Baruch Meerson

We extend the N branching Brownian motions model of population invasion to higher-order asexual reproduction. Increasing reproduction order leads to qualitative changes: Invasion fronts generically cease to exist beyond binary reproduction, and in the binary case itself, their speed becomes diffusion independent. Ternary reproduction shows critical behavior, with collapse into a strongly localized "invasion bullet" in the supercritical regime, diffusive spreading in the subcritical regime, and a continuous family of fronts at criticality. These results suggest that the dominance of division and binary reproduction in nature reflects fundamental constraints on invasion dynamics.

We study the statistics of extinction and blowup times in well-mixed systems of stochastically reacting particles. We focus on the short-time tail, T→0, of the extinction- or blowup-time distribution P_{m}(T), where m is the number of particles at t=0. This tail often exhibits an essential singularity at T=0, and we show that the singularity is captured by a time-dependent WKB (Wentzel-Kramers-Brillouin) approximation applied directly to the master equation. This approximation, however, leaves undetermined a large pre-exponential factor. We show how to calculate this factor by applying a leading- and a subleading-order WKB approximation to the Laplace-transformed backward master equation. Accurate asymptotic results can be obtained when this WKB solution can be matched to another approximate solution (the "inner" solution), valid for not too large m. We demonstrate and verify this method on three examples of reactions which are also solvable without approximations.

Fractional Brownian motion (fBm) is an important scale-invariant Gaussian non-Markovian process with stationary increments, which serves as a prototypical example of a system with long-range temporal correlations and anomalous diffusion. The fBm is traditionally defined for the Hurst exponent H in the range 0 < H < 1 . Here we extend this definition to the regime − 1 / 2 < H < 0 . The extended fBm is not a pointwise process, so we regularize it via a local temporal averaging with a narrow kernel. The resulting process is both very rough and persistent, that is long-range positively correlated. In addition, this process is stationary. The stationarity implies that diffusion is completely suppressed in this region of H . We also study another closely related Gaussian process: the stationary fractional Ornstein–Uhlenbeck (fOU) process, extended to the range − 1 / 2 < H < 0 and smoothed in the same way as the fBm. Remarkably, the smoothed fOU process is asymptotically insensitive to the strength of the confining potential. Finally, we determine the optimal paths of the fBm and fOU processes for − 1 / 2 < H < 0 , conditioned on reaching a specified value when starting from zero. In the marginal case H = 0 , our results match continuously with known results for the traditionally defined fBm and fOU processes.

We develop an age-structured hydrodynamic (HD) theory that describes the collective behavior of N ≫ 1 anomalously diffusing particles under stochastic renewal resetting. The theory treats the age of a particle—the time since its last reset—as an explicit dynamical variable and allows for resetting rules that introduce global interparticle correlations. The anomalous diffusion is modeled by the scaled Brownian motion (sBm): a Gaussian process with independent increments, characterized by a power-law time dependence of the diffusion coefficient, D ( t ) ∼ t 2 H − 1 , where H > 0 . We apply this theory to three different resetting protocols: independent resetting to the origin (model A), resetting to the origin of the particle farthest from it (model B), and a scaled-diffusion extension of the “Brownian bees” model of Berestycki []. In all these models, nonequilibrium steady states are reached at long times, and we determine the steady-state densities. For model A, the (normalized to unity) steady-state density coincides with the steady-state probability density of a single particle undergoing sBM with resetting to the origin. For model B, and for the scaled Brownian bees, the HD steady-state densities are markedly different: In particular, they have compact supports for all H > 0 . The age-structured HD formalism can be extended to other anomalous diffusion processes with renewal resetting protocols that introduce global interparticle correlations.

