We experimentally study a system of N=4 two-dimensional Brownian particles, each confined in a harmonic trap with identical stiffness. The stiffness switches simultaneously between two values at random Poissonian times. This collective switching drives the system into a nonequilibrium stationary state with strong long-range correlations between the positions of the particles. Remarkably, we find that, despite the presence of hydrodynamic interactions between the particles mediated by the surrounding fluid, the statistics of some observables are insensitive to hydrodynamic interactions and are well described by the noninteracting theory. Comparing with exact theoretical predictions for noninteracting particles, we observe excellent agreement between theory and experiments for three such observables, namely, the correlations between particles, extreme value, and order statistics (maxima, minima, and ranked positions) and the full counting statistics (i.e., the distribution of the number of particles in a finite interval [-L,L] around the trap center).

We introduce proxitaxis, a simple search strategy where the searcher has only information about the distance from the target but not the direction. The strategy consists of three crucial components: (i) local adaptive moves with a distance-dependent diffusion coefficient, (ii) intermittent long-range returns via stochastic resetting to a certain location R[over ⃗]_{0}, and (iii) an inspection move where the searcher dynamically updates the resetting position R[over ⃗]_{0}. We compute analytically the capture probability of the target within this strategy and show that it can be maximized by an optimal choice of the control parameters of this strategy. Moreover, the optimal strategy undergoes multiple phase transitions as a function of the control parameters. These phase transitions are generic and occur in all dimensions.

We consider a discrete-time continuous-space random walk, with a symmetric jump distribution, under stochastic resetting. Associated with the random walker are cost functions for jumps and resets, and we calculate the distribution of the total cost for the random walker up to the first passage to the target. By using the backward master equation approach we demonstrate that the distribution of the total cost up to the first passage to the target can be reduced to a Wiener-Hopf integral equation. The resulting integral equation can be exactly solved (in Laplace space) for arbitrary cost functions for the jump and selected functions for the reset cost. We show that the large cost behaviour is dominated by resetting or the jump distribution according to the choice of the jump distribution. In the important case of a Laplace jump distribution, which corresponds to run-and-tumble particle dynamics, and linear costs for jumps and resetting, the Wiener-Hopf equation simplifies to a differential equation which can easily be solved.

In this paper, we introduce a new stochastic process of N interacting particles on the line that evolve via Dyson Brownian motion (DBM) with Dyson's index β>0 and undergo simultaneous resetting to their initial positions at a constant rate r. We call this process the resetting Dyson Brownian motion (RDBM) with a parameter β>0, in short the β-RDBM process. For β=1,2,4, the positions of the particles in the RDBM can be interpreted as the eigenvalues of a new random matrix ensemble where the entries of an N×N Gaussian matrix evolve as simultaneously resetting Brownian motions (with rate r) in the presence or absence of a harmonic trap. For r=0 and in the presence of a harmonic trap, this system reaches an equilibrium Gibbs-Boltzmann state of the so called Dyson log-gas. However, the stochastic resetting drives the system at long time to a nonequilibrium stationary state (NESS). We compute exactly the joint distribution of the positions of the particles in this NESS for all β>0 and calculate several macroscopic and microscopic observables in the large N limit. These include the average density profile of the gas, the extreme value statistics, the spacing between two consecutive particles and the full counting statistics, i.e., the distribution of the number of particles in an interval [-L,L]. We show that a nonzero resetting rate r>0 drastically changes the nature of the fluctuations in the stationary state: while the log-gas (r=0) is rather rigid, the β-RDBM in its NESS becomes fluffy, i.e., the fluctuations of different observables are of the same order as their mean. In the absence of a harmonic trap, our results for the β=2 RDBM can be related to nonintersecting Brownian motions (vicious walkers) in the presence of resetting. Our model demonstrates interesting effects arising from the interplay between the eigenvalue repulsion and the all-to-all attraction (generated by stochastic resetting) in an interacting particle system. Numerical simulations are in excellent agreement with our analytical results.

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.

