The higher-order heat-type equation ∂u/∂t = ±∂ n u/∂x n has been investigated by many authors. With this equation is associated a pseudo-process (X t ) t 0 which is governed by a signed measure. In the even-order case, Krylov, [9], proved that the classical arc-sine law of Paul Lévy for standard Brownian motion holds for the pseudo-process (X t ) t 0 , that is, if T t is the sojourn time of (X t ) t 0 in the half line (0, +∞) up to time t, then P(T t ∈ ds) = ds π √ s(t−s) , 0 < s < t. Orsingher, [13],and next Hochberg and Orsingher, [7], obtained a counterpart to that law in the odd cases n = 3, 5, 7. Actually Hochberg and Orsingher proposed a more or less explicit expression for that new law in the odd-order general case and conjectured a quite simple formula for it. The distribution of T t subject to some conditioning has also been studied by Nikitin & Orsingher,[11], in the cases n = 3, 4. In this paper, we prove that the conjecture of Hochberg and Orsingher is true and we extend the results of Nikitin & Orsingher for any integer n. We also investigate the distributions of maximal and minimal functionals of (X t ) t 0 , as well as the distribution of the last time before becoming definitively negative up to time t.
Consider the high-order heat-type equation ∂u/∂t = ±∂ N u/∂x N for an integer N > 2 and introduce the related Markov pseudo-process (X(t)) t 0 . In this paper, we study several functionals related to (X(t)) t 0 : the maximum M (t) and minimum m(t) up to time t; the hitting times τ + a and τ − a of the half lines (a, +∞) and (−∞, a) respectively. We provide explicit expressions for the distributions of the vectors (X(t), M (t)) and (X(t), m(t)), as well as those of the vectors (τ + a , X(τ + a )) and (τ − a , X(τ − a )).AMS 2000 subject classifications: primary 60G20; secondary 60J25. Key words and phrases: pseudo-process, joint distribution of the process and its maximum/minimum, first hitting time and place, Multipoles, Spitzer identity.where κ N = (−1) 1+N/2 if N is even and κ N = ±1 if N is odd. Let p(t; z) be the fundamental solution of Eq. (1.1) and put p(t; x, y) = p(t; x − y).The function p is characterized by its Fourier transform +∞ −∞ e iuξ p(t; ξ) dξ = e κ N t(−iu) N .( 1.2) With Eq. (1.1) one associates a Markov pseudo-process (X(t)) t 0 defined on the real line and governed by a signed measure P, which is not a probability measure, according to the usual rules of ordinary stochastic processes: P x {X(t) ∈ dy} = p(t; x, y) dy
Abstract. We study the probability distribution of the location of a particle performing a cyclic random motion in R d . The particle can take n possible directions with different velocities and the changes of direction occur at random times. The speed-vectors as well as the support of the distribution form a polyhedron (the first one having constant sides and the other expanding with time t). The distribution of the location of the particle is made up of two components: a singular component (corresponding to the beginning of the travel of the particle) and an absolutely continuous component.We completely describe the singular component and exhibit an integral representation for the absolutely continuous one. The distribution is obtained by using a suitable expression of the location of the particle as well as some probability calculus together with some linear algebra. The particular case of the minimal cyclic motion (n = d + 1) with Erlangian switching times is also investigated and the related distribution can be expressed in terms of hyper-Bessel functions with several arguments.Mathematics Subject Classification. 33E99, 60K99, 62G30.
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