A diffusion approximation for systems with frequent weak resetting

Abstract: We develop a diffusion approximation for systems subject to fast random resetting by small amplitudes. Equivalently, this describes systems with frequent but small catastrophes. We demonstrate the validity of the approximation by computing the stationary distribution and mean first-passage times of simple one-dimensional systems. The approximation captures dynamically induced correlations in multi-particle systems, and it can be used to generalise the conditionally independent and identically distributed structure recently found in systems with full resetting. Finally, we show that resetting can induce cycles and patterns, which can be characterised using the diffusion approximation.

• The paper introduces a novel diffusion approximation to capture the cumulative effects of frequent, weak resets by mapping them to extrinsic Gaussian noise.
• It leverages a Kramers–Moyal expansion to derive effective stochastic differential equations that closely match simulation results for stationary distributions and first-passage times.
• The framework reveals induced correlations, noise-driven cycles, and spatial patterns in both single and multi-particle systems, bridging theoretical insights with empirical observations.

Diffusion Approximation for Stochastic Systems with Frequent Weak Resetting

Introduction

The study develops a diffusion approximation framework for dynamical systems subject to frequent but weak stochastic resetting events, or equivalently, systems experiencing small, frequent catastrophes. Unlike conventional treatments which often consider resetting as rare or of significant magnitude, this approach approximates the cumulative effect of a high frequency of small resets by a continuous Gaussian noise, resulting in an effective stochastic differential equation (SDE). The methodology is applicable to both single and multi-particle systems, including individual-based population models, and captures dynamically induced phenomena such as correlations, quasi-cycles, and spatial patterns that are not predicted in deterministic or low-frequency resetting regimes.

Diffusion Approximation Formulation

Consider a system evolving according to

$$\dot{x}(t) = f(x(t)) + \xi(t), \qquad \langle \xi(t)\xi(t')\rangle = 2D\delta(t-t')$$

interrupted by resets occurring as $x \to a(x)x$, with $a(x) \in [0,1]$. Resetting is parametrized by $s\ll 1$, $a(x) = 1-sg(x)$, and frequency $r = \lambda/s$, so individual resets are weak but frequent. This is formalized by a Kramers–Moyal expansion of the underlying Master Equation, yielding the core Fokker–Planck equation (to leading order in $s$):

$$\frac{\partial}{\partial t}P(x,t) = D\frac{\partial^2}{\partial x^2}P(x,t) + \frac{1}{2}\lambda s \frac{\partial^2}{\partial x^2}\left[g(x)^2P(x,t)\right] - \frac{\partial}{\partial x}\{[f(x) - \lambda g(x)] P(x,t)\}$$

The corresponding SDE is:

$$\dot{x} = f(x) - \lambda g(x) + \xi + \sqrt{\lambda s}\,g(x)\,\eta(t)$$

where $\eta$ is an extrinsic Gaussian white noise capturing reset-induced fluctuations. Unlike intrinsic noise, this source expresses extrinsically imposed randomness, an aspect neglected by naïve deterministic corrections ($f(x) \mapsto f(x) - r(1-a)x$). This formalism is distinct from classic chemical Langevin frameworks, which only account for intrinsic fluctuations.

Figure 1: Realizations of the logistic equation with discrete proportional resets (a: $s=0.5$, b: $s=0.02$, showing emergence of effective SDE in (b)).

Validation and Stationary Distributions

The diffusion approximation is validated via comparison with explicit simulations and analytical results in the linear case ($f(x) = \alpha + \gamma x$, $g(x) = x$) and classic first-passage problems. For $s$ sufficiently small (e.g., $s=0.1$), stationary distributions of the effective SDE closely match those obtained from the underlying discrete-resetting system. The linear-noise approximation (LNA) is also seen to be accurate when replacing multiplicative by additive noise, but deterministic approximations ignoring reset randomness deviate substantially.

Figure 2: (a) Comparison of stationary distributions in the linear model: simulation, SDE, LNA, and deterministic approximations; (b) Mean first-passage times in a random walker with partial resetting, showing the existence of an optimal reset rate for both full and partial resetting models.

The framework captures nontrivial transport properties, such as mean first-passage times, including the phenomenon that optimal search rates persist for partial resets ($s<1$), although quantitative details differ from the $s=1$ (full reset) regime.

