Central Limit Theorem¶for Stochastic Hamilton–Jacobi Equations

We study the asymptotic behavior of , where u solves the Hamilton–Jacobi equation u _t +H(x,u _x ) ≡ 0 with H a stationary ergodic process in the x-variable. It was shown in Rezakhanlou–Tarver [RT] that u ^ɛ converges to a deterministic function provided H(x,p) is convex in p and the convex conjugate of H in the p-variable satisfies certain growth conditions. In this article we establish a central limit theorem for the convergence by showing that for a class of examples, u ^ɛ(x,t) can be (stochastically) represented as , where Z(x,t) is a suitable random field. In particular we establish a central limit theorem when the dimension is one and , where ω is a random function that enjoys some mild regularity. Received: 15 February 1999 / Accepted: 14 December 1999     

Central Limit Theorem for Branching Brownian Motions in Random Environment

We introduce a model of branching Brownian motions in time-space random environment associated with the Poisson random measure. We prove that, if the randomness of the environment is moderated by that of the Brownian motion, the population density satisfies a central limit theorem and the growth rate of the population size is the same as its expectation with strictly positive probability. We also characterize the diffusive behavior of our model in terms of the decay rate of the replica overlap. On the other hand, we show that, if the randomness of the environment is strong enough, the growth rate of the population size is strictly less than its expectation almost surely. To do this, we use a connection between our model and the model of Brownian directed polymers in random environment introduced by Comets and Yoshida. Partly supported by the Global COE program at Department of Mathematics and Research Institute for Mathematical Sciences, Kyoto University.     

The Becker–Döring Process: Pathwise Convergence and Phase Transition Phenomena

In this note, we study an infinite reaction network called the stochastic Becker–Döring process, a sub-class of the general coagulation–fragmentation models. We prove pathwise convergence of the process towards the deterministic Becker–Döring equations which improves classical tightness-based results. Also, we show by studying the asymptotic behavior of the stationary distribution, that the phase transition property of the deterministic model is also present in the finite stochastic model. Such results might be interpreted closed to the so-called gelling phenomena in coagulation models. We end with few numerical illustrations that support our results.

     

Generalized Central Limit Theorem and Renormalization Group

We introduce a simple instance of the renormalization group transformation in the Banach space of probability densities. By changing the scaling of the renormalized variables we obtain, as fixed points of the transformation, the Lévy strictly stable laws. We also investigate the behavior of the transformation around these fixed points and the domain of attraction for different values of the scaling parameter. The physical interest of a renormalization group approach to the generalized central limit theorem is discussed.     

A Liouville-type Theorem for Schrödinger Operators

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P ₁, such that a nonzero subsolution of a symmetric nonnegative operator P ₀ is a ground state. Particularly, if P _j : = ?Δ + V _j , for j = 0,1, are two nonnegative Schrödinger operators defined on \(\Omega\subseteq \mathbb{R}^d\) such that P ₁ is critical in Ω with a ground state φ, the function \(\psi\nleq 0\) is a subsolution of the equation P ₀ u = 0 in Ω and satisfies \(\psi_+\leq C\varphi\) in Ω, then P ₀ is critical in Ω and \(\psi\) is its ground state. In particular, \(\psi\) is (up to a multiplicative constant) the unique positive supersolution of the equation P ₀ u = 0 in Ω. Similar results hold for general symmetric operators, and also on Riemannian manifolds.     

The Gallavotti–Cohen Fluctuation Theorem for a Nonchaotic Model

We test the applicability of the Gallavotti–Cohen fluctuation formula on a nonequilibrium version of the periodic Ehrenfest wind-tree model. This is an one-particle system whose dynamics is rather complex (e.g., it appears to be diffusive at equilibrium), but its Lyapunov exponents are nonpositive. For small applied field, the system exhibits a very long transient, during which the dynamics is roughly chaotic, followed by asymptotic collapse on a periodic orbit. During the transient, the dynamics is diffusive, and the fluctuations of the current are found to be in agreement with the fluctuation formula, despite the lack of real hyperbolicity. These results also constitute an example which manifests the difference between the fluctuation formula and the Evans–Searles identity.     

