Planar Symmetric Markov Random Flight

• Planar symmetric Markov random flight is a finite-velocity stochastic process in R² characterized by piecewise-linear trajectories and isotropic or orthogonal directional changes.
• The model employs telegraph-type partial differential equations and explicit density formulas that separate singular boundary components from continuous interior distributions.
• Extensions include non-homogeneous switching rates, orthogonal directional symmetries, and scale-invariant dynamics on Poisson line networks, offering diverse asymptotic behaviors.

Searching arXiv for recent and foundational papers on planar symmetric Markov random flights and closely related finite-velocity random motions. Searching for "planar symmetric Markov random flight". Planar symmetric Markov random flight denotes a class of finite-velocity stochastic motions in $\mathbb R^2$ in which a particle moves along linear segments and changes direction at random times, with symmetry imposed either by isotropic resampling on the unit circle or by an unbiased rule on a structured direction set. In the classical isotropic model, the particle starts at the origin, moves with constant speed $c>0$, and changes direction at the jump times of a homogeneous Poisson process of rate $\lambda>0$, each new direction being chosen uniformly on the unit circumference; the resulting law is supported in the disc of radius $ct$ and decomposes into a singular boundary component and an absolutely continuous interior component (Kolesnik, 2017, Gregorio et al., 2011). Subsequent work has extended the notion to non-homogeneous switching, orthogonal-direction motions with higher-order hyperbolic PDEs, and scale-invariant network-constrained flights on Poisson line Scale-Invariant Random Spatial Networks (SIRSN), while preserving a symmetric Markovian flight structure (Garra et al., 2014, Cinque et al., 2021, Kendall, 2019).

1. Classical isotropic planar model

The standard planar symmetric Markov random flight is the motion of a particle that starts at the origin at time $t=0$, moves with constant speed $v>0$ or $c>0$, and changes its direction at the jump times of a homogeneous Poisson process with rate $\lambda>0$; at each Poissonian instant, the direction is resampled uniformly on the unit circle (Kolesnik, 2017, Kolesink, 2013). In planar notation one writes the unit direction as $n(\theta)=(\cos\theta,\sin\theta)$, $\theta\in[0,2\pi)$, so that the velocity is piecewise constant and the trajectory is a concatenation of straight segments.

The process is isotropic because directions are chosen uniformly on $c>0$0, and it is Markov because the inter-turn times are i.i.d. exponential and the new direction after each turn is independent of the past (Kolesink, 2013). The position $c>0$1 satisfies

$c>0$2

so finite propagation speed is built into the model (Kolesnik, 2017).

Conditional on a fixed number $c>0$3 of direction changes, the planar law is explicitly computable. In the formulation with constant speed $c>0$4, one has

$c>0$5

for $c>0$6, whereas for $c>0$7 the position is uniform on the circle $c>0$8 (Gregorio et al., 2011). This conditional law already displays a characteristic structural feature: a purely absolutely continuous interior distribution once at least one switch has occurred, and a singular ballistic component when no switch occurs.

A recurrent misconception concerns the scope of the term. In the classical isotropic setting it refers to Pearson-type random flights with constant speed and Poisson reorientation, but later planar symmetric flights may replace the unit-circle direction law by orthogonal directions or by line-network constraints while retaining symmetry in the transition rule (Cinque et al., 2021, Kendall, 2019). Thus the phrase does not identify a single geometry, only a family of symmetric finite-velocity Markovian motions in the plane.

2. Governing equations, exact densities, and characteristic-function structure

Let $c>0$9 be the joint density of position and direction in the classical isotropic model. It satisfies the forward Kolmogorov–Feller equation

$\lambda>0$0

with $\lambda>0$1 (Kolesnik, 2017). After elimination of the angular variable, the spatial density $\lambda>0$2 satisfies the planar damped-wave or telegraph-type equation

$\lambda>0$3

with $\lambda>0$4 and $\lambda>0$5 (Kolesnik, 2017). By isotropy, $\lambda>0$6 with $\lambda>0$7, and the PDE reduces to a radial form on $\lambda>0$8.

The exact transition density in the plane has the form

$\lambda>0$9

where the first term is the singular wavefront on $ct$0 and the second is the absolutely continuous component on $ct$1 (Kolesnik, 2017). Equivalent formulas appear in the classical planar literature with $ct$2 in place of $ct$3 (Gregorio et al., 2011). The singular mass equals $ct$4, exactly the probability of no Poisson switches up to time $ct$5.

The planar law also admits an isotropic Fourier-Hankel description. For the multidimensional symmetric Markov random flight, two series representations of the characteristic function are available; in dimension $ct$6, the time-series specialization yields

$ct$7

with coefficients generated recursively from

$ct$8

and the singular boundary term specialized to the plane becomes

$ct$9

(Kolesnik, 2023). This representation is consistent with the decomposition into a circular singular component and an interior absolutely continuous density.

