Heterogeneous diffusion with stochastic resetting

Abstract

We study a heterogeneous diffusion process with position-dependent diffusion coefficient and Poissonian stochastic resetting. We find exact results for the mean squared displacement and the probability density function. The nonequilibrium steady state reached in the long time limit is studied. We also analyze the transition to the non-equilibrium steady state by finding the large deviation function. We found that similarly to the case of the normal diffusion process where the diffusion length grows like $t^{1⁄2}$ while the length scale ξ(t) of the inner core region of the nonequilibrium steady state grows linearly with time t, in the heterogeneous diffusion process with diffusion length increasing like $t^{p⁄2}$ the length scale ξ(t) grows like $t^{p}$. The obtained results are verified by numerical solutions of the corresponding Langevin equation.

Stochastic resetting of a population of random walks with resetting-rate-dependent diffusivity

“…The response to the external field is thus linear to leading order, with a regular (i.e., cubic) subleading correction that we do not evaluate explicitly -this would require to compute the cubic response in Eq. (12). Note that the h-dependence of D(h) does not modify here the linear response with respect to the case of a constant diffusion coefficient D 0 ; it only contributes to non-linear corrections at order h 3 and higher.…”

“…At odds with standard random walks, a random walk with stochastic resetting to the origin converges to a stationary statistical state even in an unbounded domain [7,8]. This minimal model has been extended in many different directions, including arbitrary spatial dimensions [9], bounded domains [10], Langevin dynamics [11], spacedependent diffusivity [12], time-dependent resetting rate [13] or non-Poissonian resetting dynamics [14]. Anomalous diffusion with stochastic resetting has also been considered [15,16], notably in the context of record statistics [17].…”

Efficiency of random search with space-dependent diffusivity

“…State-dependent diffusivity has been considered to describe particles moving between nearly parallel plates [3], biologically motivated problems [4][5][6][7], and stock markets [8], among many others. Recently, heterogeneous diffusion processes (HDP) have been also investigated within the stochastic resetting scenario [9]. In one-dimension, a single trajectory x(t) can be modeled by the following stochastic process ẋ = 2D(x) η(t), (1) where x is the spatial coordinate (or other state variable, such as chemical coordinate or stock prize), D(x) > 0 is the diffusion coefficient, η(t) is a zero-mean white noise with delta-correlation η(t + t )η(t) = δ(t ).…”

“…In Sec. II, we use the backward Fokker-Planck equation, with arbitrary A, to obtain the first passage time distribution and the search efficiency when the position-dependent diffusivity has a power-law form, which has been used in different frameworks [1,9,48]. In Sec.…”

Expediting Feller process with stochastic resetting

“…Tuning such parameters, it is (in principle) possible to invert such effect of resetting on the dynamics [28][29][30][31][32][33][34][35]. Due to its appearance in a plethora of natural systems and drastic effect on the dynamics, study of first-passage problems with resetting has gained overwhelming attention in recent years [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. Surprisingly, the effect of resetting on Feller process still remains scarcely explored.…”