Multi-Particle Systems and Dynamically Induced Correlations

Extending to $N$-particle systems with simultaneous (partial) reset, the diffusion approximation introduces a common extrinsic noise component $\sqrt{\lambda s}g(x_i)\eta(t)$ acting collectively on all variables. While, conditional on a realization of the common noise, the particles are independent, averaging over the noise induces correlations even though particles are otherwise noninteracting.

Explicit calculation for linear dynamics reveals nonzero steady-state correlations:

$$\lim_{t\to\infty} \left[ \langle x_i^2(t)x_j^2(t)\rangle - \langle x_i^2(t)\rangle^2 \right] = \frac{8 D^2 \lambda s}{(2\mu-3\lambda s)(2\mu-\lambda s)^2}$$

demonstrating that the diffusion approximation faithfully characterizes dynamically induced collective effects previously identified experimentally and theoretically for full resetting ($a=0$) but now generalized to arbitrary partial reset amplitudes.

Figure 3: (a) Time series from the individual-based population model with catastrophes; (b) Quasi-stationary distribution comparison between simulations (agent-based, diffusion approximation), and linear-noise theory.

Individual-Based Population Dynamics and Quasi-Stationarity

In discrete population models with stochastic birth, death, and catastrophe events (random proportional removal of individuals), a joint expansion in $1/\Omega$ (population size) and $s$ leads to an effective SDE including both intrinsic and extrinsic noise. The quasi-stationary distribution—measured before certain extinction—matches closely between agent-based simulations and the diffusion approximation for sufficiently large populations and weak resets. Both intrinsic stochasticity (birth/death) and extrinsic reset noise appear as distinct additive noise terms in the effective equations.

Resetting-Induced Cycles and Patterns

A salient result is that, analogous to demographic noise, frequent weak resetting drives dynamical oscillations and pattern formation even in parameters where the deterministic system is fixed-point stable. In a stochastic Lotka–Volterra predator–prey model with resetting affecting only the prey, the diffusion approximation predicts quasi-cycles whose amplitude and spectrum can be computed analytically. Amplitude scales as $\sqrt{\lambda s}$, in sharp contrast with the deterministic behavior.

Figure 4: (a) Oscillatory (predator) time series generated by reset-induced noise; (b) Power spectra for different reset amplitudes, with theoretical spectra from the diffusion approximation.

On spatial domains, as exemplified by the Levin–Segel plankton–herbivore model, spatially heterogeneous patterns arise purely due to the extrinsic reset noise, resulting in robust patchiness even in systems stable to Turing-type deterministic patterning.

Figure 5: Reset-induced spatial pattern in plankton abundance (2D), with the inset showing the nontrivial spectrum of density fluctuations (1D model) evidencing structure at finite wavenumber $k$.

Implications and Future Directions

The diffusion approximation for extrinsic weak resetting noise provides a tractable approach to analyze previously analytically intractable systems across statistical physics, population dynamics, and spatial ecology. It enables:

• Accurate prediction of steady-state and transient properties in systems with reset noise.
• Quantification of dynamically induced collective phenomena and correlations in agent ensembles.
• Characterization of oscillatory and pattern-forming behavior in spatially extended biological and ecological models.

The separation of intrinsic and extrinsic noise sources makes explicit the role of environmental or exogenous perturbations beyond classical demographic stochasticity or molecular noise.

Experimentally, recent realizations of resetting dynamics in colloidal systems and other contexts allow for quantitative comparison of theoretical predictions from the diffusion approximation and motivate the design of further empirical tests, especially regarding patterns and cycles observed in large stochastic systems.

Further development could focus on systems with distributed or correlated reset amplitudes, higher-order reset–induced correlations (batch resets, heterogeneous ensembles), or extend to complex networks or non-Markovian reset mechanisms.

Conclusion

The diffusion approximation for frequent, weak resetting offers an analytically solvable and numerically tractable framework for studying the impact of extrinsic stochastic reset events on complex dynamical systems. It not only replicates known results from the full-resetting regime but generalizes them to arbitrary reset amplitudes and provides insight into emergent collective phenomena such as correlation induction, noise-driven cycles, and pattern formation. This positions the formalism as a versatile tool for the study of stochastic systems subjected to extrinsic catastrophe-like perturbations across scientific domains (2602.21635).