A Kohno–Drinfeld Theorem for the Monodromy of Cyclotomic KZ Connections

We compute explicitly the monodromy representations of “cyclotomic” analogs of the Knizhnik–Zamolodchikov differential system. These are representations of the type B braid group B_n¹{B_n^1} . We show how the representations of the braid group B _n obtained using quantum groups and universal R-matrices may be enhanced to representations of B_n¹{B_n^1} using dynamical twists. Then, we show how these “algebraic” representations may be identified with the above “analytic” monodromy representations.     

Metastable states for the Becker-Döring cluster equations

The Becker-Döring equations, in whichc _l (t) can represent the concentration ofl-particle clusters or droplets in (say) a condensing vapour at timet, are $$\begin{array}{*{20}c} {{{dc_l (t)} \mathord{\left/ {\vphantom {{dc_l (t)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = J_{l - 1} (t) - J_l (t)} & {(l = 2,3,...)} \\ \end{array} $$ with $$J_l (t): = a_l c_1 (t)c_l (t) - b_{l + 1} c_{l + 1} (t)$$ and eitherc ₁=const. (‘case A’) or \(\rho : = \sum\limits_1^\infty {lc_l } \) =const. (‘case B’). The equilibrium solutions arec _l =Q _l z ^l , where \(Q_l : = \prod\limits_2^l {({{a_{r - 1} } \mathord{\left/ {\vphantom {{a_{r - 1} } {b_r }}} \right. \kern-0em} {b_r }})} \) . The density of the saturated vapour, defined as \(\rho _s : = \sum\limits_1^\infty {lQ_l z_s ^l } \) , wherez _s is the radius of convergence of the series, is assumed finite. It is proved here that, subject to some further plausible conditions on the kinetic coefficientsa _l andb _l , there is a class of “metastable” solutions of the equations, withc ₁?z _s small and positive, which take an exponentially long time to decay to their asymptotic steady states. (An “exponentially long time” means one that increases more rapidly than any negative power of the given value ofc ₁?z _s (or, in caseB,ρ?ρ _s ) as the latter tends to zero). The main ingredients in the proof are (i) a time-independent upper bound on the solution of the kinetic equations (this upper bound is a steady-state solution of case A of the equations, of the type used in the Becker-Döring theory of nucleation), and (ii) an upper bound on the total concentration of particles in clusters greater than a certain critical size, which (with suitable initial conditions) remains exponentially small until the time becomes exponentially large.     

A Class of Counterexamples to Jeans' Theorem for the Vlasov–Einstein System

The Vlasov–Poisson and Vlasov–Einstein systems model the motion of a self gravitating system such as a galaxy. The Vlasov–Poisson system is nonrelativistic. Jeans' theorem states that every spherically symmetric solution of the Vlasov–Poisson system that is independent of time may be expressed as a function of the two invariants, energy and angular momentum. This paper shows this is not the case for the Vlasov–Einstein system. Received: 2 November 1998 / Accepted: 24 December 1998     

A Self-Consistent Ornstein–Zernike Approximation for the Edwards–Anderson Spin-Glass Model

We propose a self-consistent Ornstein–Zernike approximation for studying the Edwards–Anderson spin glass model. By performing two Legendre transforms in replica space, we introduce a Gibbs free energy depending on both the magnetizations and the overlap order parameters. The correlation functions and the thermodynamics are then obtained from the solution of a set of coupled partial differential equations. The approximation becomes exact in the limit of infinite dimension and it provides a potential route for studying the stability of the high-temperature phase against replica-symmetry breaking fluctuations in finite dimensions. As a first step, we present the predictions for the freezing temperature T _f and for the zero-field thermodynamic properties and correlation length above T _f as a function of dimensionality.     

Torelli Theorem for the Deligne–Hitchin Moduli Space

Fix integers g ≥ 3 and r ≥ 2, with r ≥ 3 if g = 3. Given a compact connected Riemann surface X of genus g, let denote the corresponding Deligne–Hitchin moduli space. We prove that the complex analytic space determines (up to an isomorphism) the unordered pair , where is the Riemann surface defined by the opposite almost complex structure on X.     

Rarefaction Pulses for the Nonlinear Schrödinger Equation in the Transonic Limit

We investigate the properties of finite energy travelling waves to the nonlinear Schrödinger equation with nonzero conditions at infinity for a wide class of nonlinearities. In space dimension two and three we prove that travelling waves converge in the transonic limit (up to rescaling) to ground states of the Kadomtsev–Petviashvili equation. Our results generalize an earlier result of Béthuel et al. for the two-dimensional Gross–Pitaevskii equation, and provide a rigorous proof to a conjecture by C. Jones and P. H. Roberts about the existence of vortexless travelling waves with high energy and momentum in dimension three.     