We study a one-dimensional run-and-tumble particle (RTP), which is a prototypical model for active systems, moving within an arbitrary external potential. Using backward Fokker-Planck equations, we derive the differential equation satisfied by its mean first-passage time (MFPT) to an absorbing target, which, without any loss of generality, is placed at the origin. Depending on the shape of the potential, we identify four distinct "phases," with a corresponding expression for the MFPT in every case, which we derive explicitly. To illustrate these general expressions, we derive explicit formulas for two specific cases which we study in detail: a double-well potential and a logarithmic potential. We then present different applications of these general formulas to (1) the generalization of the Kramers escape law for an RTP in the presence of a potential barrier, (2) the "trapping" time of an RTP moving in a harmonic well, and (3) characterizing the efficiency of the optimal search strategy of an RTP subjected to stochastic resetting. Our results reveal that the MFPT of an RTP in an external potential exhibits a far more complex and, at times, counterintuitive behavior compared to that of a passive particle (e.g., Brownian) in the same potential.

We study the relaxation of a diffusive particle confined in an arbitrary external potential and subject to a non-Markovian resetting protocol. With a constant rate r, a previous time tau between the initial time and the present time t is chosen from a given probability distribution K(tau,t), and the particle is reset to the position that it occupied at time tau. Depending on the shape of K(tau,t), the particle either relaxes toward the Gibbs-Boltzmann distribution or toward a non-trivial stationary distribution that breaks ergodicity and depends on the initial position and the resetting protocol. From a general asymptotic theory, we find that if the kernel K(tau,t) is sufficiently localized near tau = 0, i.e. mostly the initial part of the trajectory is remembered and revisited, the steady state is non-Gibbs-Boltzmann. Conversely, if K(tau,t) decays slowly enough or increases with tau, i.e. recent positions are more likely to be revisited, the probability distribution of the particle tends toward the Gibbs-Boltzmann state at large times. In the latter case, however, the temporal approach to the stationary state is generally anomalously slow, following for instance an inverse power law or a stretched exponential, if K(tau,t) is not too strongly peaked at the current time t. These findings are verified by the analysis of several exactly solvable cases and by numerical simulations.

We study a continuous time branching process where an individual splits into two daughters with rate b and dies with rate a, starting from a single individual at t = 0. This model can be mapped exactly to a random walk problem where the population size N(t) performs a random walk on a positive semi-infinite lattice. The hopping rate of this random walker out of a site labeled n is proportional to n, i.e. the walker gets more and more 'active' or 'agitated' as it moves further and further away from the origin-we call this an 'agitated random walk'. We demonstrate that this random walk problem is particularly suitable to obtain exact explicit results on the extreme value statistics, namely, on the distribution of the maximal population size M(t)=max0 <=tau <= t[N(tau)] up to time t. This extreme value distribution displays markedly different behaviors in the three phases: (i) subcritical (b < a) (ii) critical (b = a) and (iii) supercritical (b>a). In the subcritical phase, Q(L,t) becomes independent of time t for large t and the stationary distribution Q(L,infinity) decays exponentially with increasing L. At the critical point b = a, the distribution again becomes independent of t for large t, but the stationary distribution Q(L,infinity)similar to 1/L2 has a power law tail for large L. For finite but large t, the distribution at the critical point exhibits a scaling form Q(L,t)similar to fc(L/at)/L2 where the scaling function fc(z) has a nontrivial shape that we compute analytically. In the supercritical phase, the distribution Q(L,t) has a 'fluid' part that becomes independent of t for large t and a 'condensate' part (a delta peak centered at e (b-a)t ) which gets disconnected from the 'fluid' part and moves rapidly to infinity as time increases. We also verify our analytical predictions via numerical simulations finding excellent agreement. Our results can be applied to characterize the time dependence of the size of the maximally infected population up to a fixed time during an epidemic spread.

