Stochastic representation of processes with resetting

In this paper we introduce a general stochastic representation for an important class of processes with resetting. It allows to describe any stochastic process intermittently terminated and restarted from a predefined random or nonrandom point. Our approach is based on stochastic differential equations called jump-diffusion models. It allows to analyze processes with resetting both, analytically and using Monte Carlo simulation methods. To depict the strength of our approach, we derive a number of fundamental properties of Brownian motion with Poissonian resetting, such as the Itô lemma, the moment-generating function, the characteristic function, the explicit form of the probability density function, moments of all orders, various forms of the Fokker-Planck equation, infinitesimal generator of the process, and its adjoint operator. Additionally, we extend the above results to the case of time-nonhomogeneous Poissonian resetting. This way we build a general framework for the analysis of any stochastic process with intermittent random resetting.

Figure 1

An exemplary plot of a resetting process. The process starts at $x_0=0$ and then diffuses normally until it moves instantaneously to $x_R=2$ in every consecutive resetting event.

Figure 2

A plot of the derived mean of the resetting process (top left), MGF (top right), the real part of the Fourier transform (bottom left), and the imaginary part of the Fourier transform (bottom right) at a different time points (0.1, 0.5, 1.5), as a blue, red, and yellow lines, respectively. The process starts at 0 and resets with rate 1 to point 5. We can also see the unusual behavior of the MGF at the point $\sqrt{2}$ due to the divergence of the function.

Figure 3

A histogram of $X_t$ obtained using Monte Carlo simulations compared with the derived analytical PDF of the process at $t=0.1$. We observe perfect agreement between both results. The process starts at 0 and resets to point 3 with rate 1.

Figure 4

A Monte Carlo simulation results (solid lines) compared with the theoretical asymptotics (dotted lines) of the MSD of Brownian motion under nonhomogeneous Poisson resetting with $r\left(t\right)=r·{(t+1)}^p$, calculated for different $p\le 0$. The parameter $\mu$ here is the corresponding power-law exponent of the theoretical asymptotic MSD. The time horizon of the simulations was of order ${10}^2$, while the number of samples was equal to ${10}^4$. We can observe the convergence of the anomalous diffusion exponents to the theoretical ones.

Figure 5

A Monte Carlo simulation (solid lines) of the MSD of the process with nonhomogeneous Poisson resetting with intensity function $r\left(t\right)=r·{(t+1)}^p$, calculated for different $p>0$. Asterisks are the corresponding theoretical values of the MSD obtained via numerical evaluation of (23). We can see the convergence of the MSD to 0, indicating the convergence to the degenerate distribution concentrated in $x_R=0$. The convergence rate increases with the increase of the $p$ parameter.