Fractional generalized Cauchy process

This paper presents a fractional generalized Cauchy process (FGCP) with an additive and a multiplicative Gaussian white noise for describing subordinated anomalous fluctuations. The FGCP displays intermittent dynamics on random time durations, whose analytical representation is given by the $\mathrm{It}\widehat{\mathrm{o}}$ stochastic integral. The associated probability density function is given by a generalized Cauchy distribution at the stationary state. A fractional Feynman-Kac formula is utilized to show that weak ergodicity breaking of the FGCP depends on the existence of the subordinator and/or the divergence of variance.

Figure 1

Realizations of the FGCP in the case of finite variance (left side, $\gamma =1.2$) and infinite variance (right side, $\gamma =0.7$). The noise intensities and fractional index are fixed as $D_a=1.0,D_m=0.5$ and $\alpha =0.7$. The top figures are the realizations of the FGCP on physical time $t$. The middle figures are the realizations of the generalized Cauchy process on intrinsic time $\tau$. The bottom figures are the random processes of the subordinator $S_t$ defined by inversion of a positive one-sided Lévy process.

Figure 2

The generalized Cauchy distributions derived from Eqs. (16) and (18) on stationary states. The solid line shows the case of finite variance with $\gamma =1.2, D_a=1.0$, and $D_m=0.5$. The dashed line shows the case of infinite variance with $\gamma =0.7, D_a=1.0$, and $D_m=0.5$, corresponding to the realizations in Fig. 1. It is shown that the tails of the latter case are broader than those of the former case.

Figure 3

Autocorrelation functions of the FGCP (solid line) and the generalized Cauchy process (dashed line) with the parameters $\gamma =0.7, D_m=0.5$, and $\alpha =0.7$. Relaxation of the FGCP is slower than that of the generalized Cauchy process because of the power-law tail.