Local time of Lévy random walks: A path integral approach

The local time of a stochastic process quantifies the amount of time that sample trajectories $x\left(\tau \right)$ spend in the vicinity of an arbitrary point $x$. For a generic Hamiltonian, we employ the phase-space path-integral representation of random walk transition probabilities in order to quantify the properties of the local time. For time-independent systems, the resolvent of the Hamiltonian operator proves to be a central tool for this purpose. In particular, we focus on the local times of Lévy random walks (Lévy flights), which correspond to fractional diffusion equations.

Figure 1

(a) Sample paths $x\left(\tau \right)$ of a Lévy random walk for $\lambda =2$ (Brownian motion), $\lambda =1.5$, and $\lambda =1$ (Cauchy walk). (b) Local-time profiles $L\left(x\right)$ corresponding to the sample paths $x\left(\tau \right)$.

Figure 2

The Lévy stable distribution $P_{\lambda }(x,t)$ for $\lambda =2$ (Gaussian), $\lambda =1.5$, and $\lambda =1$ (Cauchy). The dimensionless quantity ${\overline{P}}_{\lambda }\left(\overline{x}\right)={\left(D_{\lambda }t\right)}^{1/\lambda }{P}_{\lambda }(x,t)$ is plotted, where $\overline{x}=x/{\left(D_{\lambda }t\right)}^{1/\lambda }$.

Figure 3

The resolvent $R_{\lambda }(x,-E)$ for $\lambda =1,1.5,2$. The dimensionless quantity ${\overline{R}}_{\lambda }\left(\overline{x}\right)={(D_{\lambda }/E)}^{1/\lambda }E{R}_{\lambda }(x,-E)$ is plotted, where $\overline{x}=x/{(D_{\lambda }/E)}^{1/\lambda }$.

Figure 4

One-point distribution of local time at point $x=x_a$ for fixed endpoint $x_b=x_a$. The dimensionless quantity ${\overline{W}}_{\lambda }\left(\overline{L}\right)=t{W}_{\lambda }(L;x_a)/{\left(D_{\lambda }t\right)}^{1/\lambda }$ is plotted, where $\overline{L}={\left(D_{\lambda }t\right)}^{1/\lambda }L/t$.

Figure 5

One-point distribution of local time at point $x=x_a$ for paths with arbitrary $x_b$. The dimensionless quantity $\overline{W_{\lambda }^{*}}\left(\overline{L}\right)=t{W}_{\lambda }^{*}(L;x_a)/{\left(D_{\lambda }t\right)}^{1/\lambda }$ is plotted, where $\overline{L}={\left(D_{\lambda }t\right)}^{1/\lambda }L/t$.

Figure 6

The mean of the local time at point $x$ for the Lévy Hamiltonian $H_{\lambda }$ and fixed endpoint $x_b=x_a$. The dimensionless quantity $\overline{{\mu }_{\lambda }}\left(\overline{x}\right)={\left(D_{\lambda }t\right)}^{1/\lambda }{\mu }_{\lambda }\left(x\right)/t$ is plotted, where $\overline{x}=(x-x_a)/{\left(D_{\lambda }t\right)}^{1/\lambda }$.

Figure 7

The mean of the local time at point $x$ for the Lévy Hamiltonian $H_{\lambda }$ and “free” endpoint $x_b$. The dimensionless quantity $\overline{{\mu }_{\lambda }^{*}}\left(\overline{x}\right)={\left(D_{\lambda }t\right)}^{1/\lambda }{\mu }_{\lambda }^{*}\left(x\right)/t$ is plotted, where $\overline{x}=(x-x_a)/{\left(D_{\lambda }t\right)}^{1/\lambda }$.