We study a system of N noninteracting bosons in a harmonic trap subjected to repeated quantum quenches, where the trap frequency is switched from one value to another after a random time duration drawn from an exponential distribution. Each cycle contains two steps: (i) changing the trap frequency to enable unitary evolution under a Hamiltonian, and (ii) reapplying the original trap at stochastic times to cool the gas back to its initial state. This protocol effectively makes it an open quantum system and drives it into a unique nonequilibrium steady state (NESS). We analytically and numerically characterize the NESS, uncovering a conditionally independent and identically distributed (CIID) structure in the joint probability density function (JPDF) of the positions. The JPDF in the CIID structure is a product of Gaussians with a common random variance, which is then averaged with respect to its distribution, making the JPDF non-factorizable, giving rise to long-range emergent dynamical correlations. The average density profile of the gas shows significant deviations from the initial Gaussian shape. We further compute the order and the gap statistics, revealing universal scaling in both bulk and edge regimes. We also analyze the full counting statistics, exposing rich parameter-dependent structure. Our results demonstrate how stochastic quenches can generate nontrivial correlations in quantum many-body systems.

We study a gas of N Brownian particles in the presence of a common stochastic diffusivity D(t)=B-2(t), where B(t) represents a one-dimensional Brownian motion at time t. Starting from all the particles localized at the origin, the gas expands with a ballistic scaling x similar to t. We show that because of the common stochastic diffusivity, the expanding gas becomes dynamically correlated, and the joint probability density function of the position of the particles has a conditionally independent and identically distributed (CIID) structure that was recently found in several other systems. This special CIID structure allows us to compute the average density profile of the gas, extreme and order statistics, the gap distribution between successive particles, and the full counting statistics (FCS) that describe the probability density function (PDF) H(kappa,t) of the fraction of particles kappa in a given region [-L,L]. Interestingly, the position fluctuations of the central particles and the average density profiles are described by the same scaling function. The PDF describing the FCS has an essential singularity near kappa = 0, indicating the presence of particles inside the box [-L,L] at all times. Near the upper limit kappa = 1, the scaling function H(kappa,t) has a rather unusual behavior: H(kappa,t)similar to(1-kappa)(beta(t)), where the exponent beta(t) changes continuously with time. At early times, beta(t) is negative, indicating a divergence of H(kappa,t) as kappa -> 1, whereas beta(t) becomes positive for t>t(c), where t(c) is computed exactly. For t>t(c), the scaling function H(kappa,t) vanishes as kappa -> 1, indicating that it is highly unlikely to have all the particles in the interval [-L,L]. Exactly at t=t(c), beta = 0, indicating that the PDF approaches a non-zero constant as kappa -> 1. Thus, as a function of t, the FCS exhibits an interesting shape transition. We also obtain the PDFs of the first-passage time to a given position x and first-exit time from a box [-L,L], by any one of the particles, and find that both PDFs are described by the same scaling function.

We provide a uniform framework to compute the exact distribution of the number of minima/maxima in three different random walk landscape models in one dimension. The landscape is generated by the trajectory of a discrete-time continuous space random walk with arbitrary symmetric and continuous jump distribution at each step. In model I, we consider a 'free' random walk of N steps. In model II, we consider a 'meander landscape' where the random walk, starting at the origin, stays non-negative up to N steps. In model III, we study a 'first-passage landscape' which is generated by the trajectory of a random walk that starts at the origin and stops when it crosses the origin for the first time. We demonstrate that while the exact distribution of the number of minima is different in the three models, for each model it is universal for all N, in the sense that it does not depend on the jump distribution as long as it is symmetric and continuous. In the last two cases we show that this universality follows from a non trivial mapping to the Sparre Andersen theorem known for the first-passage probability of discrete-time random walks with symmetric and continuous jump distribution. Our analytical results are in excellent agreement with our numerical simulations.

