[Paper Review] Random motions with space-varying velocities

Roberto Garra, Enzo Orsingher|arXiv (Cornell University)|Sep 15, 2016

Diffusion and Search Dynamics7 references3 citations

TL;DR

This paper investigates random motions on the line and plane with space-varying velocities, deriving explicit probability distributions for particle positions by solving generalized telegraph and Euler-Poisson-Darboux (EPD) equations. The key contribution is the analytical solution of the damped wave equation with spatially dependent speeds, revealing that the support of the distribution takes the form of superellipses (e.g., astroids), with exact solutions for polynomial velocity profiles and fractional EPD-type equations.

ABSTRACT

Random motions on the line and on the plane with space-varying velocities are considered and analyzed in this paper. On the line we investigate symmetric and asymmetric telegraph processes with space-dependent velocities and we are able to present the explicit distribution of the position $\mathcal{T}(t)$, $t>0$, of the moving particle. Also the case of a non-homogeneous Poisson process (with rate $λ= λ(t)$) governing the changes of direction is analyzed in three specific cases. For the special case $λ(t)= α/t$ we obtain a random motion related to the Euler-Poisson-Darboux (EPD) equation which generalizes the well-known case treated e.g. in Foong and Van Kolck (1992), Garra and Orsingher (2016) and Rosencrans (1973). A EPD--type fractional equation is also considered and a parabolic solution (which in dimension $d=1$ has the structure of a probability density) is obtained. Planar random motions with space--varying velocities and infinite directions are finally analyzed in Section 5. We are able to present their explicit distributions and for polynomial-type velocity structures we obtain the hyper and hypo-elliptic form of their support (of which we provide a picture).

Motivation & Objective

• To extend classical telegraph processes to include space-dependent velocities on the line and plane.
• To derive explicit probability laws for particle positions under non-homogeneous Poisson processes governing direction changes.
• To analyze the geometric structure of the support of planar random motions under polynomial velocity functions.
• To generalize the Euler-Poisson-Darboux equation to include space-varying speeds and fractional forms.
• To establish connections between random flight models and damped wave equations with variable coefficients.

Proposed method

• Transform the variable-coefficient telegraph equation into a classical form using the change of variable $ y = \int_0^x \frac{dx'}{c(x')} $, enabling explicit solution derivation.
• Use the method of characteristics and Bessel function identities to solve the one-dimensional telegraph equation with space-dependent $ c(x) $, yielding a probability density involving $ I_0 $ and Dirac delta functions.
• Apply a similar transformation $ u = \int_0^x \frac{dw}{c_1(w)}, v = \int_0^y \frac{dz}{c_2(z)} $ to reduce the planar damped wave equation with spatially varying $ c_1(x), c_2(y) $ to the standard form.
• Derive the conditional characteristic function of the position vector $ (X(t), Y(t)) $ given $ N(t) = n $, using integrals over time intervals and Bessel functions of the first kind.
• Construct the full probability density by summing over all possible numbers of direction changes $ n $, leading to a solution in terms of a series involving $ J_0 $ functions.
• Analyze the support $ \mathcal{D}_{\gamma,\beta} $ of the distribution for power-law velocity functions $ c_j(x) \propto |x|^\gamma $, showing it forms a superellipse (Lamé curve).

Experimental results

Research questions

• RQ1 How does the inclusion of space-varying velocity $ c(x) $ affect the probability distribution of a one-dimensional telegraph process?
• RQ2 What is the geometric structure of the support of planar random motions when velocities depend on position in each coordinate?
• RQ3 Can the Euler-Poisson-Darboux equation be generalized to include spatially dependent speeds, and what are the resulting solutions?
• RQ4 How do non-homogeneous Poisson processes with time-dependent rates $ \lambda(t) $ influence the motion's distribution when combined with space-dependent velocities?
• RQ5 What is the role of fractional derivatives in modeling random motions with space-varying velocities, and can they yield non-negative solutions?

Key findings

• For the one-dimensional symmetric telegraph process with space-dependent velocity $ c(x) $, the position density is explicitly given by a formula involving the modified Bessel function $ I_0 $, Dirac deltas, and a characteristic function over domain $ D $, where $ D $ is defined by $ \left| \int_0^x \frac{dx'}{c(x')} \right| < t $.
• When $ \lambda(t) = \alpha / t $, the model yields a solution related to the Euler-Poisson-Darboux equation, generalizing known results from earlier works.
• For planar motions with velocities $ c_1(x), c_2(y) $, the support $ \mathcal{D}_{\gamma,\beta} $ is a superellipse defined by $ \left( \frac{c_1 |x|^{1-\gamma}}{1-\gamma} \right)^2 + \left( \frac{c_2 |y|^{1-\beta}}{1-\beta} \right)^2 < t^2 $, which includes astroids for $ \gamma = \beta = 2/3 $.
• The probability density for $ d $-dimensional motion with separable space-varying speeds $ c_j(x_j) $ is given by a closed-form expression involving $ \Gamma $-functions and a power of the time-distance term, valid when $ \sum_{j=1}^d \left| \int_0^{x_j} \frac{du_j}{c_j(u_j)} \right|^2 < t^2 $.
• For $ c_1(x) = c_1 $, $ c_2(y) = c_2 $, the support reduces to an ellipse $ c_1^2 |x|^2 + c_2^2 |y|^2 < t^2 $, confirming consistency with known results.
• In the case $ c_j(x) \propto |x|^{2/3} $, the support boundary becomes an astroid, and for $ c_1 = c_2 = 1/3 $, the boundary is exactly the astroid $ 9|x|^{2/3} + 9|y|^{2/3} < t^2 $.