Composition of processes and related partial differential equations

In this paper different types of compositions involving independent fractional Brownian motions B^j_{H_j}(t), t>0, j=1,$ are examined. The partial differential equations governing the distributions of I_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|), t>0 and J_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|^{1/H_1}), t>0 are derived by different methods and compared with those existing in the literature and with those related to B^1(|B^2_{H_2}(t)|), t>0. The process of iterated Brownian motion I^n_F(t), t>0 is examined in detail and its moments are calculated. Furthermore for J^{n-1}_F(t)=B^1_{H}(|B^2_H(...|B^n_H(t)|^{1/H}...)|^{1/H}), t>0 the following factorization is proved J^{n-1}_F(t)=\prod_{j=1}^{n} B^j_{\frac{H}{n}}(t), t>0. A series of compositions involving Cauchy processes and fractional Brownian motions are also studied and the corresponding non-homogeneous wave equations are derived.Comment: 32 page

Random flights governed by Klein-Gordon-type partial differential equations

In this paper we study random flights in R^d with displacements possessing Dirichlet distributions of two different types and uniformly oriented. The randomization of the number of displacements has the form of a generalized Poisson process whose parameters depend on the dimension d. We prove that the distributions of the point X(t) and Y(t), t \geq 0, performing the random flights (with the first and second form of Dirichlet intertimes) are related to Klein-Gordon-type partial differential equations. Our analysis is based on McBride theory of integer powers of hyper-Bessel operators. A special attention is devoted to the three-dimensional case where we are able to obtain the explicit form of the equations governing the law of X(t) and Y(t). In particular we show that that the distribution of Y(t) satisfies a telegraph-type equation with time-varying coefficients

Travelling Randomly on the Poincar\'e Half-Plane with a Pythagorean Compass

A random motion on the Poincar\'e half-plane is studied. A particle runs on the geodesic lines changing direction at Poisson-paced times. The hyperbolic distance is analyzed, also in the case where returns to the starting point are admitted. The main results concern the mean hyperbolic distance (and also the conditional mean distance) in all versions of the motion envisaged. Also an analogous motion on orthogonal circles of the sphere is examined and the evolution of the mean distance from the starting point is investigated

Hitting spheres on hyperbolic spaces

For a hyperbolic Brownian motion on the Poincar\'e half-plane $\mathbb{H}^2$, starting from a point of hyperbolic coordinates $z=(\eta, \alpha)$ inside a hyperbolic disc $U$ of radius $\bar{\eta}$, we obtain the probability of hitting the boundary $\partial U$ at the point $(\bar \eta,\bar \alpha)$. For $\bar{\eta} \to \infty$ we derive the asymptotic Cauchy hitting distribution on $\partial \mathbb{H}^2$ and for small values of $\eta$ and $\bar \eta$ we obtain the classical Euclidean Poisson kernel. The exit probabilities $\mathbb{P}_z\{T_{\eta_1}<T_{\eta_2}\}$ from a hyperbolic annulus in $\mathbb{H}^2$ of radii $\eta_1$ and $\eta_2$ are derived and the transient behaviour of hyperbolic Brownian motion is considered. Similar probabilities are calculated also for a Brownian motion on the surface of the three dimensional sphere. For the hyperbolic half-space $\mathbb{H}^n$ we obtain the Poisson kernel of a ball in terms of a series involving Gegenbauer polynomials and hypergeometric functions. For small domains in $\mathbb{H}^n$ we obtain the $n$-dimensional Euclidean Poisson kernel. The exit probabilities from an annulus are derived also in the $n$-dimensional case

Random flights related to the Euler-Poisson-Darboux equation

This paper is devoted to the analysis of random motions on the line and in the space R^d (d > 1) performed at finite velocity and governed by a non-homogeneous Poisson process with rate \lambda(t). The explicit distributions p(x,t) of the position of the randomly moving particles are obtained solving initial-value problems for the Euler- Poisson-Darboux equation when \lambda(t) = \alpha/t, t > 0. We consider also the case where \lambda(t) = \lambda coth \lambda t and \lambda(t) = \lambda tanh \lambda t, where some Riccati differential equations emerge and the explicit distributions are obtained for d = 1. We also examine planar random motions with random velocities by projecting random flights in R^d onto the plane. Finally the case of planar motions with four orthogonal directions is considered and the corresponding higher-order equations with time-varying coefficients obtained