---
title: Planar Symmetric Markov Random Flight
url: https://www.emergentmind.com/topics/planar-symmetric-markov-random-flight
type: topic
---

# Planar Symmetric Markov Random Flight

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 [1708.08793, 1107.5913]. 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 [1411.0648, 2108.10027, 1908.08481].

## 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 [1708.08793, 1309.6459]. 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 $S_1$, 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 [1309.6459]. The position $X(t)\in\mathbb R^2$ satisfies
$$
\operatorname{supp} X(t)\subset \mathcal B_{vt}^2:=\{x\in\mathbb{R}^2:\|x\|\le vt\},
$$
so finite propagation speed is built into the model [1708.08793].

Conditional on a fixed number $N(t)=n$ of direction changes, the planar law is explicitly computable. In the formulation with constant speed $c$, one has
$$
p_{\underline{\bf X}_2}(\underline{\bf x}_2,t;n)=\frac{n}{2\pi (ct)^n}\Big(c^2t^2-||\underline{\bf x}_2||^2\Big)^{\frac n2-1},\quad ||\underline{\bf x}_2||<ct,
$$
for $n\ge 1$, whereas for $n=0$ the position is uniform on the circle $||x||=ct$ [1107.5913]. 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 [2108.10027, 1908.08481]. 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 $u(x,\theta,t)$ be the joint density of position and direction in the classical isotropic model. It satisfies the forward Kolmogorov–Feller equation
$$
\frac{\partial u}{\partial t}(x,\theta,t)+v\,n(\theta)\cdot\nabla_x u(x,\theta,t)=-\lambda\,u(x,\theta,t)+\frac{\lambda}{2\pi}\int_0^{2\pi}u(x,\phi,t)\,d\phi,
$$
with $u(x,\theta,0)=\delta(x)\,(2\pi)^{-1}$ [1708.08793]. After elimination of the angular variable, the spatial density $p(x,t)$ satisfies the planar damped-wave or telegraph-type equation
$$
\frac{\partial^2 p}{\partial t^2}(x,t)+2\lambda\,\frac{\partial p}{\partial t}(x,t)=v^2\,\Delta p(x,t),
$$
with $p(x,0)=\delta(x)$ and $\partial_t p(x,t)|_{t=0}=0$ [1708.08793]. By isotropy, $p(x,t)=p(r,t)$ with $r=\|x\|$, and the PDE reduces to a radial form on $0<r<vt$.

The exact transition density in the plane has the form
$$
p(x,t)=\frac{e^{-\lambda t}}{2\pi v t}\,\delta\!\left(v^2 t^2-r^2\right)
+\frac{\lambda}{2\pi v}\,
\frac{\exp\!\left(-\lambda t+\frac{\lambda}{v}\sqrt{v^2 t^2-r^2}\right)}{\sqrt{v^2 t^2-r^2}}\,
\Theta(vt-r),\qquad t>0,
$$
where the first term is the singular wavefront on $r=vt$ and the second is the absolutely continuous component on $r<vt$ [1708.08793]. Equivalent formulas appear in the classical planar literature with $c$ in place of $v$ [1107.5913]. The singular mass equals $e^{-\lambda t}$, exactly the probability of no Poisson switches up to time $t$.

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 $d=2$, the time-series specialization yields
$$
\phi_2(k,t)=e^{-\lambda t}\sum_{n=0}^\infty \frac{Y_{n+1}(k)}{n!}t^n,
$$
with coefficients generated recursively from
$$
O_{2r}=0,\qquad O_{2r+1}=(-1)^r\frac{(ck)^{2r}}{2r+1},
$$
and the singular boundary term specialized to the plane becomes
$$
p^{(s)}(r,t)=e^{-\lambda t}\frac{\delta(r-ct)}{2\pi ct}
$$
[2311.08102]. 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 $t>0$:
$$
\mathbb{E}\,X(t)=0.
$$
The velocity autocorrelation is exponential,
$$
\mathbb{E}\,[V(t)\cdot V(s)]=v^2\,e^{-\lambda|t-s|},
$$
and the planar mean squared displacement is
$$
\mathbb{E}\,\|X(t)\|^2
=\frac{2v^2}{\lambda}\,t-\frac{2v^2}{\lambda^2}\left(1-e^{-\lambda t}\right)
$$
[1708.08793]. 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
$$
\frac{\partial u}{\partial t}=\rho^2\,\Delta u,
$$
which produces unbounded Gaussian spreading [1708.08793]. By contrast, the slow diffusion conditions
$$
\lambda\to 0,\qquad v\to 0,\qquad t\to\infty,\qquad \lambda t\to a>0,\qquad vt\to \rho>0
$$
lead to a stationary limit with compact support:
$$
q(x)=\frac{e^{-a}}{2\pi \rho}\,\delta\!\left(\rho^2-r^2\right)
+\frac{a}{2\pi \rho}\,
\frac{\exp\!\left(-a+\frac{a}{\rho}\sqrt{\rho^2-r^2}\right)}{\sqrt{\rho^2-r^2}}\,
\Theta(\rho-r),
$$
where $r=\|x\|$ [1708.08793]. The boundary component has mass $e^{-a}$ and the interior mass $1-e^{-a}$.

