Showing papers in "Probability Theory and Related Fields in 1995"

Martingale and stationary solutions for stochastic Navier-Stokes equations

Abstract: We prove the existence of martingale solutions and of stationary solutions of stochastic Navier-Stokes equations under very general hypotheses on the diffusion term. The stationary martingale solutions yield the existence of invariant measures, when the transition semigroup is well defined. The results are obtained by a new method of compactness.

Exchangeable and partially exchangeable random partitions

Abstract: Call a random partition of the positive integerspartially exchangeable if for each finite sequence of positive integersn 1,...,n k, the probability that the partition breaks the firstn 1+...+nk integers intok particular classes, of sizesn 1,...,nk in order of their first elements, has the same valuep(n 1,...,nk) for every possible choice of classes subject to the sizes constraint. A random partition is exchangeable iff it is partially exchangeable for a symmetric functionp(n 1,...nk). A representation is given for partially exchangeable random partitions which provides a useful variation of Kingman's representation in the exchangeable case. Results are illustrated by the two-parameter generalization of Ewens' partition structure.

Solution of forward-backward stochastic differential equations

Abstract: In this paper, we study the existence and uniqueness of the solution to forward-backward stochastic differential equations without the nondegeneracy condition for the forward equation. Under a certain “monotonicity” condition, we prove the existence and uniqueness of the solution to forward-backward stochastic differential equations.

Hammersley's interacting particle process and longest increasing subsequences

Abstract: In a famous paper [8] Hammersley investigated the lengthL n of the longest increasing subsequence of a randomn-permutation. Implicit in that paper is a certain one-dimensional continuous-space interacting particle process. By studying a hydrodynamical limit for Hammersley's process we show by fairly “soft” arguments that limn ′1/2 EL n =2. This is a known result, but previous proofs [14, 11] relied on hard analysis of combinatorial asymptotics.

Arbitrage possibilities in Bessel processes and their relations to local martingales

Abstract: We show that, if we allow general admissible integrands as trading strategies, the three dimensional Bessel process, Bes3, admits arbitrage possibilities. This is in contrast with the fact that the inverse process is a local martingale and hence is arbitrage free. This leads to some economic interpretation for the analysis of the property of arbitrage in foreign exchange rates. This notion (relative to general admissible integrands) does depend on the fact, which of the two currencies under consideration is chosen as numeraire. The results rely on a general construction of strictly positive local martingales. The construction is related to the Follmer measure of a positive super-martingale.

Superconvergence to the central limit and failure of the Cramér theorem for free random variables

Abstract: We show that convergence of the semicircle law in the free central limit theorem for bounded random variables is much better than expected. Thus, the distributions which tend to the semicircle become absolutely continuous in finite time, and the densities converge in a very strong sense. We also show that the semicircle law is the free convolution of laws which are not semicircular, thus proving that Cramer's classical result for the normal distribution does not have a free counterpart.

Small values of Gaussian processes and functional laws of the iterated logarithm

Abstract: We estimate small ball probabilities for locally nondeterministic Gaussian processes with stationary increments, a class of processes that includes the fractional Brownian motions. These estimates are used to prove Chung type laws of the iterated logarithm.

The scaling limit for a stochastic PDE and the separation of phases

Abstract: We investigate the problem of singular perturbation for a reaction-diffusion equation with additive noise (or a stochastic partial differential equation of Ginzburg-Landau type) under the situation that the reaction term is determined by a potential with double-wells of equal depth. As the parameter e (the temperature of the system) tends to 0, the solution converges to one of the two stable phases and consequently the phase separation is formed in the limit. We derive a stochastic differential equation which describes the random movement of the phase separation point. The proof consists of two main steps. We show that the solution stays near a manifoldM e of minimal energy configurations based on a Lyapunov type argument. Then, the limit equation is identified by introducing a nice coordinate system in a neighborhood ofM e.

Perfect cocycles through stochastic differential equations

Abstract: We prove that if ϕ is a random dynamical system (cocycle) for whicht→ϕ(t, ω)x is a semimartingale, then it is generated by a stochastic differential equation driven by a vector field valued semimartingale with stationary increment (helix), and conversely. This relation is succinctly expressed as “semimartingale cocycle=exp(semimartingale helix)”. To implement it we lift stochastic calculus from the traditional one-sided time ℝ to two-sided timeT=ℝ and make this consistent with ergodic theory. We also prove a general theorem on the perfection of a crude cocycle, thus solving a problem which was open for more than ten years.

