Physica A: Statistical Mechanics and its Applications
Transitions induced by bounded noise
Introduction
Traditionally, stochastic dynamical systems used in the physical sciences have involved Gaussian noise. In recent times, however, it has been recognized that the assumption of Gaussianity is not appropriate in some cases (see Ref. [1] for a review). The Gaussian noise is unbounded, i.e., there exists a positive chance of having very large values. Strictly speaking, this fact contradicts the very nature of a real physical quantity which is always bounded [2]. Studies of dynamical systems with non-Gaussian continuous noise are much more complicated, especially analytically. Only recently, some results in this area have been obtained, see Refs. [1], [3], [4], [5], [6].
In this paper we consider one of the simplest nonlinear stochastic flows, viz., the so-called genetic model:which appears in population genetics or in a chemical reaction modeling [7], [8]. Two forms of the multiplicative noise are considered. The first one is the Gaussian process (Ornstein–Uhlenbeck (OU) noise) with zero mean and the correlationwhere is the correlation time and D denotes the noise intensity. The second one is the process , with standing for the standard Wiener process. It is important to note that the noise is non-Gaussian because absolute values of all its trajectories are bounded by . Let us call it the sine-Wiener (SW) noise. Using the well-known properties of the Wiener process and the Euler representation of the sine function one can easily show that Therefore the noises and have the same mean and stationary correlation. Moreover, their odd moments are zero. Our aim is to show that despite the above facts the probability density functions (PDFs) for Eq. (1) with OU and SW noises can differ essentially: the stationary PDF in the SW noise case can be bimodal even if the PDF in the OU noise case is unimodal. The transitions between the unimodal and the bimodal stationary PDF are referred to as the noise-induced transitions [7], [9]. In this paper, we obtain the PDFs for Eq. (1) by a new method based on approximations of these functions via solutions of hyperbolic systems. We believe that this approach represents a new technique for studying colored-noise-driven dynamical systems.
Recently, another type of non-Gaussian noise has been used in the study of transitions for genetic model [6]. This interesting noise is a particular case of Markov diffusion process (see Refs. [1], [3], [6], [10] for details). Its stationary PDF is similar to the generalized thermostatistics proposed by Tsallis [11]. Although that noise is phenomenologically richer than the SW noise, its influence on dynamical systems is more difficult to analyze.
Section snippets
Hyperbolic systems: OU noise
Extending the non-Markovian solutions of Eq. (1) to the pairs and we obtain Markov processes whose joint PDFs satisfy Fokker–Planck equations [12]. For example, the time evolution of the joint PDF of the process is given by the two-dimensional Fokker–Planck equationwith the condition Our interest lies in the system dynamics of alone but Eq. (3) contains
Hyperbolic systems: SW noise
Now let us consider Eq. (5) with SW noise. Of course is a functional of the Wiener process , and therefore we shall write , . To disentangle the correlation we use the following relation:which follows from the Cameron–Martin formula for the density of the Wiener measure under translation [19], [20]. In what follows, equality (8) will play a role similar to Eq. (6) in the OU noise case. Since
Probability density functions
Hierarchies (7) and (9) cannot be solved analytically in general case. Nevertheless they can be useful in asymptotic analysis. For example, one can obtain asymptotic expansions for the PDF in powers of the correlation time using the approach from Ref. [17]. These expansions have both the regular and the boundary layer parts. If the correlation time is not small we analyze the hyperbolic systems which are obtained after the truncation of the hierarchies. No analytical solutions are known in
Acknowledgements
The authors thank O. Hryniv for a revision of the manuscript and ESF for support.
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