176 research outputs found
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
Random motions in inhomogeneous media
Space inhomogeneous random motions of particles on the line and in the plane are considered in the paper. The changes of the movement direction are driven by a Poisson process. The particles are assumed to move according to a finite velocity field that depends on a spatial argument. The explicit distribution of particles is obtained in the paper for the case of dimension 1 in terms of characteristics of the governing equations. In the case of dimension 2, the distribution is obtained if a rectifying diffeomorphism exists. © 2008 American Mathematical Society
Random motions at finite velocity in a non-Euclidean space
In this paper telegraph processes on geodesic lines of the Poincaré half-space and Poincaré disk are introduced and the behavior of their hyperbolic distances examined. Explicit distributions of the processes are obtained and the related governing equations derived. By means of the processes on geodesic lines, planar random motions (with independent components) in the Poincaré half-space and disk are defined and their hyperbolic random distances studied. The limiting case of one-dimensional and planar motions together with their hyperbolic distances is discussed with the aim of establishing connections with the well-known stochastic representations of hyperbolic Brownian motion. Extensions of motions with finite velocity to the three-dimensional space are also hinted at, in the final section
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