavevectors are entirely associated with the essential part of the spectrum.The lower spectral bound is a consequence of the internal recombination dynamics of the stochastic states of the system, and corresponds to the largest decay exponent (with reversed sign) of the overall probability density function P(x, t), for an initial configuration with all the particles located at the same point and possessing the same velocity.
In a broader perspective, the analysis of SEOs outlined in this article can be extended to other relevant physical phenomenologies.This is the case for the linear response of stochastic systems possessing anomalous behavior, addressed in [62][63][64][65][66] by either using Continuous Time Random Walk scalings or approximate SEOs involving fractional derivative operators.The same problem can by framed within the age formalism of LWs introduced in Section 3, by keeping the same definitions and evolution equations ( 7)- (10), and modifying the boundary condition eq. ( 11) in the form of where the transition probabilities A ±,± (t), A ±,± (t) ≥ 0, A +,± (t) + A −,± (t) = 1 account for the effect of the forcing field f (t) and are defined according to [62,65] as where 0 < µ < 1 and | f (t)| ≤ 1.In the analysis of this problem the spectral results outlined in this manuscript are of limited use, as the simplest way to tackle this problem from physical grounds is to consider the partial moment hierarchy m x n p ± (x, t; τ) dx, which will be addressed in forthcoming works.
Getting back to our probabilistic model with a spectrum of velocities, its quantum extension displays two main qualitative properties: (i) contrarily to the purely probabilistic case, for any k ∈ R an eigenvalue of the quantum operator L q exists; (ii) the boundedness in the propagation velocity characterizing L q implies in the quantum case a linear dispersive relation of the energy E with respect to |k| for high values of k.This result is a direct consequence of the undulatory propagation, as from quantization E = h ω, and the ω is related to the group velocity v g , by dω/dk = v g , implying E ∼ k.The quantum model analyzed in paragraph 4 is a simple example of a sub-quantum model possessing internal ("hidden") degrees of freedom (in this case the internal variable β ∈ [−1, 1]) and providing the same emergent results of classical quantum theory.

Figure 1 .
Figure 1.Real (a) and imaginary (b) part of the eigenvalue spectrum µ * vs k * of the 1d-biased GPK model at different values of the parameter r.Panel (a): line (a) refers to r = 0, line (b) to r = 0.1, line (c) to r = 0.5, line (d) to r = 0.9.Panel (b): the arrow indicates increasing values of r, the same as in panel (a).

Figure 3 .
Figure 3. Real part of the spectrum vs the wavevector k * for the 2d GPK model discussed in the text with N stochastic states.(a): N = 4; (b): N = 20.

Figure 7 .
Figure 7. Eigenvalue branch µ * vs the wavenumber k * (line a).Line (b) represents the maximum relaxation exponent γ * = 1, associated with the essential spectrum of the evolution operator.Symbol (•) and ( ) are the result of the numerical simulations of eq.(57), considering the decay of the L 2 -norm of the solution.

Figure 8 .
Figure 8. Profile of the real part of the eigenfunction φ k (β) vs β at k * = 1.5.The line represents the graph of eq.(56), symbols (•) the profile obtained from the relaxation of eq.(57), integrated numerically.

Figure 13 .
Figure13.Dispersion curve −ε/D h vs the wavenumber |k| for a free particle in the sub-quantum model(65).Solid line (a) is the classical quantum result −ε/D h = k 2 .Solid line (b) represents the linear scaling −ε/D h ∼ |k|.The other lines depict the graph of the function at the r.h.s of eq.(83) for increasing values of σ = 10 2 , 10 4 , 10 6 , 10 8 , indicated by the direction of the arrow.