Cauchy noise and affiliated stochastic processes

Abstract

By departing from the previous attempt (Phys. Rev. E 51, 4114, (1995)) we give a detailed construction of conditional and perturbed Markov processes, under the assumption that the Cauchy law of probability replaces the Gaussian law (appropriate for the Wiener process) as the model of primordial noise. All considered processes are regarded as probabilistic solutions of the so-called Schrödinger interpolation problem, whose validity is thus extended to the jump-type processes and their step process approximants.

Lévy flights in confining environments: Random paths and their statistics

“…3. This convergence is much slower than in the Gaussian (harmonic confinement) case which is not a surprise: (8) and (11)) indicate that the present rate of convergence should be logarithmically slower. Fig.…”

“…4 the oscillations are more noticeable and persist even for 200000 trajectories and more. This is related to much slower decay of transition rates (11) (determined by slowdecaying asymptotic pdf (10)) as compared to those for Gaussian case (8).…”

“…According to the step (iii) of the simulation algorithm, we must evaluate integrals with transition rates (8) in the intervals [−ε 2 , −ε 1 ] and [ε 1 , ε 2 ]. To execute the step (iv) of the algorithm, we employ the Mersenne-Twister random number generator, [23].…”

“…An upper bound is less innocent and its effects need to be controlled by long tailed pdfs which stands for a distinctive feature of Lévy flights, see a discussion of Lévy stable limits of step processes in Ref. [8]. There is also pertinent discussion of a long time behavior of (unconfined, e.g.…”

“…[5]- [7]. It has next been expanded in various directions, with a special emphasis on so-called Lévy-Schrödinger semigroup reformulation of the original probability density function (pdf) dynamics, [6,15,16] and [8]- [13], c.f. also [14][15][16].…”