These formulas encode the defining analytical signature of the classical planar symmetric flight: finite-speed propagation, an interior transport law governed by a telegraph equation, and a boundary wavefront carrying the no-switch probability. They also provide the starting point for essentially all planar extensions.

3. Moments, scaling regimes, and long-time behavior

Because of isotropy, the mean displacement vanishes for all $t=0$0:

$t=0$1

The velocity autocorrelation is exponential,

$t=0$2

and the planar mean squared displacement is

$t=0$3

(Kolesnik, 2017). These relations quantify the crossover between ballistic transport at short times and diffusive scaling under suitable high-frequency turning limits.

Two asymptotic regimes are distinguished in the literature. Under Kac’s fast-diffusion scaling, symmetric random flights converge to Brownian motion; in the plane the limiting PDE is

$t=0$4

which produces unbounded Gaussian spreading (Kolesnik, 2017). By contrast, the slow diffusion conditions

$t=0$5

lead to a stationary limit with compact support:

$t=0$6

where $t=0$7 (Kolesnik, 2017). The boundary component has mass $t=0$8 and the interior mass $t=0$9.

A different long-time statement concerns the coordinate marginals. Although the projection of the planar flight onto an axis has random velocity $v>0$0 or $v>0$1, its density is asymptotically equivalent to that of the one-dimensional Goldstein–Kac telegraph process. For fixed $v>0$2 with $v>0$3,

$v>0$4

where $v>0$5 is the marginal density of the planar flight and $v>0$6 is the telegraph density (Kolesnik, 10 Jul 2025). The result is pointwise and applies to interior points; it does not include the telegraph process’s delta masses at $v>0$7.

This contrast between compactly supported finite-speed laws, Brownian limits, stationary slow-diffusion profiles, and telegraph-like marginal asymptotics is central to the planar theory. It shows that the same finite-velocity mechanism yields distinct macroscopic behaviors depending on the scaling of $v>0$8, $v>0$9, and $c>0$0.

4. Non-homogeneous rates and orthogonal-direction symmetries

The homogeneous Poisson model extends naturally to non-homogeneous switching rates $c>0$1. In the isotropic planar case the spatial density continues to satisfy a telegraph-type equation,

$c>0$2

with $c>0$3 and $c>0$4 (Garra et al., 2014). For the Euler–Poisson–Darboux class, one writes

$c>0$5

corresponding in the plane to $c>0$6 with $c>0$7, and the explicit density on $c>0$8 becomes

$c>0$9

Since $\lambda>0$0, the zero-switch probability is $\lambda>0$1, so there is no boundary mass for any $\lambda>0$2 (Garra et al., 2014). The same mechanism suppresses boundary singularities for $\lambda>0$3, whereas $\lambda>0$4 retains a nonzero boundary mass because $\lambda>0$5 (Garra et al., 2014).

A different planar symmetry arises when motion is restricted to the four orthogonal directions $\lambda>0$6. In the standard orthogonal model with non-homogeneous rate $\lambda>0$7, the accessible set is

$\lambda>0$8

and, after the $\lambda>0$9 rotation

$n(\theta)=(\cos\theta,\sin\theta)$0

the planar density factorizes as

$n(\theta)=(\cos\theta,\sin\theta)$1

where $n(\theta)=(\cos\theta,\sin\theta)$2 and $n(\theta)=(\cos\theta,\sin\theta)$3 are independent one-dimensional symmetric telegraph processes with speed $n(\theta)=(\cos\theta,\sin\theta)$4 and rate $n(\theta)=(\cos\theta,\sin\theta)$5 (Cinque et al., 2021). The total density in this geometry satisfies a fourth-order hyperbolic PDE rather than the second-order telegraph equation of the isotropic disk-supported model.

In the symmetric orthogonal case of the clockwise/counterclockwise model with parameter $n(\theta)=(\cos\theta,\sin\theta)$6, the decomposition becomes

$n(\theta)=(\cos\theta,\sin\theta)$7

with $n(\theta)=(\cos\theta,\sin\theta)$8 and $n(\theta)=(\cos\theta,\sin\theta)$9 i.i.d. telegraph processes of speed $\theta\in[0,2\pi)$0 and rate $\theta\in[0,2\pi)$1, and the hydrodynamic limit is

$\theta\in[0,2\pi)$2

(Marchione et al., 2024). A symmetric planar Markov random flight therefore need not have circular support: in orthogonal-direction models the support is diamond-shaped and the governing PDE is of higher order, yet the symmetry principle remains the same.

5. Marginals, radial observables, and inter-flight distances

For the classical isotropic planar flight, the coordinate projections $\theta\in[0,2\pi)$3 and $\theta\in[0,2\pi)$4 are identically distributed, supported on $\theta\in[0,2\pi)$5, and absolutely continuous. Their common density on $\theta\in[0,2\pi)$6 is

$\theta\in[0,2\pi)$7

where $\theta\in[0,2\pi)$8 is the modified Bessel function of order $\theta\in[0,2\pi)$9 and $c>0$00 is the modified Struve function of order $c>0$01 (Kolesnik, 10 Jul 2025). Unlike the full planar law, the marginal has no delta masses at $c>0$02; instead it has an integrable square-root divergence generated by the no-switch arc contribution.