We study the nonequilibrium stationary state (NESS) induced by quantum resetting of a system of N noninteracting bosons in a harmonic trap. Our protocol consists of preparing initially the system in the ground state of a harmonic oscillator centered at +a, followed by a rapid quench where the center is shifted to -a and the system is allowed to evolve unitarily up to a random Poissonian time tau distributed via re-r tau. Then the trap center is reset to +a again and the system is assumed to cool instantaneously to the initial ground state. The system is again allowed to evolve unitarily in the trap centered at -a up to a random time, and the procedure is repeated. Under repeated resetting, the system reaches a NESS where the positions of bosons get strongly correlated due to simultaneous resetting induced by the trap. We fully characterize the steady state by analytically computing several physical observables such as the average density, extreme value statistics, order and gap statistics, and also the distribution of the number of particles in a region [-L,L], known as the full counting statistics (FCS). In particular, we show that in the large N limit, the scaling function describing the FCS exhibits a striking feature: it is supported over a nontrivial finite interval, and moreover is discontinuous at an interior point of the support. Our results are supported by numerical simulations. This is a rare example of a strongly correlated quantum many-body NESS where various observables can be exactly computed.

We consider three global chracteristic times for a one-dimensional Brownian motion x(τ) in the interval τ∈ [0,t]: the occupation time t_ o denoting the cumulative time where x(τ)>0, the time t_ m at which the process achieves its global maximum in [0,t] and the last-passage time t_l through the origin before t. All three random variables have the same marginal distribution given by Lévy's arcsine law. We compute exactly the pairwise joint distributions of these three times and show that they are quite different from each other. The joint distributions display rather rich and nontrivial correlations between these times. Our analytical results are verified by numerical simulations.

We study the height distribution of a one-dimensional Edwards-Wilkinson interface in the presence of a stochastic diffusivity D(t)=B-2(t), where B(t) represents a one-dimensional Brownian motion at time t. The height distribution at a fixed point in space is computed analytically. The typical height h(x,t) at a given point in space is found to scale as t(3/4) and the distribution G(H) of the scaled height H=h/t(3/4) is symmetric but with a nontrivial shape: while it approaches a nonzero constant quadratically as H -> 0, it has a non-Gaussian tail that decays exponentially for large H. We show that this exponential tail is rather robust and holds for a whole family of linear interface models parametrized by a dynamical exponent z > 1, with z = 2 corresponding to the Edwards-Wilkinson model.

We study the nonequilibrium steady state of a two-dimensional Coulomb gas under the action of an anisotropic harmonic trapping potential along with a nonconservative rotational force. In the case without rotation, the equilibrium (zero temperature) steady state has a uniform density supported over a static elliptical droplet. The addition of a rotational force drives the system into a nonequilibrium steady state where the density is still uniform inside an ellipse, but the ellipse gets tilted by a fixed angle compared to the nonrotational case. In addition, a nonzero current is generated inside the droplet which runs concentrically to the droplet boundary. For large rotational force, the droplet develops a purely circular form. Our results are predicted by a simple hydrodynamic calculation, confirmed by numerical simulations, and provide a full understanding of a driven nonequilibrium state in a strongly interacting system.

We provide an exact formula for the mean first-passage time (MFPT) to a target at the origin for a single particle diffusing on a d-dimensional hypercubic lattice starting from a fixed initial position R[over ⃗]_{0} and resetting to R[over ⃗]_{0} with a rate r. Previously known results in the continuous space are recovered in the scaling limit r→0, R_{0}=|R[over ⃗]_{0}|→∞ with the product sqrt[r]R_{0} fixed. However, our formula is valid for any r and any R[over ⃗]_{0} that enables us to explore a much wider region of the parameter space that are inaccessible in the continuum limit. For example, we have shown that the MFPT, as a function of r for fixed R[over ⃗]_{0}, diverges in the two opposite limits r→0 and r→∞ with a unique minimum in between, provided the starting point is not a nearest neighbor of the target. In this case, the MFPT diverges as a power law ∼r^{ϕ} as r→∞, but very interestingly with an exponent ϕ=(|m_{1}|+|m_{2}|+...+|m_{d}|)-1 that depends on the starting point R[over ⃗]_{0}=a(m_{1},m_{2},...,m_{d}) where a is the lattice spacing and m_{i}'s are integers. If, on the other hand, the starting point happens to be a nearest neighbor of the target, then the MFPT decreases monotonically with increasing r, approaching a universal limiting value 1 as r→∞, indicating that the optimal resetting rate in this case is infinity. We provide a simple physical reason and a simple Markov-chain explanation behind this somewhat unexpected universal result. These interesting results on a lattice are not captured by the continuum theory. Our analytical predictions are verified in numerical simulations on lattices up to 50 dimensions. Finally, in the absence of a target, we also compute exactly the position distribution of the walker in the nonequlibrium stationary state that also displays interesting lattice effects not captured by the continuum theory.

