147 research outputs found
Marcus versus Stratonovich for Systems with Jump Noise
The famous It\^o-Stratonovich dilemma arises when one examines a dynamical
system with a multiplicative white noise. In physics literature, this dilemma
is often resolved in favour of the Stratonovich prescription because of its two
characteristic properties valid for systems driven by Brownian motion: (i) it
allows physicists to treat stochastic integrals in the same way as conventional
integrals, and (ii) it appears naturally as a result of a small correlation
time limit procedure. On the other hand, the Marcus prescription [IEEE Trans.
Inform. Theory 24, 164 (1978); Stochastics 4, 223 (1981)] should be used to
retain (i) and (ii) for systems driven by a Poisson process, L\'evy flights or
more general jump processes. In present communication we present an in-depth
comparison of the It\^o, Stratonovich, and Marcus equations for systems with
multiplicative jump noise. By the examples of areal-valued linear system and a
complex oscillator with noisy frequency (the Kubo-Anderson oscillator) we
compare solutions obtained with the three prescriptions.Comment: 14 pages, 4 figure
Non-standard Skorokhod convergence of Levy-driven convolution integrals in Hilbert spaces
We study the convergence in probability in the non-standard Skorokhod
topology of the Hilbert valued stochastic convolution integrals of the type
to a process driven
by a L\'evy process . In Banach spaces we introduce strong, weak and product
modes of -convergence, prove a criterion for the -convergence in
probability of stochastically continuous c\`adl\`ag processes in terms of the
convergence in probability of the finite dimensional marginals and a good
behaviour of the corresponding oscillation functions, and establish criteria
for the convergence in probability of L\'evy driven stochastic convolutions.
The theory is applied to the infinitely dimensional integrated
Ornstein--Uhlenbeck processes with diagonalisable generators.Comment: 34 pages, 1 figur
Hyperbolic constant mean curvature one surfaces: Spinor representation and trinoids in hypergeometric functions
We present a global representation for surfaces in 3-dimensional hyperbolic
space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic
spinors. This is a modification of Bryant's representation.
It is used to derive explicit formulas in hypergeometric functions for CMC-1
surfaces of genus 0 with three regular ends which are asymptotic to catenoid
cousins (CMC-1 trinoids).Comment: 29 pages, 9 figures. v2: figures of cmc1-surfaces correcte
L\'evy Ratchet in a Weak Noise Limit: Theory and Simulation
We study the motion of a particle embedded in a time independent periodic
potential with broken mirror symmetry and subjected to a L\'evy noise
possessing L\'evy stable probability law (L\'evy ratchet). We develop
analytical approach to the problem based on the asymptotic probabilistic method
of decomposition proposed by P. Imkeller and I. Pavlyukevich [J. Phys. A
{\bf39}, L237 (2006); Stoch. Proc. Appl. {\bf116}, 611 (2006)]. We derive
analytical expressions for the quantities characterizing the particle motion,
namely the splitting probabilities of first escape from a single well, the
transition probabilities and the particle current. A particular attention is
devoted to the interplay between the asymmetry of the ratchet potential and the
asymmetry (skewness) of the L\'evy noise. Intensive numerical simulations
demonstrate a good agreement with the analytical predictions for sufficiently
small intensities of the L\'evy noise driving the particle.Comment: 14 pages, 11 figures, 63 reference
Cooling down Levy flights
Let L(t) be a Levy flights process with a stability index \alpha\in(0,2), and
U be an external multi-well potential. A jump-diffusion Z satisfying a
stochastic differential equation dZ(t)=-U'(Z(t-))dt+\sigma(t)dL(t) describes an
evolution of a Levy particle of an `instant temperature' \sigma(t) in an
external force field. The temperature is supposed to decrease polynomially
fast, i.e. \sigma(t)\approx t^{-\theta} for some \theta>0. We discover two
different cooling regimes. If \theta<1/\alpha (slow cooling), the jump
diffusion Z(t) has a non-trivial limiting distribution as t\to \infty, which is
concentrated at the potential's local minima. If \theta>1/\alpha (fast cooling)
the Levy particle gets trapped in one of the potential wells
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