Asymmetric Telegraph Process

• The asymmetric telegraph process is a two-state random evolution model characterized by unequal switching intensities and directional speeds.
• It extends the standard two-state Markov chain to include finite-velocity motions, nonstandard boundary conditions, and resetting mechanisms to model persistent random behaviors.
• Recent studies have incorporated history dependence, multi-state generalizations, and advanced statistical inference techniques to enhance its applicability in fields like physics and signal processing.

The asymmetric telegraph process is a class of two-state finite-velocity or two-level random evolutions in which the two directional states are not probabilistically equivalent. In the canonical formulation, asymmetry is introduced through unequal switching intensities between the two states, unequal velocities, or both. In broader usage, the term also covers telegraph-type models with directional speed imbalance, state-dependent switching, nonstandard boundary mechanisms, multi-state on/off structures, damping, resetting, and circular wrapping. Across these formulations, the common core is persistent motion or occupation in one of two regimes, together with random regime changes whose parameters need not be symmetric (Khasminskii et al., 2015, Cinque, 2022, Marchione et al., 6 May 2026, Gregorio et al., 2020).

1. Canonical two-state formulation

In its most standard continuous-time form, the asymmetric telegraph process is a two-state Markov chain with unequal directional rates. One common representation is a hidden signal $Y(t)\in\{y_1,y_2\}$ with generator

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$

where $\lambda$ is the transition intensity from $y_1$ to $y_2$ and $\mu$ is the transition intensity from $y_2$ to $y_1$. Under stationarity,

$$\mathbf P\{Y(t)=y_1\}=\frac{\mu}{\lambda+\mu},\qquad \mathbf P\{Y(t)=y_2\}=\frac{\lambda}{\lambda+\mu},$$

and the transition probabilities are

$$P_{11}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},\qquad P_{12}(t)=\frac{\lambda}{\lambda+\mu}-\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},$$

with the analogous formulas for $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$0 and $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$1. The stationary mean and covariance are

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$2

This is the basic unequal-rate telegraph structure used in hidden-signal estimation problems (Khasminskii et al., 2015).

A signal-processing version writes the hidden state as $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$3 with transition rates $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$4 for $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$5 and $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$6 for $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$7. In that notation, the state-conditioned dwell times are exponential,

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$8

and the stationary occupancies are

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$9

When the observed levels are $\lambda$0, the stationary mean is $\lambda$1. In the special case $\lambda$2, asymmetry enters both through the mean and through the single-exponential relaxation rate $\lambda$3 (Lambert et al., 2020).

These formulas isolate the canonical meaning of asymmetry: the process still alternates between two states, but the stationary weights, mean level, and persistence properties are skewed because the two directional escape rates differ.

2. Finite-velocity telegraph motion on the line

A geometrically richer formulation treats the telegraph process as integrated velocity. In one basic asymmetric version, the particle moves with one of two constant velocities $\lambda$4, and the reversal rate depends on the current velocity. Writing

$\lambda$5

with $\lambda$6, the transition density $\lambda$7 satisfies

$\lambda$8

The support at time $\lambda$9 is the interval $y_1$0, and there are singular masses at the endpoints corresponding to trajectories with no reversal. Conditional on the initial velocity and on the number of switches, the interior law is explicit and parity-sensitive: for an odd number $y_1$1 of switches,

$y_1$2

where $y_1$3 is the generalized Mittag-Leffler function. For even parity, the endpoint powers are different according to the starting speed. This makes explicit how unequal rates and unequal velocities jointly tilt the law toward one side of the propagation cone (Cinque, 2022).

The same framework yields an exact conditional evolution formula. If $y_1$4, $y_1$5, and the number of switches up to time $y_1$6 is $y_1$7, then

$y_1$8

where $y_1$9 is the current velocity after $y_2$0 reversals. This identifies the current velocity as the hidden state that restores Markovianity. In the equal-rate asymmetric case, conditioning only on $y_2$1 produces an additional correction term $y_2$2, because the position alone does not determine the current velocity. A plausible implication is that the asymmetry of telegraph motion is not only directional but also informational: hidden-velocity uncertainty survives in the position process itself (Cinque, 2022).

