Random Flights in Higher Spaces

Isotropic Random Motion at Finite Speed with K-Erlang Distributed Direction Alternations

“…Uniformly distributed direction of motion or isotropic motion with motion driven by a homogeneous Poisson process has been studied by Pinsky [5] on a Riemannian manifold and by Orsingher and De Gregorio in higher dimensions [6], see also [7] and references therein. A recent study on multidimensional random motion shows that the evolutions are driven by hyperparabolic operators composed of the telegraph operator and their integer powers [8].…”

Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

“…where, for x 2 + y 2 < c 2 t 2 (see [15]), Pr {X(t) ∈ dx, Y (t) ∈ dy|N (t) = n} dx dy = n 2n(ct) n c 2 t 2 − x 2 + y The factor 2 appearing in (4.81) and (4.83) can be interpreted as follows. The displacements generated by an even number of Poisson events are disregarded and replaced by displacements produced by an odd number of deviations.…”

“…Random flights in R n have been studied (Orsingher and De Gregorio [15]) but their distributions are not related to higher-dimensional telegraph equations.…”

“…(1.12) We then take the inverse L ν (t), t > 0, to the process H ν (t), t > 0, defined as 13) whose distribution is related to that of H ν (t), t > 0, by means of the formula 15) where the fractional derivatives appearing in (1.15) are again in the RiemannLiouville sense. We are able to give explicit forms of the Laplace transforms of h ν (x, t) and l ν (x, t) in terms of Mittag-Leffler functions for all values of 0…”

“…r(x, y, 0) = δ(x, y), r t (x, y, 0) = 0, (1.27) corresponds to the distribution r(x, y, t) of the vector T (t) = (X(t), Y (t)) related to a planar motion described in Orsingher and De Gregorio [15]. This random motion T (t), t > 0, is performed at finite velocity c, possesses sample paths composed by segments whose orientation is uniform in (0, 2π), and with changes of direction at Poisson times.…”