Macroscopic Limit of a Bipartite Curie–Weiss Model: A Dynamical Approach

We analyze the Glauber dynamics for a bi-populated Curie–Weiss model. We obtain the limiting behavior of the empirical averages in the limit of infinitely many particles. We then characterize the phase space of the model in absence of magnetic field and we show that several phase transitions in the inter-groups interaction strength occur.     

A Self-Consistent Ornstein–Zernike Approximation for the Random Field Ising Model

We extend the self-consistent Ornstein–Zernike approximation (SCOZA), first formulated in the context of liquid-state theory, to the study of the random field Ising model. Within the replica formalism, we treat the quenched random field just as another spin variable, thereby avoiding the usual average over the random field distribution. This allows us to study the influence of the distribution on the phase diagram in finite dimensions. The thermodynamics and the correlation functions are obtained as solutions of a set a coupled partial differential equations with magnetization, temperature, and disorder strength as independent variables. A preliminary analysis based on high-temperature and 1/d series expansions shows that the theory can predict accurately the dependence of the critical temperature on disorder strength (no sharp transition, however, occurs for d4). For the bimodal distribution, we find a tricritical point which moves to weaker fields as the dimension is reduced. For the Gaussian distribution, a tricritical point may appear for d around 4.     

A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics

We extend the work of Kurchan on the Gallavotti–Cohen fluctuation theorem, which yields a symmetry property of the large deviation function, to general Markov processes. These include jump processes describing the evolution of stochastic lattice gases driven in the bulk or through particle reservoirs, general diffusive processes in physical and/or velocity space, as well as Hamiltonian systems with stochastic boundary conditions. For dynamics satisfying local detailed balance we establish a link between the average of the action functional in the fluctuation theorem and the macroscopic entropy production. This gives, in the linear regime, an alternative derivation of the Green–Kubo formula and the Onsager reciprocity relations. In the nonlinear regime consequences of the new symmetry are harder to come by and the large deviation functional difficult to compute. For the asymmetric simple exclusion process the latter is determined explicitly using the Bethe ansatz in the limit of large N.     

A Riemann–Roch Theorem¶for One-Dimensional Complex Groupoids

We consider a smooth groupoid of the form Σ⋊Γ, where Σ is a Riemann surface and Γ a discrete pseudogroup acting on Σ by local conformal diffeomorphisms. After defining a K-cycle on the crossed product C ₀(Σ)⋊Γ generalising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using the index theorem of Connes and Moscovici. This involves in particular a generalisation of the Euler class constructed from the modular automorphism group of the von Neumann algebra L ^∞(Σ)⋊Γ. Received: 1 February 2000 / Accepted: 3 December 2000     

Levinson's Theorem for the Schrödinger Equation in One Dimension

Levinson's theorem for the one-dimensional Schrödinger equation with asymmetric potential which decays at infinity faster thanx ^–2 is established by theSturm-Liouville theorem. The critical case where the Schrödinger equation hasa finite zero-energy solution is also analyzed. It is demonstrated that the numberof bound states with even (odd) parityn ₊(n _–) is related to the phase shift ₊ (0)[_– (0)] of the scattering states with the same parity at zero momentum as ₊ (0)+ /2 =n ₊ and _– (0) =n _– for the noncritical case, and ₊ (0) =n ₊ and_– (0) – /2 =n _– for the critical case.     

A Perron–Frobenius Type of Theorem for Quantum Operations

We define a special class of quantum operations we call Markovian and show that it has the same spectral properties as a corresponding Markov chain. We then consider a convex combination of a quantum operation and a Markovian quantum operation and show that under a norm condition its spectrum has the same properties as in the conclusion of the Perron–Frobenius theorem if its Markovian part does. Moreover, under a compatibility condition of the two operations, we show that its limiting distribution is the same as the corresponding Markov chain. We apply our general results to partially decoherent quantum random walks with decoherence strength \(0 \le p \le 1\). We obtain a quantum ergodic theorem for partially decoherent processes. We show that for \(0 < p \le 1\), the limiting distribution of a partially decoherent quantum random walk is the same as the limiting distribution for the classical random walk.