A different long-time statement concerns the coordinate marginals. Although the projection of the planar flight onto an axis has random velocity $c\cos\alpha$ or $c\sin\alpha$, its density is asymptotically equivalent to that of the one-dimensional Goldstein–Kac telegraph process. For fixed $x$ with $|x|<ct$,
$$
|f(x,t)-g(x,t)|=O(t^{-1}),\qquad t\to\infty,
$$
where $g$ is the marginal density of the planar flight and $f$ is the telegraph density [2507.07525]. The result is pointwise and applies to interior points; it does not include the telegraph process’s delta masses at $\pm ct$.

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 $c$, $\lambda$, and $t$.

## 4. Non-homogeneous rates and orthogonal-direction symmetries

The homogeneous Poisson model extends naturally to non-homogeneous switching rates $\lambda(t)$. In the isotropic planar case the spatial density continues to satisfy a telegraph-type equation,
$$
\frac{\partial^2 p}{\partial t^2}+2\,\lambda(t)\,\frac{\partial p}{\partial t}=c^2\,\Delta p,
$$
with $p(x,0)=\delta(x)$ and $\partial_t p(x,t)|_{t=0}=0$ [1411.0648]. For the Euler–Poisson–Darboux class, one writes
$$
\frac{\partial^2 p}{\partial t^2}+\frac{1+2\gamma}{t}\,\frac{\partial p}{\partial t}=c^2\,\Delta p,
$$
corresponding in the plane to $\lambda(t)=\alpha/t$ with $\gamma=\alpha-\tfrac12>0$, and the explicit density on $r<ct$ becomes
$$
p(x,t)=\frac{\gamma}{\pi}\,\frac{1}{(ct)^{2\gamma}\,\big(c^2t^2-r^2\big)^{\gamma-1}}.
$$
Since $\Lambda(t)=\int_0^t \alpha/s\,ds=+\infty$, the zero-switch probability is $e^{-\Lambda(t)}=0$, so there is no boundary mass for any $t>0$ [1411.0648]. The same mechanism suppresses boundary singularities for $\lambda(t)=\lambda\coth(\lambda t)$, whereas $\lambda(t)=\lambda\tanh(\lambda t)$ retains a nonzero boundary mass because $e^{-\Lambda(t)}=1/\cosh(\lambda t)$ [1411.0648].

A different planar symmetry arises when motion is restricted to the four orthogonal directions $\{(\pm1,0),(0,\pm1)\}$. In the standard orthogonal model with non-homogeneous rate $\lambda(t)$, the accessible set is
$$
S_{ct}=\{(x,y)\in\mathbb R^2:\ |x|+|y|\le ct\},
$$
and, after the $45^\circ$ rotation
$$
u=\frac{x+y}{2},\qquad v=\frac{x-y}{2},
$$
the planar density factorizes as
$$
p(x,y,t)=2\,p_U(u,t)\,p_V(v,t),
$$
where $U$ and $V$ are independent one-dimensional symmetric telegraph processes with speed $c/2$ and rate $\lambda(t)/2$ [2108.10027]. 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 $p=\tfrac12$, the decomposition becomes
$$
X(t)=U(t)+V(t),\qquad Y(t)=U(t)-V(t),
$$
with $U$ and $V$ i.i.d. telegraph processes of speed $c/2$ and rate $\lambda/2$, and the hydrodynamic limit is
$$
\partial_t f=\frac12\Delta f,\qquad 
f(x,y;t)\to \frac{1}{2\pi t}\exp\!\left(-\frac{x^2+y^2}{2t}\right)
$$
[2408.01825]. 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 $X_1(t)$ and $X_2(t)$ are identically distributed, supported on $[-ct,ct]$, and absolutely continuous. Their common density on $(-ct,ct)$ is
$$
g(x,t)=\frac{e^{-\lambda t}}{\pi \sqrt{c^2 t^2-x^2}}
+\frac{\lambda e^{-\lambda t}}{2 c}\left[I_0(z)+L_0(z)\right],
\qquad 
z=\frac{\lambda}{c}\sqrt{c^2 t^2-x^2},
$$
where $I_0$ is the modified Bessel function of order $0$ and $L_0$ is the modified Struve function of order $0$ [2507.07525]. Unlike the full planar law, the marginal has no delta masses at $\pm ct$; 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 $Z_1(t)$ and $Z_2(t)$ have parameters $(c_1,\lambda_1)$ and $(c_2,\lambda_2)$, then
$$
\rho(t)=\|Z_1(t)-Z_2(t)\|
$$
takes values in $(0,(c_1+c_2)t)$, and its law is absolutely continuous on this support [1309.6459]. If $R_i(t)=\|Z_i(t)\|$ and $\Phi(t)$ is the acute angle between the position vectors, then $\Phi(t)$ is uniformly distributed on $(0,\pi)$ and independent of $R_1(t),R_2(t)$, and conditional on $R_1(t)=s$, $R_2(t)=u$ the law of cosines gives
$$
\rho(t)=\sqrt{s^2+u^2-2su\cos\Phi(t)}\in(|s-u|,s+u).
$$
The exact distribution function is obtained by conditioning on whether each flight has made zero or at least one turn by time $t$, producing four contributions corresponding to $(N_1,N_2)\in\{0,\ge1\}^2$ [1309.6459].