3. Boundary value problems, confinement, and absorption

On bounded intervals, asymmetry appears both in the bulk dynamics and in the boundary mechanism. A direction-dependent finite-velocity model on $y_2$3 assigns rightward velocity $y_2$4 with switching rate $y_2$5 and leftward velocity $y_2$6 with switching rate $y_2$7. For the right-exit probabilities

$y_2$8

the backward equations are

$y_2$9

with boundary conditions $\mu$0, $\mu$1. The unconditional exit probability $\mu$2 satisfies

$\mu$3

The same reduction holds for the mean exit time $\mu$4,

$\mu$5

The asymmetry parameter

$\mu$6

controls the hydrodynamic limit: under $\mu$7, $\mu$8, and $\mu$9, these exit equations converge to the classical Brownian equations with drift $y_2$0 (Marchione et al., 6 May 2026).

Other confined models replace standard reflection by more elaborate rules. On $y_2$1, one asymmetric telegraph process uses velocities $y_2$2, $y_2$3, switching rates $y_2$4, an absorbing lower boundary at $y_2$5, and a reflecting-with-delay upper boundary at $y_2$6: when the process hits $y_2$7, it stays at $y_2$8 until the switch $y_2$9 occurs. Writing

$y_1$0

the coupled transport equations are

$y_1$1

$y_1$2

with boundary conditions

$y_1$3

The analysis proceeds through Laplace transforms in space and time, and the lower-boundary condition yields the closure relation

$y_1$4

where $y_1$5 and $y_1$6. The law is then recovered through inverse Laplace transforms (Pospelov et al., 2015).

A different bounded model on $y_1$7 retains equal speed magnitude $y_1$8 but allows asymmetric switching rates $y_1$9 upward and $$\mathbf P\{Y(t)=y_1\}=\frac{\mu}{\lambda+\mu},\qquad \mathbf P\{Y(t)=y_2\}=\frac{\lambda}{\lambda+\mu},$$0 downward, with absorption probability $$\mathbf P\{Y(t)=y_1\}=\frac{\mu}{\lambda+\mu},\qquad \mathbf P\{Y(t)=y_2\}=\frac{\lambda}{\lambda+\mu},$$1 at each boundary hit and reflection otherwise. The particle alternates between four endpoint-to-endpoint phases, and the analysis reduces boundary events to first-crossing problems for a compound Poisson process. In the asymmetric case,

$$\mathbf P\{Y(t)=y_1\}=\frac{\mu}{\lambda+\mu},\qquad \mathbf P\{Y(t)=y_2\}=\frac{\lambda}{\lambda+\mu},$$2

and the expected absorption time $$\mathbf P\{Y(t)=y_1\}=\frac{\mu}{\lambda+\mu},\qquad \mathbf P\{Y(t)=y_2\}=\frac{\lambda}{\lambda+\mu},$$3 is expressed through the expected cycle lengths $$\mathbf P\{Y(t)=y_1\}=\frac{\mu}{\lambda+\mu},\qquad \mathbf P\{Y(t)=y_2\}=\frac{\lambda}{\lambda+\mu},$$4 and the same hitting probabilities. This makes the boundary problem explicitly solvable even though the underlying motion is not symmetric (Crescenzo et al., 2020).

On the half-line, a further generalization treats absorption at the origin through a random number $$\mathbf P\{Y(t)=y_1\}=\frac{\mu}{\lambda+\mu},\qquad \mathbf P\{Y(t)=y_2\}=\frac{\lambda}{\lambda+\mu},$$5 of visits before final killing. The absorption time is decomposed as

$$\mathbf P\{Y(t)=y_1\}=\frac{\mu}{\lambda+\mu},\qquad \mathbf P\{Y(t)=y_2\}=\frac{\lambda}{\lambda+\mu},$$6

where $$\mathbf P\{Y(t)=y_1\}=\frac{\mu}{\lambda+\mu},\qquad \mathbf P\{Y(t)=y_2\}=\frac{\lambda}{\lambda+\mu},$$7 is the first hit from $$\mathbf P\{Y(t)=y_1\}=\frac{\mu}{\lambda+\mu},\qquad \mathbf P\{Y(t)=y_2\}=\frac{\lambda}{\lambda+\mu},$$8 and the $$\mathbf P\{Y(t)=y_1\}=\frac{\mu}{\lambda+\mu},\qquad \mathbf P\{Y(t)=y_2\}=\frac{\lambda}{\lambda+\mu},$$9 are i.i.d. interarrival times between visits to the origin. When $$P_{11}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},\qquad P_{12}(t)=\frac{\lambda}{\lambda+\mu}-\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},$$0 is light-tailed, the moment generating function is

$$P_{11}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},\qquad P_{12}(t)=\frac{\lambda}{\lambda+\mu}-\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},$$1

and, under the appropriate domain condition, $$P_{11}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},\qquad P_{12}(t)=\frac{\lambda}{\lambda+\mu}-\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},$$2 satisfies an LDP with rate

$$P_{11}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},\qquad P_{12}(t)=\frac{\lambda}{\lambda+\mu}-\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},$$3

This is not a directional asymmetry paper in the narrow sense; rather, it shows how nonstandard boundary behavior can induce asymptotic structures usually studied in asymmetric telegraph models (Iuliano et al., 2022).

