Ornstein–Uhlenbeck–Cauchy process

Garbaczewski, Piotr; Olkiewicz, Robert

doi:10.1063/1.1290054

Cited by 59 publications

(75 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The angular variable ϕ is a deterministic functional (integral) of the stochastic variable θ, and in [43] it is pointed out that for fixed initial θ 0 Eq. (17) represents a Markov process.…”

Section: Ornstein-uhlenbeck Process For the Cauchy Distributionmentioning

confidence: 99%

Diffusion of active particles with stochastic torques modeled asα-stable noise

Nötel¹,

Sokolov²,

Schimansky‐Geier³

2016

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We investigate the stochastic dynamics of an active particle moving at a constant speed under the influence of a fluctuating torque. In our model the angular velocity is generated by a constant torque and random fluctuations described as a Lévy-stable noise. Two situations are investigated.First, we study white Lévy noise where the constant speed and the angular noise generate a persistent motion characterized by the persistence time τ D . At this time scale the crossover from ballistic to normal diffusive behavior is observed. The corresponding diffusion coefficient can be obtained analytically for the whole class of symmetric α-stable noises. As typical for models with noise-driven angular dynamics, the diffusion coefficient depends non-monotonously on the angular noise intensity. As second example, we study angular noise as described by an OrnsteinUhlenbeck process with correlation time τ c driven by the Cauchy white noise. We discuss the asymptotic diffusive properties of this model and obtain the same analytical expression for the diffusion coefficient as in the first case which is thus independent on τ c . Remarkably, for τ c > τ D the crossover from a non-Gaussian to a Gaussian distribution of displacements takes place at a time τ G which can be considerably larger than the persistence time τ D .

show abstract

Section: Ornstein-uhlenbeck Process For the Cauchy Distributionmentioning

confidence: 99%

Diffusion of active particles with stochastic torques modeled asα-stable noise

Nötel¹,

Sokolov²,

Schimansky‐Geier³

2016

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…Numerical methods [17,20] for such equations are more sophisticated than for differential equations [25] and for stochastic differential equations with Gaussian noises [26]. In particular, the nonexistence of variance for stable variables makes the problem much more complicated to tackle both, numerically [17,20] and analytically [27,28]. It turns out, however, that with the use of suitable statistical estimation techniques, computer simulation procedures and numerical discretization methods it is possible to construct relevant approximations of stochastic in-tegrals with stable measures as integrators [20,21].…”

Section: Introductionmentioning

confidence: 99%

Resonant activation in the presence of nonequilibrated baths

Dybiec

Gudowska–Nowak

2004

Phys. Rev. E

View full text Add to dashboard Cite

We study the generic problem of the escape of a classical particle over a fluctuating barrier under the influence of non-Gaussian noise mimicking the effects of not-fully equilibrated bath. Our attention focuses on the effect of the stable noises on the mean escape time and on the phenomenon of resonant activation (RA). Possible physical interpretation of the occurrence of Lévy noises in the system of interest is discussed and the connection between the Tsallis statistics and the Fractional Fokker Planck Equation is addressed.PACS numbers: 05.10.-a, 05.90+m, 02.50.-r, 82.20.-w

show abstract

“…(9) in Ref. [8] A characteristic function of this density reads −F (p) = −σ|p| and gives account of a non-thermal fluctuationdissipation balance. The modified noise intensity parameter σ is a ratio of an intensity parameter λ of the free Cauchy noise and of the friction coefficient γ.…”

Section: A Ornstein-uhlenbeck Processmentioning

confidence: 99%

“…[8,18,19]. The pseudo-differential Fokker-Planck equation, which corresponds to the fractional Hamiltonian (10) and the fractional semigroup exp(−tĤ µ ) = exp(−λ|∆| µ/2 ), reads…”

Section: A Lévy Drivermentioning

confidence: 99%

Cauchy flights in confining potentials

Garbaczewski¹

2010

Physica A: Statistical Mechanics and its Applications

Self Cite

View full text Add to dashboard Cite

We analyze confining mechanisms for Lévy flights evolving under an influence of external potentials. Given a stationary probability density function (pdf), we address the reverse engineering problem: design a jump-type stochastic process whose target pdf (eventually asymptotic) equals the preselected one. To this end, dynamically distinct jump-type processes can be employed. We demonstrate that one "targeted stochasticity" scenario involves Langevin systems with a symmetric stable noise. Another derives from the Lévy-Schrödinger semigroup dynamics (closely linked with topologically induced super-diffusions), which has no standard Langevin representation. For computational and visualization purposes, the Cauchy driver is employed to exemplify our considerations.

show abstract

Ornstein–Uhlenbeck–Cauchy process

Cited by 59 publications

References 26 publications

Diffusion of active particles with stochastic torques modeled asα-stable noise

Diffusion of active particles with stochastic torques modeled asα-stable noise

Resonant activation in the presence of nonequilibrated baths

Cauchy flights in confining potentials

Contact Info

Product

Resources

About