The Brownian snake and solutions of Δu=u2 in a domain

Abstract: We investigate the connections between the path-valued process called the Brownian snake and nonnegative solutions of the partial differential equation Δu=u2 in a domain of ℝ d . In particular, we prove two conjectures recently formulated by Dynkin. The first one gives a complete characterization of the boundary polar sets, which correspond to boundary removable singularities for the equation Δu=u2. The second one establishes a one-to-one correspondence between nonnegative solutions that are bounded above by a harmonic function, and finite measures on the boundary that do not charge polar sets. This correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the Brownian snake. Our proofs combine probabilistic and analytic arguments. An important role is played by a new version of the special Markov property, which is of independent interest.

Exact large deviation bounds up toT c for the Ising model in two dimensions

Abstract: We prove an upper large deviation bound for the block spin magnetization in the 2D Ising model in the phase coexistence region. The precise rate (given by the Wulff construction) is shown to hold true for all β > βc. Combined with the lower bounds derived in [I] those results yield an exact second order large deviation theory up to the critical temperature.

Limit theorems and Markov approximations for chaotic dynamical systems

Abstract: We prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients. We also develop techniques of Markov approximations for dynamical systems. We apply our results to expanding interval maps, Axiom A diffeomorphisms, chaotic billiards and hyperbolic attractors.

Stochastic p.d.e.'s arising from the long range contact and long range voter processes

Abstract: A long range contact process and a long range voter process are scaled so that the distance between sites decreases and the number of neighbors of each site increases. The approximate densities of occupied sites, under suitable tine scaling, converge to continuous space time densities which solve stochastic p.d.e.'s. For the contact process the limiting equation is the Kolmogorov-Petrovskii-Piscuinov equation driven by branching white noise. For the voter process the limiting equation is the heat equation driven by Fisher-Wright white noise.

Large deviations for Langevin spin glass dynamics

Abstract: We study the asymptotic behaviour of asymmetrical spin glass dynamics in a Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove that the annealed law of the empirical measure on path space of these dynamics satisfy a large deviation principle in the high temperature regime. We study the rate function of this large deviation principle and prove that it achieves its minimum value at a unique probability measureQ which is not markovian. We deduce that the quenched law of the empirical measure converges to δ Q . Extending then the preceeding results to replicated dynamics, we investigate the quenched behavior of a single spin. We get quenched convergence toQ in the case of a symmetric initial law and even potential for the free spin.

Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces

Abstract: A class of stochastic evolution equations with additive noise and weakly continuous drift is considered. First, regularity properties of the corresponding Ornstein-Uhlenbeck transition semigroupRt are obtained. We show thatRt is a compactC0-semigroup in all Sobolev spacesWn,p which are built on its invariant measure μ. Then we show the existence, uniqueness, compactness and smoothing properties of the transition semigroup for semilinear equations inLp(μ) spaces and spacesW1,p. As a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to μ. It is shown also that the density of this measure with respect to μ is inLp(μ) for allp≧1.

On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE's

Abstract: We investigate asymptotic properties of the maximum likelihood estimators for parameters occurring in parabolic SPDEs of the form $$du(t,x) = (A_0 + \theta A_1 )u(t,x)dt + dW(t,x),$$ whereA 0 andA 1 are partial differential operators andW is a cylindrical Brownian motion. We introduce a spectral method for computing MLEs based on finite dimensional approximations to solutions of such systems, and establish criteria for consistency, asymptotic normality and asymptotic efficiency as the dimension of the approximation goes to infinity. We derive the asymptotic properties of the MLE from a condition on the order of the operators. In particular, the MLE is consistent if and only if ord(A 1)≧1/2(ord(A 0+θA 1)−d).

A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation

Abstract: A system ofN particles inR d with mean field interaction and diffusion is considered. Assuming adiabatic elimination of the momenta the positions satisfy a stochastic ordinary differential equation driven by Brownian sheets (microscopic equation), where all coefficients depend on the position of the particles and on the empirical mass distribution process. This empirical mass distribution process satisfies a quasilinear stochastic partial differential equation (SPDE). This SPDE (mezoscopic equation) is solved for general measure valued initial conditions by “extending” the empirical mass distribution process from point measure valued initial conditions with total mass conservation. Starting with measures with densities inL 2(R d ,dr), wheredr is the Lebesgue measure, the solution will have densities inL 2(R d ,dr) and strong uniqueness (in the Ito sense) is obtained. Finally, it is indicated how to obtain (macroscopic) partial differential equations as limits of the so constructed SPDE's.

