Excessive kernels and Revuz measures

We consider a proper submarkovian resolvent of kernels on a Lusin measurable space and a given excessive measure ξ. With every quasi bounded excessive function we associate an excessive kernel and the corresponding Revuz measure. Every finite measure charging no ξ–polar set is such a Revuz measure, provided the hypothesis (B) of Hunt holds. Under a weak duality hypothesis, we prove the Revuz formula and characterize the quasi boundedness and the regularity in terms of Revuz measures. We improve results of Azéma [2] and Getoor and Sharpe [20] for the natural additive functionals of a Borel right process. Received: 30 April 1997 / Revised version: 17 September 1999 /?Published online: 11 April 2000     

Almost sure asymptotics for the continuous parabolic Anderson model

We consider the parabolic Anderson problem ∂ _t u = κΔu + ξ(x)u on ℝ₊×ℝ ^d with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order term of the almost sure asymptotics of u(t, 0) as t→∞. Received: 26 July 1999 / Revised version: 6 April 2000 / Published online: 22 November 2000     

The log-Sobolev inequality for weakly coupled lattice fields

We consider a ferromagnetic spin system with unbounded interactions on the d-dimensional integer lattice (d > 1). Under mild assumptions on the one-body interactions (so that arbitrarily deep double wells are allowed), we prove that if the coupling constants are small enough, then the finite volume Gibbs states satisfy the log-Sobolev inequality uniformly in the volume and the boundary condition. Received: 11 November 1997 / Revised version: 17 July 1998     

Discrete schemes for processes in random media

We present discrete schemes for processes in random media. We prove two results. The first one is the convergence of Sinai's random walks in random environments to the Brox model. The second one is the convergence of random walks in media with random “gates” to a continuous process in a Poisson potential. The proofs are based on the following idea: we consider the discrete media as random potentials for continuous models. Received: 6 May 1999 / Revised version: 18 October 1999 / Published online: 20 October 2000     

Elliptic equations for measures on infinite dimensional spaces and applications

We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models. Received: 23 January 2000 / Revised version: 4 October 2000 / Published online: 5 June 2001     

Lifschitz tail in a magnetic field: coexistence of classical and quantum behavior in the borderline case

We establish the exact low-energy asymptotics of the integrated density of states (Lifschitz tail) in a homogeneous magnetic field and Poissonian impurities with a repulsive single-site potential of Gaussian decay. It has been known that the Gaussian potential tail discriminates between the so-called “classical” and “quantum” regimes, and precise asymptotics are known in these cases. For the borderline case, the coexistence of the classical and quantum regimes was conjectured. Here we settle this last remaining open case to complete the full picture of the magnetic Lifschitz tails. Received: 28 March 2000 / Revised version: 22 December 2000 / Published online: 24 July 2001     

Equilibrium fluctuations of asymmetric simple exclusion processes in dimension d≥3

We consider an asymmetric exclusion process in dimension d≥ 3 under diffusive rescaling starting from the Bernoulli product measure with density 0 < α < 1. We prove that the density fluctuation field Y ^N _t converges to a generalized Ornstein–Uhlenbeck process, which is formally the solution of the stochastic differential equatin dY _t = ?Y _t dt + dB ^∇ _t , where ? is a second order differential operator and B ^∇ _t is a mean zero Gaussian field with known covariances. Received: 31 May 1999 / Revised version: 15 June 2000 / Published online: 24 January 2001     

The characterizations of a semicircle law by the certain freeness in a C*-probability space

In usual probability theory, various characterizations of the Gaussian law have been obtained. For instance, independence of the sample mean and the sample variance of independently identically distributed random variables characterizes the Gaussian law and the property of remaining independent under rotations characterizes the Gaussian random variables. In this paper, we consider the free analogue of such a kind of characterizations replacing independence by freeness. We show that freeness of the certain pair of the linear form and the quadratic form in freely identically distributed noncommutative random variables, which covers the case for the sample mean and the sample variance, characterizes the semicircle law. Moreover we give the alternative proof for Nica's result that the property of remaining free under rotations characterizes a semicircular system. Our proof is more direct and straightforward one. Received: 12 February 1997 / Revised version: 16 June 1998     

Local limit theorems for transition densities of Markov chains converging to diffusions

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Local limit theorems for transition densities are proved. Received: 28 August 1998 / Revised version: 6 September 1999 / Published online: 14 June 2000     

On the distributions of the lengths of the longest monotone subsequences in random words

We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. (In the limit as k→∞ these become the corresponding distributions for permutations on N letters.) We find Toeplitz determinant representations for the exponential generating functions (on N) of these distribution functions and show that they are expressible in terms of solutions of Painlevé V equations. We show further that in the weakly increasing case the generating unction gives the distribution of the smallest eigenvalue in the k×k Laguerre random matrix ensemble and that the distribution itself has, after centering and normalizing, an N→∞ limit which is equal to the distribution function for the largest eigenvalue in the Gaussian Unitary Ensemble of k×k hermitian matrices of trace zero. Received: 9 September 1999 / Revised version: 24 May 2000 / Published online: 24 January 2001     

On homogenization of time-dependent random flows

We study a diffusion with a random, time dependent drift. We prove the invariance principle when the spectral measure of the drift satisfies a certain integrability condition. This result generalizes the results of [13, 7]. Received: 25 February 2000 / Revised version: 11 December 2000 /?Published online: 14 June 2001     

Singular perturbations of some nonlinear problems

We deal with singular perturbations of nonlinear problems depending on a small parameter ε > 0. First we consider the abstract theory of singular perturbations of variational inequalities involving some nonlinear operators, defined in Banach spaces, and describe the asymptotic behavior of these solutions as ε → 0. Then these abstract results are applied to some boundary value problems. Bibliography: 15 titles.     

Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media

The Central Limit Theorem for a model of discrete-time random walks on the lattice ℤ^ν in a fluctuating random environment was proved for almost-all realizations of the space-time nvironment, for all ν > 1 in [BMP1] and for all ν≥ 1 in [BBMP]. In [BMP1] it was proved that the random correction to the average of the random walk for ν≥ 3 is finite. In the present paper we consider the cases ν = 1,2 and prove the Central Limit Theorem as T→∞ for the random correction to the first two cumulants. The rescaling factor for theaverage is for ν = 1 and (ln T), for ν=2; for the covariance it is , ν = 1,2. Received: 25 November 1999 / Revised version: 7 June 2000 / Published online: 15 February 2001     

Contour lines of the two-dimensional discrete Gaussian free field

We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant λ > 0 such that when h is an interpolation of the discrete Gaussian free field on a Jordan domain—with boundary values −λ on one boundary arc and λ on the complementary arc—the zero level line of h joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are −a < 0 on the first arc and b > 0 on the complementary arc, then the convergence is to SLE(4; a/λ - 1, b/λ - 1), a variant of SLE(4).     

Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations

We consider adaptive estimating the value of a linear functional from indirect white noise observations. For a flexible approach, the problem is embedded in an abstract Hilbert scale. We develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collection of balls in the Hilbert scale. It is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functionals. The case of discretized indirect white noise observations is studied, and the adaptive estimator in this setting is developed. Received: 5 May 1999 / Revised version: 25 October 1999 / Published online: 5 September 2000     

Instability,complexity, and evolution

In this paper, we consider a new class of random dynamical systems that contains, in particular, neural networks and complicated circuits. For these systems, we consider the viability problem: we suppose that the system survives only the system state is in a prescribed domain Π of the phase space. The approach developed here is based on some fundamental ideas proposed by A. Kolmogorov, R. Thom, M. Gromov, L. Valiant, L. Van Valen, and others. Under some conditions it is shown that almost all systems from this class with fixed parameters are unstable in the following sense: the probability P _t to leave Π within the time interval [0, t] tends to 1 as t → ∞. However, it is allowed to change these parameters sometimes (“evolutionary” case), then it may happen that P _t  < 1 − δ  < 1 for all t (“stable evolution”). Furthermore, we study the properties of such a stable evolution assuming that the system parameters are encoded by a dicsrete code. This allows us to apply complexity theory, coding, algorithms, etc. Evolution is a Markov process of modification of this code. Under some conditions we show that the stable evolution of unstable systems possesses the following general fundamental property: the relative Kolmogorov complexity of the code cannot be bounded by a constant as t → ∞. For circuit models, we define complexity characteristics of these circuits. We find that these complexities also have a tendency to increase during stable evolution. We give concrete examples of stable evolution. Bibliography: 80 titles. To the memory of A. N. Livshitz Published in Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 31–69.     

Mean field bounds on Lyapunov exponents in Z d at the critical energy

We further develop the supersymmetric formalism initiated in [W1] (see also [SjW]). We obtain the optimal mean field bounds at the critical energy for Lyapunov exponents of random walks in random potentials in Z ^d at weak disorder. This extends some of the results in [W1]. Received: 9 December 1999 / Revised version: 8 May 2000 /?Published online: 15 February 2001     

Fractional Normal Inverse Gaussian Process

Normal inverse Gaussian (NIG) process was introduced by Barndorff-Nielsen (Scand J Statist 24:1–13, 1997) by subordinating Brownian motion with drift to an inverse Gaussian process. Increments of NIG process are independent and are stationary. In this paper, we introduce dependence between the increments of NIG process, by subordinating fractional Brownian motion to an inverse Gaussian process and call it fractional normal inverse Gaussian (FNIG) process. The basic properties of this process are discussed. Its marginal distributions are scale mixtures of normal laws, infinitely divisible for the Hurst parameter 1/2 ≤ H < 1 and are heavy tailed. First order increments of the process are stationary and possess long-range dependence (LRD) property. It is shown that they have persistence of signs LRD property also. A generalization to an n-FNIG process is also discussed, which allows Hurst parameter H in the interval (n − 1, n). Possible applications to mathematical finance and hydraulics are also pointed out.     

Stochastic functional differential equations on manifolds

In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters. Received: 6 July 1999 / Revised version: 19 April 2000 /?Published online: 14 June 2001     

A note on the decay of correlations under δ-pinning

We prove that for a class of massless ∇φ interface models on ℤ², introduction of an arbitrarily small pinning self-potential leads to exponential decay of correlation, or, in other words, to creation of mass. Received: 23 November 1998 / Revised version: 14 June 1999