Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

The thermodynamic description of many-particle systems rests on the assumption of ergodicity, the ability of a system to explore all allowed configurations in the phase space. Recent studies on many-body localization have revealed the existence of systems that strongly violate ergodicity in the presence of quenched disorder. Here, we demonstrate that ergodicity can be weakly broken by a different mechanism, arising from the presence of special eigenstates in the many-body spectrum that are reminiscent of quantum scars in chaotic non-interacting systems. In the single-particle case, quantum scars correspond to wavefunctions that concentrate in the vicinity of unstable periodic classical trajectories. We show that many-body scars appear in the Fibonacci chain, a model with a constrained local Hilbert space that has recently been experimentally realized in a Rydberg-atom quantum simulator. The quantum scarred eigenstates are embedded throughout the otherwise thermalizing many-body spectrum but lead to direct experimental signatures, as we show for periodic recurrences that reproduce those observed in the experiment. Our results suggest that scarred many-body bands give rise to a new universality class of quantum dynamics, opening up opportunities for the creation of novel states with long-lived coherence in systems that are now experimentally realizable.

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Weak ergodicity breaking from quantum many-body scars

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past 25 years a new universality class has emerged to describe a host of important physical and probabilistic models (including one-dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar–Parisi–Zhang (KPZ) universality class and underlying it is, again, a continuum object — a non-linear stochastic partial differential equation — known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with narrow wedge initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.

The kardar-parisi-zhang equation and universality class

We consider the solution of the stochastic heat equation @TZ D 1 @ 2 X ZZ P W with delta function initial condition Z.T D0;X/ DiXD0 whose logarithm, with appropriate normalization, is the free energy of the con- tinuum directed polymer, or the Hopf-Cole solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We obtain explicit formulas for the one-dimensional marginal distributions, the crossover distributions, which interpolate between a standard Gaussian dis- tribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest-descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion process with antishock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes. © 2010 Wiley Periodicals, Inc.

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Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions

We consider two strictly related models: a solid on solid interface growth model and the weakly asymmetric exclusion process, both on the one dimensional lattice. It has been proven that, in the diffusive scaling limit, the density field of the weakly asymmetric exclusion process evolves according to the Burgers equation and the fluctuation field converges to a generalized Ornstein-Uhlenbeck process. We analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the (non linear) Burgers equation with a random noise on the density current. For the solid on solid model, we prove that the fluctuation field of the interface profile, if suitably rescaled, converges to the Kardar–Parisi–Zhang equation. This provides a microscopic justification of the so called kinetic roughening, i.e. the non Gaussian fluctuations in some non-equilibrium processes. Our main tool is the Cole-Hopf transformation and its microscopic version. We also develop a mathematical theory for the macroscopic equations.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1158328658

Stochastic Burgers and KPZ equations from particle systems

We study, in one space dimension, the heat equation with a random potential that is a white noise in space and time. This equation is a linearized model for the evolution of a scalar field in a space-time-dependent random medium. It has also been related to the distribution of two-dimensional directed polymers in a random environment, to the KPZ model of growing interfaces, and to the Burgers equation with conservative noise. We show how the solution can be expressed via a generalized Feynman-Kac formula. We then investigate the statistical properties: the two-point correlation function is explicitly computed and the intermittence of the solution is proven. This analysis is carried out showing how the statistical moments can be expressed through local times of independent Brownian motions.

/pdf/the-stochastic-heat-equation-feynman-kac-formula-and-3poqpsvuko.pdf

The stochastic heat equation: Feynman-Kac formula and intermittence

We study Burgers Equation perturbed by a white noise in space and time. We prove the existence of solutions by showing that the Cole-Hopf transformation is meaningful also in the stochastic case. The problem is thus reduced to the anaylsis of a linear equation with multiplicative half white noise. An explicit solution of the latter is constructed through a generalized Feynman-Kac formula. Typical properties of the trajectories are then discussed. A technical result, concerning the regularizing effect of the convolution with the heat kernel, is proved for stochastic integrals.

