Showing papers by "Enzo Orsingher published in 2019"
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TL;DR: In this article, the explicit conditional distributions of the position of the moving particle when the number of switches of directions is fixed were derived and expressed in terms of Bessel functions, and the governing equations are derived and given as products of D'Alembert operators.
Abstract: A cyclic random motion at finite velocity with orthogonal directions is considered in the plane and in $\mathbb{R}^3$. We obtain in both cases the explicit conditional distributions of the position of the moving particle when the number of switches of directions is fixed. The explicit unconditional distributions are also obtained and are expressed in terms of Bessel functions. The governing equations are derived and given as products of D'Alembert operators. The limiting form of the equations is provided in the Euclidean space $\mathbb{R}^d$ and takes the form of a heat equation with infinitesimal variance $1/d$.
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TL;DR: In this article, the authors studied the sojourn time on the positive half-line up to time $ t $ of a drifted Brownian motion with starting point $ u $ and subject to the condition that $ \min 0\leq z \leq l} B(z)> v $ with $ u > v $.
Abstract: In this paper we study the sojourn time on the positive half-line up to time $ t $ of a drifted Brownian motion with starting point $ u $ and subject to the condition that $ \min_{ 0\leq z \leq l} B(z)> v $, with $ u > v $. This process is a drifted Brownian meander up to time $ l $ and then evolves as a free Brownian motion. We also consider the sojourn time of a bridge-type process, where we add the additional condition to return to the initial level at the end of the time interval. We analyze the weak limit of the occupation functional as $ u \downarrow v $. We obtain explicit distributional results when the barrier is placed at the zero level, and also in the special case when the drift is null.