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Multidimensional random motions with a natural number of finite velocities

Fabrizio Cinque, Mattia Cintoli

TL;DR

This work develops a unified framework for multidimensional random motions with a finite number of finite velocities, introducing minimal and canonical motions and detailing how the position process X(t) evolves within a convex polytope that grows with time. By expressing X(t) as an affine function of the cumulative sojourn times T_{(h)}(t) and establishing a bijection between minimal motions and their time allocations, the authors derive explicit joint distributions of position, velocity counts, and current velocity, including boundary behavior. They show that higher-dimensional motions can be projected onto lower-dimensional spaces without loss of essential distributional properties, enabling dimension reduction and connection across spaces, and extend the analysis to non-homogeneous Poisson dynamics, yielding PDE systems for the absolutely continuous component and singularity results that relate to reduced-dimensional motions. The approach unifies cyclic, complete, and canonical motion families and provides a tractable analytic framework for analyzing finite-velocity stochastic processes in arbitrary dimensions, with potential applications in physics, geology, and finance where finite-velocity models are relevant.

Abstract

We present a detailed analysis of random motions moving in higher spaces with a natural number of velocities. In the case of the so-called minimal random dynamics, under some wide assumptions, we show the joint distribution of the position of the motion (both for the inner part and the border of the support) and the number of displacements performed with each velocity. Explicit results for cyclic and complete motions are derived. We establish useful relationships between motions moving in different spaces and we derive the form of the distribution of the movements in arbitrary dimension. Finally, we investigate further properties for stochastic motions governed by non-homogeneous Poisson processes.

Multidimensional random motions with a natural number of finite velocities

TL;DR

This work develops a unified framework for multidimensional random motions with a finite number of finite velocities, introducing minimal and canonical motions and detailing how the position process X(t) evolves within a convex polytope that grows with time. By expressing X(t) as an affine function of the cumulative sojourn times T_{(h)}(t) and establishing a bijection between minimal motions and their time allocations, the authors derive explicit joint distributions of position, velocity counts, and current velocity, including boundary behavior. They show that higher-dimensional motions can be projected onto lower-dimensional spaces without loss of essential distributional properties, enabling dimension reduction and connection across spaces, and extend the analysis to non-homogeneous Poisson dynamics, yielding PDE systems for the absolutely continuous component and singularity results that relate to reduced-dimensional motions. The approach unifies cyclic, complete, and canonical motion families and provides a tractable analytic framework for analyzing finite-velocity stochastic processes in arbitrary dimensions, with potential applications in physics, geology, and finance where finite-velocity models are relevant.

Abstract

We present a detailed analysis of random motions moving in higher spaces with a natural number of velocities. In the case of the so-called minimal random dynamics, under some wide assumptions, we show the joint distribution of the position of the motion (both for the inner part and the border of the support) and the number of displacements performed with each velocity. Explicit results for cyclic and complete motions are derived. We establish useful relationships between motions moving in different spaces and we derive the form of the distribution of the movements in arbitrary dimension. Finally, we investigate further properties for stochastic motions governed by non-homogeneous Poisson processes.
Paper Structure (12 sections, 8 theorems, 67 equations)

This paper contains 12 sections, 8 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Random motions with a natural number of finite velocities
  3. Minimal random motions
  4. Probability law of the position of minimal motions
  5. Distribution on the boundary of the support
  6. Random motions with a finite number of velocities
  7. Motions in $\mathbb{R}^D$ with $D$-dimensional state space
  8. Random motions governed by non-homogeneous Poisson process
  9. Appendix. Proof of Lemma \ref{['lemmaProiezione']}
  10. Appendix. Complete canonical random motion
  11. Probability mass of the singularity
  12. PDE governing the absolutely continuous component

Key Result

Lemma 1.1

Let $v_0,\dots,v_M\in \mathbb{R}^D$ such that $\text{dim}\Bigl(\text{Conv}(v_0,\dots,v_M)\Bigr) = R$. For $k=0,\dots, M$, there exists the set $I^{R,k}$ of the indexes of the first $R$ linearly independent rows of the matrix $\Bigl[v_h-v_k\Bigr]_{\substack{h=0,\dots,M\\h\not=k}}$ and $I^R = I^{R,k}

Theorems & Definitions (31)

  • Example 1.1: Telegraph process and cyclic motions
  • Example 1.2: Complete random motions
  • Example 1.3: Random motion with orthogonal velocities
  • Lemma 1.1
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Theorem 2.1
  • proof
  • Remark 2.2: Canonical motion
  • ...and 21 more