Effect of multiplicative noise on stationary stochastic process

An open system that can be analyzed using the Langevin equation with multiplicative noise is considered. The stationary state of the system results from a balance of deterministic damping and random pumping simulated as noise with controlled periodicity. The dependence of statistical moments of the variable that characterizes the system on parameters of the problem is studied. A nontrivial decrease in the mean value of the main variable with an increase in noise stochasticity is revealed. Applications of the results in several physical, chemical, biological, and technical problems of natural and humanitarian sciences are discussed.

Figure 1

Contour plot of the solution to Eq. (14) with respect to $\beta$ vs $a$ and $D$ ($\gamma =3/4, b=1$).

Figure 2

Plot of the solution to Eq. (14) with respect to $\beta$ vs $a$ for several values of $\gamma$ ($b=1, D=0.05$).

Figure 3

Plot of mean value $\langle x\rangle$ vs time for quasiperiodic DTDP noise ($\varrho =0.9$) (circles) and totally random pulse noise ($\varrho =0$) (triangles) calculated at $\gamma =3/4, \beta =1, b=1, f_0=0.1, T=0.01, x\left(0\right)=5$. The inset additionally shows analytical solutions (13) for $\varrho =0$ (dashed line) and 0.9 (solid line).

Figure 4

Plot of mean value $\langle x\rangle$ vs time for quasiperiodic DTDP noise ($\varrho =0.9$) (circles) and totally random pulse noise ($\varrho =0$) (triangles) calculated at $\gamma =3/4, \beta =7/4, b=1, f_0=0.1, T=0.01, x\left(0\right)=5$. The inset additionally shows analytical solution (13) (solid line).

Figure 5

Plot of mean value $\langle x\rangle$ vs time for quasiperiodic DTDP noise ($\varrho =0.9$) (circles) and totally random pulse noise ($\varrho =0$) (triangles) calculated at $\gamma =3/4, \beta =2, b=1, f_0=0.1, T=0.01, x\left(0\right)=5$. The inset additionally shows analytical solutions (13) for $\varrho =0$ (dashed line) and 0.9 (solid line).

Figure 6

Plot of the reduced mean value ${\langle x\rangle }^{*}$ vs $D$ for three values of $\beta$ at $\gamma =3/4, a=10$, and $b=1$: analytical solutions (13) (solid lines) and results calculated using expression (6) (dots).

Figure 7

Plot of the mean value $\langle x\rangle$ vs time for two values of $\varrho$: quasiperiodic FTI noise ($\varrho =0.98$) [green (light gray) line] and FTI noise with maximal dispersion ($\varrho =0$) (black line). The parameters and initial condition are $\gamma =3/4, \beta =1, b=1, f_0=0.1, T=0.01$, and $x\left(0\right)=5$.

Figure 8

Plot of the mean value $\langle x\rangle$ vs time for two values of $\varrho$: quasiperiodic FTI noise ($\varrho =0.98$) [green (light gray) line] and FTI noise with maximal dispersion ($\varrho =0$) (black line). The parameters and initial condition are $\gamma =3/4, b=1, f_0=0.1, T=0.01$, and $x\left(0\right)=5$.