4. History dependence, resetting, and multi-state generalizations

A substantial part of the literature extends asymmetry beyond the canonical two-state Markov chain. One example is the damped telegraph process

$$P_{11}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},\qquad P_{12}(t)=\frac{\lambda}{\lambda+\mu}-\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},$$4

with initial bias $$P_{11}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},\qquad P_{12}(t)=\frac{\lambda}{\lambda+\mu}-\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},$$5, but with run-dependent switching rates $$P_{11}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},\qquad P_{12}(t)=\frac{\lambda}{\lambda+\mu}-\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},$$6 and $$P_{11}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},\qquad P_{12}(t)=\frac{\lambda}{\lambda+\mu}-\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},$$7 at the $$P_{11}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},\qquad P_{12}(t)=\frac{\lambda}{\lambda+\mu}-\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},$$8-th run. The model can have unequal velocities $$P_{11}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},\qquad P_{12}(t)=\frac{\lambda}{\lambda+\mu}-\frac{\lambda}{\lambda+\mu}e^{-(\lambda+\mu)t},$$9, unequal directional parameters $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$00, and $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$01, but its main novelty is the damping mechanism: switching becomes faster as more reversals occur. The one-time law has endpoint atoms at $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$02 and $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$03 plus an explicit interior density, and the scaled process $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$04 satisfies an LDP with rate

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$05

The rate is piecewise linear, vanishes at

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$06

and satisfies $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$07 for all $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$08, where $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$09 is the standard telegraph rate function. This suggests that history-dependent damping suppresses atypical empirical velocities more strongly than the homogeneous asymmetric telegraph process (Gregorio et al., 2013).

Another direction enlarges the latent state space. In a one-on/two-off model, the latent chain

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$10

has one active state $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$11 and two distinct inactive states $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$12. The observed process is

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$13

so Brownian diffusion is switched on only while $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$14. The occupation-time law of the on-state is derived exactly, the increment distribution is a Gaussian variance mixture, and the model reduces to the classical two-state telegraph setting when $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$15 or $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$16. Here the asymmetry is structural rather than merely directional: the two off-states create a nontrivial off-sojourn mixture (Pozdnyakov et al., 2018).

A discrete-time semi-Markov analogue is the squirrel random walk, where direction reverses only at the arrival times of a renewal process with generalized Sibuya waiting times. The model is not truly left/right asymmetric in the standard sense, because the reversal law is the same in both directions, but it exhibits three diffusion regimes depending on the tail parameter: ballistic for $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$17,

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$18

superdiffusive for $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$19,

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$20

and diffusive for $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$21,

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$22

The paper explicitly notes that this semi-Markov framework is a natural platform for asymmetric extensions with direction-dependent renewal laws (Michelitsch et al., 2022).

Resetting introduces yet another asymmetry mechanism. In a two-velocity model with $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$23, geometric-counting-process switching, and Poisson resets to the origin at rate $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$24, the reset-modified density satisfies the renewal identity

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$25

The long-time behavior depends sharply on the sign structure of the velocities. If $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$26, the reset point $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$27 lies inside the propagation interval and the density becomes unimodal at the origin in the $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$28 limit. If $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$29, the reset point lies outside the no-reset support, creating a reset-induced region and an upward jump at $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$30. The process also acquires a stationary density as $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$31, a feature absent without resetting (Crescenzo et al., 2023).

5. Statistical inference and hidden telegraph models

Inference for asymmetric telegraph processes is often formulated under incomplete or noisy observation. In one hidden-signal model,

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$32

the latent process $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$33 is a stationary asymmetric telegraph chain with rates $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$34. A method-of-moments estimator is built from $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$35 and

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$36

and then refined into a one-step MLE based on the Wonham filter

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$37

The resulting estimator-process is consistent, asymptotically normal, and asymptotically efficient: $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$38 This shows that the unequal rates of a hidden asymmetric telegraph signal can be estimated jointly from continuous-time Gaussian-noise observations (Khasminskii et al., 2015).