Stationary parabolic Anderson model and intermittency

Abstract: This paper is devoted to the analysis of the large time behavior of the solutions of the Anderson parabolic problem: $$\frac{{\partial u}}{{\partial t}} = \kappa \Delta u\xi (x)u$$ when the potential ξ(x) is a homogeneous ergodic random field on ℝ d . Our goal is to prove the asymptotic spatial intermittency of the solution and for this reason, we analyze the large time properties of all the moments of the positive solutions. This provides an extension to the continuous space ℝ d of the work done originally by Gartner and Molchanov in the case of the lattice ℤ d . In the process of our moment analysis, we show that it is possible to exhibit new asymptotic regimes by considering a special class of generalized Gaussian fields, interpolating continuously between the exponent 2 which is found in the case of bona fide continuous Gaussian fields ξ(x) and the exponent 3/2 appearing in the case of a one dimensional white noise. Finally, we also determine the precise almost sure large time asymptotics of the positive solutions.

Spitzer's condition and ladder variables in random walks

Abstract: Spitzer's condition holds for a random walk if the probabilities ρ n =P{ n > 0} converge in Cesaro mean to ϱ, where 0<ϱ<1. We answer a question which was posed both by Spitzer [12] and by Emery [5] by showing that whenever this happens, it is actually true that ρn converges to ϱ. This also enables us to give an improved version of a result in Doney and Greenwood [4], and show that the random walk is in a domain of attraction, without centering, if and only if the first ladder epoch and height are in a bivariate domain of attraction.

A new approach to the single point catalytic super-Brownian motion

Abstract: A new approach is provided to the super-Brownian motionX with a single point-catalyst δc as branching rate. We start from a superprocessU with constant branching rate and spatial motion given by the 1/2-stable subordinator. We prove that the occupation density measure λc ofX at the catalystc is distributed as the total occupation time measure ofU. Furthermore, we show thatXt is determined from λc by an explicit representation formula. Heuristically, a mass λc(ds) of “particles” leaves the catalyst at times and then evolves according to Ito's Brownian excursion measure. As a consequence of our representation formula, the density fieldx ofX satisfies the heat equation outside ofc, with a noisy boundary condition atc given by the singularly continuous random measure λc. In particular,x isC outside the catalyst. We also provide a new derivation of the singularity of the measure λc.

The diffusive phase of a model of self-interacting walks

Abstract: We consider simple random walk onZ d perturbed by a factor exp[βT −P J T], whereT is the length of the walk and $$J_T = \sum olimits_{0 \leqslant i< j \leqslant T} \delta _{\omega (i),\omega (j)} $$ . Forp=1 and dimensionsd≥2, we prove that this walk behaves diffusively for all − ∞ 0. Ford>2 the diffusion constant is equal to 1, but ford=2 it is renormalized. Ford=1 andp=3/2, we prove diffusion for all real β (positive or negative). Ford>2 the scaling limit is Brownian motion, but ford≤2 it is the Edwards model (with the “wrong” sign of the coupling when β>0) which governs the limiting behaviour; the latter arises since for $$p = \frac{{4 - d}}{2}$$ ,T −p J T is the discrete self-intersection local time. This establishes existence of a diffusive phase for this model. Existence of a collapsed phase for a very closely related model has been proven in work of Bolthausen and Schmock.

A general duality theorem for marginal problems

Abstract: Given probability spaces (Xi,Ai,Pi),i=1, 2 letM(P1,P2) denote the set of all probabilities on the product space with marginalsP1 andP2 and leth be a measurable function on (X1×X2,A1 ⊗A2). In order to determine supfh dP where the supremum is taken overP inM(P1,P2), a general duality theorem is proved. Only the perfectness of one of the coordinate spaces is imposed without any further topological or tightness assumptions. An example without any further topological or tightness assumptions. An example is given to show that the assumption of perfectness is essential. Applications to probabilities with given marginals and given supports, stochastic order and probability metrics are included.

Sharp estimates for capacities and applications to symmetric diffusions

Abstract: We give sharp estimates for the capacity of compact setF⊂X whereX is the state space of a strongly regular Dirichlet form of diffusion type. These estimates are in terms of the reference measurem and the Caratheodory metric ρ onX. For instance, $$Cap_0 F \leqq \left( {\mathop \smallint \limits_0^\infty \frac{{dr}}{{v'(r)}}} \right)^{ - 1} \leqq 2 \left( {\mathop \smallint \limits_0^\infty \frac{{r dr}}{{v(r)}}} \right)^{ - 1} ,$$ wherev(r)=m({0<ρ(x, F)

Large time behavior of interface solutions to the heat equation with Fisher-Wright white noise

Abstract: The one-dimensional heat equation driven by Fisher-Wright white noise is studied. From initial conditions with compact support, solutions retain this compact support and die out in finite time. There exist interface solutions which change from the value 1 to the value 0 in a finite region. The motion of the interface location is shown to approach that of a Brownian motion under rescaling. Solutions with a finite number of interfaces are approximated by a system of annihilating Brownian motions.

