Arnab Pal

We study a generalized geometric Brownian motion framework that incorporates both entries of new units and exit mechanisms for the current population, extending earlier stochastic resetting models where these rates are treated as identical. The model captures realistic features observed in many economic observables, which can be explained as market-driven firm entries/exits, worker inflow/outflow, and income growth/loss. This model is not conservative and, despite the asymmetry in the entry and exit rates, we find that the system eventually relaxes to a stationary distribution. Moreover, our analysis reveals three distinct dynamical regimes in the moments of the distribution, arising from the interplay between volatility, drift, entry, and exit rates. We further derive the survival probability and the mean first-passage time associated with the observed variable reaching certain threshold under the competing entry-exit processes. Interestingly, we identify an optimal exit rate that minimizes the mean first-passage time, providing insights into how entry and exit policies can influence the outcome of the system. These results should be useful for understanding the long-run behavior of economic systems in which growth, volatility, entry, and exit jointly shape the evolution of heterogeneous units.

We investigate the role of stochastic resetting in non-Markovian systems, where memory effects arise due to slow relaxation, rugged energy landscapes, disordered environments, and molecular crowding. Using the celebrated continuous-time random walk framework, we analyze first-passage processes with multiple competing outcomes and examine how resetting can selectively enhance desired outcomes. We characterize the efficiency of resetting through conditional mean first-passage times and demonstrate that its impact is highly sensitive to the underlying waiting-time statistics. Furthermore, we derive an inequality that quantifies how resetting controls fluctuations in conditional first-passage times, revealing regimes where variability is significantly suppressed. Our results provide a systematic understanding of how long-term memory influences competitive first-passage outcomes and establish resetting as a powerful control mechanism beyond the conventional Markovian setting.

Stochastic resetting has attracted significant attention in recent years due to its wide-ranging applications across physics, biology, and search processes. In most existing studies, however, resetting events are governed by an external timer and remain decoupled from the system’s intrinsic dynamics. In a recent Letter Biswas et al (2025 Phys. Rev. Lett. 135 227101), we introduced threshold resetting (TR) as an alternative, event-driven optimization strategy for target search problems. Under TR, the entire process is reset whenever any searcher reaches a prescribed threshold, thereby coupling the resetting mechanism directly to the internal dynamics. In this work, we study TR-enabled search by N non-interacting diffusive searchers in a one-dimensional box [ 0 , L ] , with the target at the origin and the threshold at L . By optimally tuning the scaled threshold distance u = x 0 / L , the mean first-passage (FP) time can be significantly reduced for N ⩾ 2 . We identify a critical population size N c ( u ) below which TR outperforms reset-free dynamics. Furthermore, we show that, for fixed u , the mean FP time depends non-monotonically on N , attaining a minimum at N opt ( u ) . We further quantify the achievable speed-up and analyse the operational cost of TR, revealing a nontrivial optimization landscape. These findings highlight TR as an efficient and realistic optimization mechanism for complex stochastic search processes.

We investigate the target search process by proteins locating specific target sites along DNA, a phenomenon fundamental to biological functions such as gene regulation, transcription, replication, recombination, and gene-editing technologies. This process proceeds through a repetitive sequence of stochastic motions: consisting of one-dimensional (1D) sliding along the DNA contour interspersed with detachment and three-dimensional (3D) excursions in the bulk, and then reattachment to a random location on DNA. Recognizing this sequence of random events as analogous to the resetting processes widely studied in statistical physics, we employ a first-passage-renewal framework and derive general expressions for both the mean and fluctuations of the total search time. Our results are completely generic and do not depend on the detailed microscopic dynamics of either the 1D or 3D phases. Quite interestingly, we find that intermittent detachment can not only accelerate the mean search but can also regulate fluctuations around it. Our analysis reveals a universal fluctuation inequality that links the variability and mean of the sliding time to the mean excursion time, thereby identifying the fundamental conditions under which target search process becomes efficient. Notably, we find that broad distributions of sliding times emerge as a universal characteristic for optimal search efficiency: a feature emanating from the slow dynamics along the DNA. Using the facilitated diffusion mechanism as a representative example, we validate the generality of our results. These findings provide a unified theoretical framework connecting stochastic search, resetting dynamics, and biological efficiency, while also highlighting the crucial role of DNA structure such as its contour length in modulating search performance.

We study the diffusion process in the presence of stochastic resetting inside a two-dimensional wedge of top angle α, bounded by two infinite absorbing edges. In the absence of resetting, the second moment of the first-passage time diverges for α>π/4, while it remains finite for α<π/4, resulting in an unbounded or bounded coefficient of variation in the respective angular regimes. Upon introducing stochastic resetting, we analyze the first-passage properties in both cases and identify the geometric configurations in which resetting consistently enhances the rate of absorption or escape through the boundaries. By deriving the expressions for the probability currents and conditional first-passage quantities such as splitting probabilities and conditional mean first-passage times, we demonstrate how resetting can be employed to bias the escape pathway through the favorable boundary. Our theoretical predictions are verified through Langevin-type numerical simulations, showing excellent agreement.

Metastable states appear as long-lived intermediate states in various natural transport phenomena, which are governed by energy landscapes. As such, these intermediate metastable states dominate the system’s dynamics at coarse grained times. Moreover, they can strongly influence the overall pathways through which the energy landscape is explored. Therefore, quantifying these metastabilities is crucial for uncovering the key details of the underlying landscape. Here, we introduce a robust method based on a generalized many-body Arrhenius law to identify metastable states in escape problems involving interacting particles with excluded volume. Experimental platforms such as colloidal transport or macromolecular translocation through biological pores can offer promising settings to validate our predictions.

We introduce a new class of first-passage time optimization driven by threshold resetting, inspired by many natural processes where crossing a critical limit triggers failure, degradation, or transition. Here, search agents are collectively reset when a threshold is reached, creating event-driven, system-coupled simultaneous resets that induce long-range interactions. We develop a unified framework to compute mean search times for these correlated stochastic processes, with ballistic and diffusive searchers as key examples uncovering diverse optimization behaviors. A cost function, akin to breakdown penalties, reveals that optimal resetting can forestall larger losses. This formalism generalizes to broader stochastic systems with multiple degrees of freedom.

The mean first-passage time (MFPT) for a Brownian particle to surmount a potential barrier of height ΔU is a fundamental quantity governing a wide array of physical and chemical processes. According to the Arrhenius law, the MFPT typically grows exponentially with increasing barrier height, reflecting the rarity of thermally activated escape events. In this work, we demonstrate that the MFPT can be significantly reduced by reshaping the original single-barrier potential into a structured energy landscape comprising multiple intermediate barriers of lower heights, while keeping the total barrier height ΔU unchanged. Furthermore, this counterintuitive result holds across both linear and nonlinear potential profiles. Our findings suggest that tailoring the energy landscape—by introducing well-placed intermediate barriers—can serve as an effective control strategy to accelerate thermally activated transitions. These predictions are amenable to experimental validation using optical trapping techniques.

Replicating efficient and adaptable microbial navigation strategies, such as run and tumble (RnT) and tactic motions to synthetic active agents has been an enduring quest. To this end, we introduce a stochastic orientational reset (SOR) protocol, in which the propulsion direction of an active Brownian particle (ABP) is reassigned to a random orientation within a defined reset-cone. When the reset-cone is aligned with the instantaneous propulsion direction, ABPs reproduce the RnT dynamics of E. coli; when set along an attractant gradient, they exhibit taxis - with extensive adaptability in persistence through the angular width of the reset-cone and reset rate. We establish the robustness of this protocol across a broad range of swimming speeds using experiments, simulations, and analytical theory.