For directly observed but noisy random telegraph signals, an LSTM-based estimator was trained on traces switching between two values, typically $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$39, with unequal rates $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$40 and $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$41. The network predicts one of the log-rates,

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$42

and the other is obtained by sign inversion $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$43. The training range was

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$44

with additive $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$45 and white noise and, in filtered experiments, a 5th-order digital Butterworth low-pass filter. The paper reports that the network remains generally robust against low-pass filtering when

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$46

and that it gives reliable results as the signal-to-noise ratio approaches one. This is an explicitly asymmetric-rate inference problem rather than a new telegraph-process theory (Lambert et al., 2020).

A related hidden-model framework arises when the latent switching is one-on/two-off rather than two-state. For discrete observations at irregular times $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$47, with increments $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$48, the likelihood is evaluated through the HMM forward algorithm: $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$49 The computational burden lies in evaluating the state-dependent increment densities, which are Gaussian variance mixtures involving the occupation-time law. This extends telegraph-process inference from binary hidden states to structurally asymmetric multi-off-state systems (Pozdnyakov et al., 2018).

6. Circular, interacting, and physical realizations

Asymmetric telegraph dynamics also appear on non-Euclidean state spaces and in composite systems. On the circle, the wrapped process

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$50

inherits asymmetry from a line process $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$51 with velocities $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$52 and $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$53 and switching rates $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$54. The asymmetric line law has support $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$55, endpoint atoms at $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$56 and $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$57, and an interior density $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$58 solving an asymmetric hyperbolic PDE with mixed derivative and first-derivative terms. After wrapping, the circular law is obtained by periodization. Under Kac-type scaling,

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$59

the process converges to a circular Brownian motion with drift,

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$60

This is the wrapped analogue of drift generated by directional imbalance (Gregorio et al., 2020).

Linear combinations of independent telegraph processes produce another generalized asymmetric class. For

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$61

with arbitrary nonzero coefficients $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$62, arbitrary speeds $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$63, and arbitrary Poisson rates $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$64, the joint densities with the $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$65 direction states satisfy a coupled first-order hyperbolic system. Eliminating the directional states yields a scalar PDE of order $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$66,

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$67

For $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$68, the sum and difference

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$69

obey an explicit fourth-order hyperbolic equation whose coefficients depend on $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$70. The same paper proves weak convergence under Kac scaling to a Wiener process with variance coefficient $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$71. This suggests that asymmetry can be generated not only at the single-particle level but also through heterogeneous superposition (Kolesnik, 2015).

Interacting systems create yet another form of effective asymmetry. In one-dimensional hard-core telegraph particle systems, each free particle follows the symmetric Goldstein-Kac dynamics, but asymmetry emerges from ordered initial positions, initial velocity configuration, collisions, and reflecting boundaries. For two particles, the four initial velocity states $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$72, $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$73, $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$74, and $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$75 lead to distinct collision-time laws; for example,

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$76

because the first pair initially moves toward each other and the second apart. Thus asymmetry can be interaction-induced even when the individual generator is symmetric (Pogorui, 2013).

A physically distinct but conceptually related realization appears in qubit decoherence. A quantum two-level fluctuator coupled to a qubit and to an environment has a classical limit described by telegraph noise $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$77. The effective telegraph process is symmetric at high temperatures and asymmetric at low temperatures because the thermal absorption and emission rates satisfy

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$78

In the classical limit, the two telegraph switching rates are $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$79 and $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$80, with total rate $$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$81, and the criterion for replacing the quantum fluctuator by a classical telegraph process is

$$Q= \begin{pmatrix} -\lambda & \lambda\ \mu & -\mu \end{pmatrix},$$82

This links asymmetric telegraph noise to detailed balance, thermal bias, and the quantum-to-classical crossover (Wold et al., 2012).

Taken together, these constructions show that the asymmetric telegraph process is less a single model than a family of persistent random evolutions organized around unequal directional structure. In the narrowest sense, it is the two-state unequal-rate telegraph chain; in the broader literature, it includes finite-velocity motions with unequal speeds, hidden and noisy observation models, nonstandard boundaries, run-dependent switching, resetting, multi-state latent structure, wrapping on compact manifolds, and interacting-particle extensions.