Asymptotic Relation for the Transition Density of the Three-Dimensional Markov Random Flight on Small Time Intervals

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• 作者：Alexander D. Kolesnik
• 关键词：Markov random flight ; Conditional density ; Characteristic function ; Asymptotic relation ; Transition density ; Small time intervals
• 刊名：Journal of Statistical Physics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：166
• 期：2
• 页码：434-452
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Statistical Physics and Dynamical Systems; Theoretical, Mathematical and Computational Physics; Physical Chemistry; Quantum Physics;
• 出版者：Springer US
• ISSN：1572-9613
• 卷排序：166

文摘

We consider the Markov random flight \(\varvec{X}(t), \; t>0,\) in the three-dimensional Euclidean space \({\mathbb {R}}^{3}\) with constant finite speed \(c>0\) and the uniform choice of the initial and each new direction at random time instants that form a homogeneous Poisson flow of rate \(\lambda >0\). Series representations for the conditional characteristic functions of \(\varvec{X}(t)\) corresponding to two and three changes of direction, are obtained. Based on these results, an asymptotic formula, as \(t\rightarrow 0\), for the unconditional characteristic function of \(\varvec{X}(t)\) is derived. By inverting it, we obtain an asymptotic relation for the transition density of the process. We show that the error in this formula has the order \(o(t^3)\) and, therefore, it gives a good approximation on small time intervals whose lengths depend on \(\lambda \). An asymptotic formula, as \(t\rightarrow 0\), for the probability of being in a three-dimensional ball of radius \(r<ct\), is also derived. Estimate of the accuracy of the approximation is analysed.