Nonlinear stochastic equations with multiplicative Lévy noise

The Langevin equation with a multiplicative Lévy white noise is solved. The noise amplitude and the drift coefficient have a power-law form. A validity of ordinary rules of the calculus for the Stratonovich interpretation is discussed. The solution has the algebraic asymptotic form and the variance may assume a finite value for the case of the Stratonovich interpretation. The problem of escaping from a potential well is analyzed numerically; predictions of different interpretations of the stochastic integral are compared.

Figure 1

(Color online) Probability distribution, calculated from Eq. (7), for the case $G\left(x\right)=x$ and $F\left(x\right)=0$ with $\alpha =1.5$ at $t=1$, compared with the distribution calculated according to Eq. (8) with a truncation at a large value of the noise, such that $a_i>0$ (upper curves for $x>0$: solid green line and dashed red line, respectively). Those distributions are identical. The distribution which follows from Eq. (8), but without any truncation, is marked by the dotted black line. The result of the Itô interpretation is also shown (dashed blue line).

Figure 2

(Color online) Stationary probability distributions, for the system given by Eq. (12), calculated by applying the transformation [Eq. (13)] (lines) and by using Eq. (20) (points) for $\alpha =1.8$, $\gamma =2.5$ and two values of $\theta$: $-0.2$ (the case with a maximum in the origin) and 0.5.

Figure 3

(Color online) Time evolution of the probability distribution in the case Eq. (12) for $\alpha =1.5$, $\theta =1$, and $\gamma =2.5$. The distribution was evaluated at the following times: 0.001, 0.01, 0.1, 0.5, and 1, which cases correspond to the rising width. The steady state is reached at $t=1$.

Figure 4

(Color online) Stationary distributions, calculated numerically, for the system given by Eq. (12) (points). The following cases are presented: (1) $\gamma =1.6$, $\alpha =1.5$, and $\theta =-0.5$; (2) $\gamma =2.5$, $\alpha =0.5$, and $\theta =1$; (3) $\gamma =2.5$, $\alpha =1.5$, and $\theta =-0.5$; (4) $\gamma =2.5$, $\alpha =1.5$, and $\theta =1$; (5) $\gamma =3.5$, $\alpha =1.5$, and $\theta =1$; (from top to bottom at the right hand side). Slopes of the straight lines follow from Eq. (19).

Figure 5

(Color online) The effective potential in the Stratonovich interpretation, calculated from Eq. (24), for $A=1$, $B=0.1$ and the following values of $\theta /\alpha$: 4/3, 2/3, 0, $-1/3$ (from top to bottom).

Figure 6

(Color online) Mean first passage time as a function of $\theta$ for $\alpha =1.5$ and 1.2, calculated for the potential [Eq. (22)] with $A=1$ and $B=0.1$. Results for both interpretations of the stochastic integral are presented.

Figure 7

(Color online) Mean first passage time in both interpretations of the stochastic integral, as a function of $\alpha$, for some values of $\theta$.