Average trajectory of returning walks

We compute the average shape of trajectories of some one-dimensional stochastic processes $x\left(t\right)$ in the $(t,x)$ plane during an excursion, i.e., between two successive returns to a reference value, finding that it obeys a scaling form. For uncorrelated random walks the average shape is semicircular, independent from the single increments distribution, as long as it is symmetric. Such universality extends to biased random walks and Levy flights, with the exception of a particular class of biased Levy flights. Adding a linear damping term destroys scaling and leads asymptotically to flat excursions. The introduction of short and long ranged noise correlations induces nontrivial asymmetric shapes, which are studied numerically.

Figure 1

Schematic representation of the average shape of a fluctuation. Thin solid lines are two realizations of the stochastic process $x\left(t\right)$, both returning for the first time at zero at time $T$. The thick solid line is the average shape computed over many realizations.

Figure 2

The factor $N\left(T\right)$ vs $T$ for Levy flights with several values of $\mu$. Notice that the exponent is $1∕2$ for $\mu ⩾2$.

Figure 3

Scaling function $f$ for for Levy flights with several values of $\mu$.

Figure 4

Comparison between $c(x,t)$ evaluated numerically for for Levy flights with steps distributed according to Eq. (21) and the ansatz (22).

Figure 5

Bias-dominated regime for Levy-distributed increments $\left(\mu =1.5\right)$ with bias $v=20$. Main: Normalized scaling function $f$ converging toward the asymptotic form $f\left(s\right)=2s$. Upper inset: The factor $N\left(T\right)$ growing as $T^{\alpha }$ with $\alpha =1$. Lower inset: First return distribution $P\left(T\right)$ decaying as $T^{-\mu -1}$.

Figure 6

Main: Scaling function $f$ for biased Levy flights with $\mu =1.5$ and $v=1$. Inset: Temporal evolution of the asymmetry parameter [Eq. (24)].

Figure 7

Main: Scaling function $f$ for biased Levy flights with $\mu =0.5$ and $v=25$. Inset: Temporal evolution of the asymmetry parameter [Eq. (24)].

Figure 8

Sketch of the average shape for positively biased flights. The crossover time $T^{*}$, depending on the value of the bias, changes the asymptotic behavior only for $1<\mu <2$. For $\mu <1$ the average trajectory is asymmetric in the intermediate regime $T⪡T^{*}$, and it takes a triangular shape in the limit $T\to \infty$ after $T^{*}\to \infty$.

Figure 9

Average excursion for a damped random walk [Eq. (38)] for $1∕\lambda =20$. From top to bottom lines are for $T=10,25,50,100,250$. Notice that the shape flattens to the constant value $\sqrt{4∕\left(\pi \lambda \right)}$ (thin line).

Figure 10

Main: Average excursion for a RAP with zero initial velocity, with uniform or exponential distribution of the noise $\eta$, showing perfect agreement with the simple form (42). Inset: Factor $N\left(T\right)$ and first-return time distribution $P\left(T\right)$ for uniform noise distribution.

Figure 11

Main: Scaling function $f$ for a RAP with finite initial velocity, with uniform distribution of the noise $\eta$.

Figure 12

Main: Scaling function $f$ for a RAP with noise $\eta$ distributed with slow decaying tails $P\left(\mid \eta \mid \right)$ and several values of $\mu$, compared with the simple form (42). Inset: Factor $N\left(T\right)$ and first-return time distribution $P\left(T\right)$ for the same values of $\mu$.

Figure 13

Left top: Small $t∕T$ tail of the scaling function $f$ for RAP with Cauchy distribution of single steps. Left bottom: Local effective exponent computed on the figure above. Right top: Small $1-t∕T$ tail of the scaling function $f$ for RAP with Cauchy distribution of single steps. Right bottom: Local effective exponent computed on the figure above.

Figure 14

Normalized average shape for a short-memory process with zero initial velocity, finite variance noise and $\tau =1000$. The dashed lines are the expected limiting curves for $T⪡\tau$ and $T⪢\tau$.

Figure 15

Plot of the average shape and standard deviation for Levy flight with $\mu =1.5$. Curves for different durations are divided by the scaling factor $T^{1∕\mu }$. The upper curves are the average shapes, the lower ones are the standard deviations.

Figure 16

Plot of the average shape and standard deviation for RAP with Gaussian noise. Curves for different durations are divided by the scaling factor $T^{3∕2}$. The upper curves are the average shapes, the lower ones are the standard deviations.