Small ball probabilities of Gaussian fields

Abstract: Lower bounds on the small ball probability are given for Brownian sheet type Gaussian fields as well as for general Gaussian fields with stationary increments in ℝ d . In particular, a sharp bound is found for the fractional Levy Brownian fields.

Random rotations of the Wiener path

Abstract: Let (W, H, μ) be an abstract Wiener space and letR(w) be a strongly measurable random variable with values in the set of isometries onH. Suppose that ∇Rh is smooth in the Sobolev sense and that it is a quasi-nilpotent operator onH for everyh∈H. It is shown that δ(R(w)h) is again a Gaussian (0, |h| 2 )-random variable. Consequently, if (e i ,i∈ℕ)⊂W * is a complete, orthonormal basis ofH, then $$\tilde w = \sum olimits_i {(\delta R(w)e_i )e_i } $$ defines a measure preserving transformation, a “rotation”, onW. It is also shown that if for some strongly measurable, operator valued (onH) random variableR, δ(R(w+k)h) is (0, |h| 2 )-Gaussian for allk, h∈H, thenR is an isometry and ∇Rh is quasi-nilpotent for allH∈H. The relation between the stochastic calculi for these Wiener pathsw and $$\tilde w$$ , as well as the conditions of the inverbibility of the map $$w \to \tilde w$$ are discussed and the problem of the absolute continuity of the image of the Wiener measure μ under Euclidean motion on the Wiener space (i.e. $$w \to \tilde w$$ composed with a shift) is studied.

On large deviations for particle systems associated with spatially homogeneous Boltzmann type equations

Abstract: We consider a dynamical interacting particle system whose empirical distribution tends to the solution of a spatially homogeneous Boltzmann type equation, as the number of particles tends to infinity. These laws of large numbers were proved for the Maxwellian molecules by H. Tanaka [Tal] and for the hard spheres by A.S. Sznitman [Szl]. In the present paper we investigate the corresponding large deviations: the large deviation upper bound is obtained and, using convex analysis, a non-variational formulation of the rate function is given. Our results hold for Maxwellian molecules with a cutoff potential and for hard spheres.

Markov properties of multiparameter processes and capacities

Abstract: We study a class of multiparameter symmetric Markov processes We prove that this class is stable by subordination in Bochner's sense We show then that for these processes, a probabilistic and an analytic potential theory correspond to each other In particular, additive functionals are associated with finite energy measures, hitting probabilities are estimated by capacities, quasicontinuity corresponds to path-continuity In the last section, examples show that many earlier results, as well as new ones, in this domain can be obtained by our method

Principal component decomposition of non-parametric tests

Abstract: Let ϕ denote an arbitrary non-parametric unbiased test for a Gaussian shift given by an infinite dimensional parameter space. Then it is shown that the curvature of its power function has a principal component decomposition based on a Hilbert-Schmidt operator. Thus every test has reasonable curvature only for a finite number of orthogonal directions of alternatives. As application the two-sided Kolmogorov-Smirnov goodnessof-fit test is treated. We obtain lower bounds for their local asymptotic relative efficiency. They converge to one as α↓0 for the directionh0(u)=sign(2u−1) of the gradient of the median test. These results are analogous to earlier results of Hajek and Sidak for one-sided Kolmogorov-Smirnov tests.

On some growth models with a small parameter

Abstract: We consider the behavior of the asymptotic speed of growth and the asymptotic shape in some growth models, when a certain parameter becomes small. The basic example treated is the variant of Richardson's growth model on ℤd in which each site which is not yet occupied becomes occupied at rate 1 if it has at least two occupied neighbors, at rate ɛ≦1 if it has exactly 1 occupied neighbor and, of course, at rate 0 if it has no occupied neighbor. Occupied sites remain occupied forever. Starting from a single occupied site, this model has asymptotic speeds of growth in each direction (as time goes to infinity) and these speeds determine an asymptotic shape in the usual sense. It is proven that as ɛ tends to 0, the asymptotic speeds scale as ɛ1/d and the asymptotic shape, when renormalized by dividing it by ɛ1/d , converges to a cube. Other similar models which are partially oriented are also studied.