Space-time-modulated stochastic processes

Starting from the physical problem associated with the Lorentzian transformation of a Poisson-Kac process in inertial frames, the concept of space-time-modulated stochastic processes is introduced for processes possessing finite propagation velocity. This class of stochastic processes provides a two-way coupling between the stochastic perturbation acting on a physical observable and the evolution of the physical observable itself, which in turn influences the statistical properties of the stochastic perturbation during its evolution. The definition of space-time-modulated processes requires the introduction of two functions: a nonlinear amplitude modulation, controlling the intensity of the stochastic perturbation, and a time-horizon function, which modulates its statistical properties, providing irreducible feedback between the stochastic perturbation and the physical observable influenced by it. The latter property is the peculiar fingerprint of this class of models that makes them suitable for extension to generic curved-space times. Considering Poisson-Kac processes as prototypical examples of stochastic processes possessing finite propagation velocity, the balance equations for the probability density functions associated with their space-time modulations are derived. Several examples highlighting the peculiarities of space-time-modulated processes are thoroughly analyzed.

Figure 1

Graph of the trajectory of a realization of a STM Poisson-Kac process (at $a=1$ and $D=1$) for different choices of the time horizon function $f(t;x)$: (a) $f(t;x)=t$, i.e., the usual Poisson-Kac process, (b) $f(t;x)=\sqrt{t}$, and (c) $f(t;x)$ expressed by Eq. (58).

Figure 2

Transition time densities $p(\theta ,n)$ vs $\theta$ at iteration $n$ for the STM process characterized by the time-horizon function (59) at $a=1$ and $D=1$: (a) $\alpha =1/2$ and (b) $\alpha =2$. The arrows indicate increasing values of $n=1,2,5,10$.

Figure 3

(a) First-order moment $\langle {\theta }_n\rangle$ vs $n$ of the transition time density at iteration $n$ for the STM process defined by the time-horizon function (59) at $a=1$ and $D=1$ for different values of $\alpha$. (b) Absolute value $|C_{\theta ,N}\left(n\right)|$ vs $n$ of the normalized transition-time correlation function. The arrows indicate increasing values of (a) $\alpha =0.5,0.8,1,2,10$ and (b) $\alpha =0.5,0.8,2,10$, since $C_{\theta ,N}\left(n\right)=0$ at $\alpha =1$. Line a in (b) represents the scaling $|C_{\theta ,N}\left(n\right)|\sim n^{-1}$.

Figure 4

Scaling exponent $\mu$ of the first-order moment $\langle {\theta }_n\rangle$ vs $n$ as a function of $\alpha$ at $a=1$ and $D=1$. Circles represent the results of stochastic simulations and the solid line depicts the graph of the function (66).

Figure 5

Transition time correlation function $C_{\theta }\left(n\right)$ vs $n$ for several values of the exponent $\alpha$, defining the time-horizon function (59), at $a=1$ and $D=1$. From line a to line c, $\alpha =0.6,2/3,0.7$. Lines d–f correspond to the scaling (67) with the exponent $\beta$ given by Eq. (68).

Figure 6

Overall probability density function $P(x,t)$ vs $x$ for the STM process defined by the time-horizon function (69), at $b=1$ and $D=1$, with $\kappa =0.8/2\pi$ for (a) $t=2$ and (b) $t=10$. Solid lines represent the results of the solution of the partial wave equations (44), with $v\left(x\right)=0$ and $Z\left(\xi \right)=\xi$, and circles the results of stochastic simulations.

Figure 7

Plot of the STM process defined by Eq. (69) with $\kappa =2$ at $b=1$ and $D=1$ for (a) $t=0.3$, (b) $t=0.5$, (c) $t=1$, and (d) $t=1.5$. Shown is a comparison of the overall probability density function $P(x,t)$ vs $x$ obtained from the solution of Eqs. (27) (solid lines) and from the stochastic simulations of the process (circles).

Figure 8

Spatial transition probability density function $p_t^{\pm }\left(x\right)$ (lines a) for the STM process defined by Eq. (70) at $a=1$ and $D=1$: (a) $p_t^{+}\left(x\right)$ and ${\eta }_{+}\left(x\right)$ and (b) $p_t^{-}\left(x\right)$ and ${\eta }_{-}\left(x\right)$. Lines a represent the functions ${\eta }_{\pm }\left(x\right)$ multiplied by a factor 3 for graphical convenience.

Figure 9

Overall probability density function $P(x,t)$ vs $x$ for the STM process defined by Eq. (70) at $a=1$ and $D=1$ for (a) $t=4$ and (b) $t=10$.

Figure 10

Mean square displacement $R^2\left(t\right)$ vs $t$ for the STM process defined by the space-dependent time-horizon function (74), at $a=1$ and $D=1$, for different values of the exponent $\alpha =0.5,1,1.5,2,3$ ($\alpha$ increases in the direction of the arrow).

Figure 11

Scaling exponent $\delta$ of the mean square displacement vs the exponent $\alpha$ characterizing the time-horizon function. Circles are the results deriving from the stochastic simulations depicted in Fig. 10 and the solid line represents Eq. (75).

Figure 12

(a) Overall probability density functions $P(x,t)$ vs $x$ for the STM process defined by (74) with $\alpha =1, a=1$, and $D=1$. The arrow indicates increasing values of time $t=100,200,500,1000$. (b) Invariant rescaling $t^{1/2}P(x,t)$ vs $x{t}^{-1/2}$ of the data depicted in (a) $t=200$ ($\square$), $t=500$ ($\circ$), and $t=1000$ ($•$).