Limiting distributions of continuous-time random walks with superheavy-tailed waiting times

We study the long-time behavior of the scaled walker (particle) position associated with decoupled continuous-time random walks which is characterized by superheavy-tailed distribution of waiting times and asymmetric heavy-tailed distribution of jump lengths. Both the scaling function and the corresponding limiting probability density are determined for all admissible values of tail indexes describing the jump distribution. To analytically investigate the limiting density function, we derive a number of different representations of this function and, in this way, establish its main properties. We also develop an efficient numerical method for computing the limiting probability density and compare our analytical and numerical results.

Figure 1

The limiting probability density at ${\alpha }_{+}=\alpha =1/2$ and ${\alpha }_{-}>{\alpha }_{+}$. The solid blue (dark gray) line represents the theoretical result (4.24), and the simulation results (see Sec. 6) are indicated by red (light gray) triangles.

Figure 2

The limiting probability density at ${\alpha }_{+}=\alpha =5/4$ and ${\alpha }_{-}>{\alpha }_{+}$. The solid line with an exponential left branch is obtained from Eq. (4.27), and triangles show the simulation results.

Figure 3

The limiting probability density at ${\alpha }_{+}={\alpha }_{-}=\alpha =1/2$ and $\epsilon =1/3$. The solid line and triangles represent the limiting probability density (4.30) and simulation data, respectively.

Figure 4

The limiting probability density at ${\alpha }_{+}={\alpha }_{-}=\alpha =5/4$ and $\epsilon =(\sqrt{2}-1)/(\sqrt{2}+1)$. As in the previous case, the solid line is obtained from Eq. (4.30), and triangles indicate the simulation results.