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 $\Pi$ on the plane with intensity
$$
\nu(\mathrm{d}v\,\mathrm{d}r\,\mathrm{d}\theta)
=\frac{\gamma-1}{2}\, v^{-\gamma}\, \mathrm{d}v\,\mathrm{d}r\,\mathrm{d}\theta,
$$
for $\gamma>2$ [1908.08481]. Here each line $\ell\in\Pi$ carries a positive speed mark $v(\ell)$, and the model is isotropic, Euclidean-motion invariant, and scale-invariant under
$$
r\mapsto a\,\bar r,\qquad v \mapsto a^{1/(\gamma-1)}\,\bar v.
$$

The continuous-time flight moves at the maximum allowable speed along a line of $\Pi$ and may change line at intersections. Sampling at line-switching times yields a discrete-time Markov chain
$$
Z_n=(\ell_{-,n},\ell_{0,n})\in S:=\{(\ell_-,\ell_0)\in\Pi\times\Pi:\ \ell_-\neq \ell_0\},
$$
with involution $(\ell_-,\ell_0)\leftrightarrow(\ell_0,\ell_-)$ representing direction reversal [1908.08481]. The model is built through an abstract scattering representation of Markov chains, and in the balanced case the directional transmission probabilities satisfy
$$
\omega_{a,+}=\omega_{a,-}=1/2.
$$

Similarity-equivariance forces the class parameter to be a power of line speed, $\kappa(\ell)=v(\ell)^\alpha$, and the Metropolis–Hastings scattering probability is
$$
s_{(\ell_1,\ell_2)}=\min\Big\{1,\Big(\frac{v(\ell_2)}{v(\ell_1)}\Big)^\alpha\Big\},
$$
with invariant measure
$$
\pi_{(\ell_1,\ell_2)}=\min\{v(\ell_1)^\alpha,v(\ell_2)^\alpha\},
$$
defined for $\alpha>\gamma-1$ [1908.08481]. In stationarity, the turning angle $\phi$ between previous and current lines has density $\tfrac12\sin\phi$ on $[0,\pi)$, and the log-relative speed has an asymmetric Laplace density.

The critical index is
$$
\alpha_c=2(\gamma-1),
$$
because the stationary mean log-speed increment is
$$
\mathbb E[U]
=
\frac{1}{\gamma-1}-\frac{1}{\alpha-(\gamma-1)}
=
\frac{\alpha-2(\gamma-1)}{(\gamma-1)(\alpha-(\gamma-1))}.
$$
At $\alpha=\alpha_c$, $\mathbb E[U]=0$ and the main theorem states speed neighborhood-recurrence:
$$
\mathbb{P}\big(|W_n-W_0|\le\varepsilon \text{ infinitely often}\big)=1,\quad \forall \varepsilon>0,
$$
so the speed neither drifts to $\infty$ nor to $0$ [1908.08481]. 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 [1908.08481]. 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 [1708.08793, 2108.10027, 1908.08481]. 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 $e^{-\lambda t}$ on the circle $r=ct$, whereas EPD-type rates with $\Lambda(t)=\infty$ remove this component entirely [1411.0648]. 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 [2108.10027]. 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 [2507.07525].

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 [1708.08793, 1309.6459, 1908.08481].

Source: https://www.emergentmind.com/topics/planar-symmetric-markov-random-flight