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The stochastic Burgers equation

Kinetically constrained spin models (KCSM) are interacting particle systems which are intensively studied in physics literature as models for systems undergoing glass or jamming transitions. KCSM leave on discrete lattices and evolve via a Glauber-like dynamics which is reversible w.r.t. a simple product measure. The key feature is that the creation/destruction of a particle at a given site can occur only if the current configuration satisfies proper local constraints. Due to the fact that creation/destruction rates can be zero, the whole analysis of the long time behavior becomes quite delicate. From the mathematical point of view, the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density remained open for most KCSM (with the exception of the East model in d=1 Aldous and P. Diaconis, J. Stat. Phys. 107(5–6):945–975 2002). Here we review a novel multi-scale approach which we have developed in Cancrini et al. (Probab. Theory Relat. Fields 140:459–504, 2008; Lecture Notes in Mathematics, vol. 1970, pp. 307–340, Springer, 2009) trough which we: (i) prove positivity of the spectral gap in the whole ergodic region for a wide class of KCSM on ℤ d , (ii) establish (sometimes optimal) bounds on the behavior of the spectral gap near the boundary of the ergodicity region and (iii) prove pure exponential decay at equilibrium for the persistence function, i.e. the probability that the occupation variable at the origin does not change before time t. Our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations. In particular (i) above establishes exponential decay of auto-correlation functions disproving the stretched exponential decay which had been conjecture for some KCSM and (ii) disproves some of the scalings which had been extrapolated from numerical simulations for the relaxation times (inverse of the spectral gap).

/pdf/kinetically-constrained-models-2uwnjkiqu2.pdf

Kinetically Constrained Models

We analyze the density and size dependence of the relaxation time for kinetically constrained spin models (KCSM) intensively studied in the physics literature as simple models sharing some of the features of the glass transition. KCSM are interacting particle systems on \({\mathbb{Z}}^d\) with Glauber-like dynamics, reversible w.r.t. a simple product i.i.d Bernoulli(p) measure. The essential feature of a KCSM is that the creation/destruction of a particle at a given site can occur only if the current configuration around it satisfies certain constraints which completely define each specific model. No other interaction is present in the model. From the mathematical point of view, the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density p remained open for most KCSM (with the notably exception of the East model in d = 1; Aldous and Diaconis in J Stat Phys 107(5–6):945–975, 2002). Here for the first time we: (i) identify the ergodicity region by establishing a connection with an associated bootstrap percolation model; (ii) develop a novel multi-scale approach which proves positivity of the spectral gap in the whole ergodic region; (iii) establish, sometimes optimal, bounds on the behavior of the spectral gap near the boundary of the ergodicity region and (iv) establish pure exponential decay at equilibrium for the persistence function, i.e. the probability that the occupation variable at the origin does not change before time t. Our techniques are flexible enough to allow a variety of constraints and our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations.

/pdf/kinetically-constrained-spin-models-3g6bdhbw5y.pdf

Kinetically constrained spin models

We consider a continuous gas in a d -dimensional rectangular box with a finite range, positive pair potential, and we construct a Markov process in which particles appear and disappear with appropriate rates so that the process is reversible w.r.t. the Gibbs measure. If the thermodynamical paramenters are such that the Gibbs specification satisfies a certain mixing condition, then the spectral gap of the generator is strictly positive uniformly in the volume and boundary condition. The required mixing condition holds if, for instance, there is a convergent cluster expansion.

/pdf/the-spectral-gap-for-a-glauber-type-dynamics-in-a-continuous-1q604chtlf.pdf

Nicoletta Cancrini

Papers

The stochastic heat equation: Feynman-Kac formula and intermittence

The stochastic Burgers equation

Kinetically Constrained Models

Kinetically constrained spin models

The spectral gap for a Glauber-type dynamics in a continuous gas☆