We study the diffusion of a particle with a time-dependent diffusion coefficient D(t) that switches between random values drawn from a distribution W(D) at a fixed rate r. Using a renewal approach, we compute exactly the moments of the position of the particle ⟨x^{2n}(t)⟩ at any finite time t, and for any W(D) with finite moments ⟨D^{n}⟩. For t≫1, we demonstrate that the cumulants ⟨x^{2n}(t)⟩_{c} grow linearly with t and are proportional to the free cumulants of a random variable distributed according to W(D). For specific forms of W(D), we compute the large deviations of the position of the particle, uncovering rich behaviors and dynamical transitions of the rate function I(y=x/t). Our analytical predictions are validated numerically with high precision, achieving accuracy up to 10^{-2000}.

We investigate the first-passage properties of a jump process with a constant drift, focusing on two key observables: the first-passage time τ and the number of jumps n before the first-passage event. By mapping the problem onto an effective discrete-time random walk, we derive an exact expression for the Laplace transform of the joint distribution of τ and n using the generalized Pollaczek–Spitzer formula. This result is then used to analyze the first-passage properties for two exactly solvable cases: (i) both the inter-jump intervals and jump amplitudes are exponentially distributed, and (ii) the inter-jump intervals are exponentially distributed while all jumps have the same fixed amplitude. We show the existence of two distinct regimes governed by the strength of the drift: (i) a survival regime , where the process remains positive indefinitely with finite probability; (ii) an absorption regime , where the first-passage eventually occurs; and (iii) a critical point at the boundary between these two phases. We characterize the asymptotic behavior of survival probabilities in each regime: they decay exponentially to a constant in the survival regime, vanish exponentially fast in the absorption regime, and exhibit power-law decay at the critical point. Furthermore, in the absorption regime, we derive large deviation forms for the marginal distributions of τ and n . The analytical predictions are validated through extensive numerical simulations.

We consider the quantum dynamical evolution of a fully-connected quantum system subjected to random projective measurements and study the first detection time of an extended subspace of the Hilbert space. Exact analytical expressions are obtained for the mean first detection time and the full first detection probability distribution as functionals of the initial state |ψ_0⟩ and the measurement rate r. Exact enumeration agrees perfectly with our analytical predictions.

In this Chapter, we consider a model of N independent random walkers, each of duration t, and each starting from the origin, on a lattice in d dimensions. We focus on two observables, namely D_N(t) and C_N(t) denoting respectively the number of distinct and common sites visited by the walkers. For large t, where the lattice random walkers converge to independent Brownian motions, we compute exactly the mean ⟨ D_N(t) ⟩ and ⟨ C_N(t) ⟩. Our main interest is on the N-dependence of these quantities. While for ⟨ D_N(t) ⟩ the N-dependence only appears in the prefactor of the power-law growth with time, a more interesting behavior emerges for ⟨ C_N(t) ⟩. For this latter case, we show that there is a “phase transition” in the (N, d) plane where the two critical line d=2 and d=d_c(N) = 2N/(N-1) separate three phases of the growth of ⟨ C_N(t)⟩. The results are extended to the mean number of sites visited exactly by K of the N walkers. Furthermore in d=1, the full distribution of D_N(t) and C_N(t) are computed, exploiting a mapping to the extreme value statistics. Extensions to two other models, namely N independent Brownian bridges and N independent resetting Brownian motions/bridges are also discussed.