We study large deviations, over a long time window T → ∞ , of the dynamical observables A n = ∫ 0 T x n ( t ) , d t , n = 3 , 4 , … , where x ( t ) is a centered stationary Gaussian process in continuous time. We show that, for short-correlated processes the probability density of A _n exhibits an anomalous scaling P ( A n , T ) ∼ exp ⁡ [ − T μ f n ( Δ A n T − ν ) ] at T → ∞ while keeping Δ A n T − ν constant. Here Δ A _n is the deviation of A _n from its ensemble average. The anomalous exponents µ and ν depend on n and are smaller than 1, whereas the rate function f n ( z ) exhibits a first-order dynamical phase transition (DPT) which resembles condensation transitions observed in many systems. The same type of anomaly and DPT, with the same µ and ν , was previously uncovered for the Ornstein–Uhlenbeck process—the only stationary Gaussian process which is also Markovian. We also uncover an anomalous behavior and a similar DPT in the long-correlated Gaussian processes. However, the anomalous exponents µ and ν are determined in this case not only by n but also by the power-law long-time decay ∼ | t | − α of the covariance function. The different anomalous scaling behavior is a consequence of a faster-than-linear scaling with T of the variance of A _n . Finally, for sufficiently long-ranged correlations, α < 2 / n , the DPT disappears, giving way to a smooth crossover between the regions of typical, Gaussian fluctuations and large deviations. The basic mechanism behind the DPT is the existence of strongly localized optimal paths of the process conditioned on very large A _n and coexistence between the localized and delocalized paths of the conditioned process. Our theoretical predictions are corroborated by replica-exchange Wang–Landau simulations where we could probe probability densities down to 10 ⁻²⁰⁰ .

We study the dynamics of an overdamped Brownian particle in a repulsive scale-invariant potential V(x)∼-x^{n+1}. For n>1, a particle starting at position x reaches infinity in a finite, randomly distributed time. We focus on the short-time tail T→0 of the probability distribution P(T,x,n) of the blowup time T for integer n>1. Krapivsky and Meerson [Phys. Rev. E 112, 024128 (2025)2470-004510.1103/1hds-9ttg] recently evaluated the leading-order asymptotics of this tail, which exhibits an n-dependent essential singularity at T=0. Here we provide a more accurate description of the T→0 tail by calculating, for all n=2,3,⋯, the previously unknown large preexponential factor of the blowup-time probability distribution. To this end, we apply a WKB (after Wentzel, Kramers and Brillouin) approximation-at both leading and subleading orders-to the Laplace-transformed backward Fokker-Planck equation governing P(T,x,n). For even n, the WKB solution alone suffices. For odd n, however, the WKB solution breaks down in a narrow boundary layer around x=0. In this case, it must be supplemented by an "internal" solution and a matching procedure between the two solutions in their common region of validity.

We consider a Brownian particle performing an overdamped motion in a power-law repulsive potential. If the potential grows with the distance faster than quadratically, the particle escapes to infinity in a finite time. We determine the average blowup time and study the probability distribution of the blowup time. In particular, we show that the long-time tail of this probability distribution decays purely exponentially, while the short-time tail exhibits an essential singularity. These qualitative features turn out to be quite universal, as they occur for all rapidly growing power-law potentials in arbitrary spatial dimensions. The quartic potential is especially tractable, and we analyze it in more detail.

The fractional Ornstein-Uhleneck (fOU) process is described by the overdamped Langevin equation x ˙ ( t ) + γ x = 2 D ξ ( t ) , where ξ ( t ) is the fractional Gaussian noise with the Hurst exponent 0 < H < 1 . For H ≠ 1 / 2 the fOU process is non-Markovian but Gaussian, and it has either vanishing (for H < 1 / 2 ), or divergent (for H > 1 / 2 ) spectral density at zero frequency. For H > 1 / 2 , the fOU is long-correlated. Here we study dynamical large deviations of the fOU process and focus on the area A n = ∫ − T T x n ( t ) d t , n = 1 , 2 , … over a long time window 2T. Employing the optimal fluctuation method, we determine the optimal path of the conditioned process, which dominates the large-A_n tail of the probability distribution of the area, P ( A n , T ) ∼ exp ⁡ [ − S ( A n , T ) ] . We uncover a nontrivial phase diagram of scaling behaviors of the optimal paths and of the action S ( A n ≡ 2 a n T , T ) ∼ T α ( H , n ) a n 2 / n on the (H, n) plane. The phase diagram includes three distinct regions: (i) H > 1 − 1 / n , where α ( H , n ) = 2 − 2 H , and the optimal paths are delocalized in time, (ii) n = 2 and H ⩽ 1 2 , where α ( H , n ) = 1 , and the optimal paths oscillate with an H-dependent frequency, and (iii) H ⩽ 1 − 1 / n and n > 2, where α ( H , n ) = 2 / n , and the optimal paths are strongly localized. We verify our theoretical predictions in large-deviation simulations of the fOU process. By combining the Wang-Landau Monte-Carlo algorithm with the circulant embedding method of generation of stationary Gaussian fields, we were able to measure probability densities as small as 10⁻¹⁷⁰. We also generalize our findings to other stationary Gaussian processes with either divergent, or vanishing spectral density at zero frequency.