A second important observable is the Euclidean distance between two independent planar symmetric Markov random flights. If $c>0$03 and $c>0$04 have parameters $c>0$05 and $c>0$06, then

$c>0$07

takes values in $c>0$08, and its law is absolutely continuous on this support (Kolesink, 2013). If $c>0$09 and $c>0$10 is the acute angle between the position vectors, then $c>0$11 is uniformly distributed on $c>0$12 and independent of $c>0$13, and conditional on $c>0$14, $c>0$15 the law of cosines gives

$c>0$16

The exact distribution function is obtained by conditioning on whether each flight has made zero or at least one turn by time $c>0$17, producing four contributions corresponding to $c>0$18 (Kolesink, 2013).

These observables show that the planar model remains tractable after projection, radialization, or pairing. Marginals preserve finite-speed support without endpoint atoms; pairwise distances eliminate the singular circle masses of the individual flights and yield a fully absolutely continuous interaction law.

6. Scale-invariant Poisson-line generalization on SIRSNs

A conceptually different planar symmetric Markov random flight is constructed on an isotropic Poisson line SIRSN, where the environment is a speed-marked improper Poisson line process $c>0$19 on the plane with intensity

$c>0$20

for $c>0$21 (Kendall, 2019). Here each line $c>0$22 carries a positive speed mark $c>0$23, and the model is isotropic, Euclidean-motion invariant, and scale-invariant under

$c>0$24

The continuous-time flight moves at the maximum allowable speed along a line of $c>0$25 and may change line at intersections. Sampling at line-switching times yields a discrete-time Markov chain

$c>0$26

with involution $c>0$27 representing direction reversal (Kendall, 2019). The model is built through an abstract scattering representation of Markov chains, and in the balanced case the directional transmission probabilities satisfy

$c>0$28

Similarity-equivariance forces the class parameter to be a power of line speed, $c>0$29, and the Metropolis–Hastings scattering probability is

$c>0$30

with invariant measure

$c>0$31

defined for $c>0$32 (Kendall, 2019). In stationarity, the turning angle $c>0$33 between previous and current lines has density $c>0$34 on $c>0$35, and the log-relative speed has an asymmetric Laplace density.

The critical index is

$c>0$36

because the stationary mean log-speed increment is

$c>0$37

At $c>0$38, $c>0$39 and the main theorem states speed neighborhood-recurrence:

$c>0$40

so the speed neither drifts to $c>0$41 nor to $c>0$42 (Kendall, 2019). The paper interprets the resulting trajectory as a “randomly-broken local geodesic” and uses the critical recurrence result to support the conjecture that true geodesics on the Poisson line SIRSN never come to a complete stop.

This network-constrained theory changes almost every classical ingredient. The state is a pair of lines rather than a position-direction pair, the speeds are power-law line marks rather than a fixed constant, and “Rayleigh” refers to the Pearson random-flight paradigm rather than to a Rayleigh speed distribution (Kendall, 2019). Yet the object remains a planar symmetric Markov random flight in the precise sense of a piecewise-linear, direction-reversible, finite-speed Markovian motion in the plane.

7. Conceptual unification and scope

Across its variants, the planar symmetric Markov random flight is characterized by finite-speed transport, piecewise-linear trajectories, and a symmetry principle that enters through either isotropic directional resampling, balanced orthogonal switching, or direction-reversal invariance on a random line network (Kolesnik, 2017, Cinque et al., 2021, Kendall, 2019). The classical isotropic model has disk support and a second-order telegraph equation; orthogonal-direction models have diamond support and fourth-order hyperbolic equations; SIRSN flights are constrained to a Poisson line network and are governed by scattering kernels rather than by uniform angular resets.

Several distinctions are therefore essential. First, boundary singularities are model- and rate-dependent: homogeneous isotropic flights carry mass $c>0$43 on the circle $c>0$44, whereas EPD-type rates with $c>0$45 remove this component entirely (Garra et al., 2014). Second, planar symmetry does not force Euclidean isotropy of support: orthogonal symmetric models are symmetric but not rotation-invariant, and their singular support lies on edges and vertices rather than on a circle (Cinque et al., 2021). Third, long-time similarity to the telegraph process may occur at the level of marginals even when the microscopic velocity set is continuous rather than binary (Kolesnik, 10 Jul 2025).

The topic therefore spans a coherent but nontrivial family of planar stochastic motions. Its core analytical themes are explicit finite-speed propagators, telegraph- and EPD-type equations, Poisson or non-Poisson switching structures, exact marginal and pairwise observables, and in the scale-invariant SIRSN setting, a rigorous connection between symmetric random flights and conjectural geodesic structure on random spatial networks (Kolesnik, 2017, Kolesink, 2013, Kendall, 2019).