Relative entropy and mixing properties of infinite dimensional diffusions

Summary. Let η be a diffusion process taking values on the infinite dimensional space T ^Z , where T is the circle, and with components satisfying the equations dη _i =σ _i (η) dW _i +b _i (η) dt for some coefficients σ _i and b _i , i∈Z. Suppose we have an initial distribution μ and a sequence of times t _n →∞ such that lim _{n →∞}μS _tn =ν exists, where S _t is the semi-group of the process. We prove that if σ _i and b _i are bounded, of finite range, have uniformly bounded second order partial derivatives, and inf _{i ,η}σ _i (η)>0, then ν is invariant. Received: 12 September 1996 / In revised form: 10 November 1997     

Stability of infinite clusters in supercritical percolation

. A recent theorem by Häggström and Peres concerning independent percolation is extended to all the quasi-transitive graphs. This theorem states that if 0<p ₁<p ₂≤1 and percolation occurs at level p ₁, then every infinite cluster at level p ₂ contains some infinite cluster at level p ₁. Consequences are the continuity of the percolation probability above the percolation threshold and the monotonicity of the uniqueness of the infinite cluster, i.e., if at level p ₁ there is a unique infinite cluster then the same holds at level p ₂. These results are further generalized to graphs with a “uniform percolation” property. The threshold for uniqueness of the infinite cluster is characterized in terms of connectivities between large balls.     

Recurrence and ergodicity of interacting particle systems

Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of their ergodic invariant measures. The question arises whether a.s.the process eventually stays close to one of these ergodic states, or if it changes between the attainable ergodic states infinitely often (“recurrence”). Under the assumption that there exists a convergence–determining class of distributions that is (strongly) preserved under the dynamics, we show that the system is in fact recurrent in the above sense. We apply our method to several interacting particle systems, obtaining new or improved recurrence results. In addition, we answer a question raised by Ed Perkins concerning the change of the locally predominant type in a model of mutually catalytic branching. Received: 22 January 1999 / Revised version: 24 May 1999     

Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously

. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p ₂>p ₁>p _c , each infinite p ₂-cluster contains an infinite p ₁-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that the probability θ _v (p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>p _c . All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group. Received: 22 December 1997 / Revised version: 1 July 1998     

Uniqueness of Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ d

Summary. Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ ^d are studied. Introducing a linear operator L ₀ defined on a space of smooth local functions, we show the uniqueness of Dirichlet forms associated with self adjoint Markovian extensions of L ₀. We also discuss the ergodicity of the reversible process associated with the Dirichlet form. Received: 18 July 1996/In revised form: 13 February 1997     

Branching random walks and contact processes on homogeneous trees

Summary. Branching random walks and contact processes on the homogeneous tree in which each site has d+1 neighbors have three possible types of behavior (for d≧ 2): local survival, local extinction with global survival, and global extinction. For branching random walks, we show that if there is local extinction, then the probability that an individual ever has a descendent at a site n units away from that individual’s location is at most d ^{− n/2} , while if there is global extinction, this probability is at most d ⁻ⁿ . Next, we consider the structure of the set of invariant measures with finite intensity for the system, and see how this structure depends on whether or not there is local and/or global survival. These results suggest some problems and conjectures for contact processes on trees. We prove some and leave others open. In particular, we prove that for some values of the infection parameter λ, there are nontrivial invariant measures which have a density tending to zero in all directions, and hence are different from those constructed by Durrett and Schinazi in a recent paper. Received: 26 April 1996/In revised form: 20 June 1996     

Convection–diffusion equation with space–time ergodic random flow

We prove the homogenization of convection-diffusion in a time-dependent, ergodic, incompressible random flow which has a bounded stream matrix and a constant mean drift. We also prove two variational formulas for the effective diffusivity. As a consequence, we obtain both upper and lower bounds on the effective diffusivity. Received: 17 December 1996/Revised revision: 9 February 1998     

Positivity of the self-diffusion matrix of interacting Brownian particles with hard core

We prove the positivity of the self-diffusion matrix of interacting Brownian particles with hard core when the dimension of the space is greater than or equal to 2. Here the self-diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle. We will do this for all activities, z>0, of Gibbs measures; in particular, for large z– the case of high density particles. A typical example of such a particle system is an infinite amount of hard core Brownian balls. Received: 22 September 1997 / Revised version: 15 January 1998     

A generalized two dimensional Emden-Fowler equation with exponential nonlinearity

In this paper we study the asymptotic behavior of two-dimensional equation in divergence form with exponential nonlinearity. We discuss also the construction of singular limits in symmetric case. Received: 27 October 1999 / Accepted: 17 July 2000 / Published online: 23 April 2001     

The number of infinite clusters in dynamical percolation

Summary. Dynamical percolation is a Markov process on the space of subgraphs of a given graph, that has the usual percolation measure as its stationary distribution. In previous work with O. H?ggstr?m, we found conditions for existence of infinite clusters at exceptional times. Here we show that for ℤ ^d , with p>p _c , a.s. simultaneously for all times there is a unique infinite cluster, and the density of this cluster is θ(p). For dynamical percolation on a general tree Γ, we show that for p>p _c , a.s. there are infinitely many infinite clusters at all times. At the critical value p=p _c , the number of infinite clusters may vary, and exhibits surprisingly rich behaviour. For spherically symmetric trees, we find the Hausdorff dimension of the set T ^k of times where the number of infinite clusters is k, and obtain sharp capacity criteria for a given time set to intersect T ^k . The proof of this capacity criterion is based on a new kernel truncation technique. Received: 5 May 1997 / In revised form: 24 November 1997     

On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order (nonlinear) coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form on D are comparable when one modifies the drift term b on a compact subset of D. This generalizes a previous result of Ancona [2], obtained under an condition on b, to a Kato condition on . Received: 21 April 1998 / in final form 26 March 1999     

Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory

We study a perturbed semilinear problem with Neumann boundary condition where is a bounded smooth domain of , , , if or if and is the unit outward normal at the boundary of . We show that for any fixed positive integer K any “suitable” critical point of the function generates a family of multiple interior spike solutions, whose local maximum points tend to as tends to zero. Received March 7, 1999 / Accepted October 1, 1999 / Published online April 6, 2000     

Some new estimates for solutions of degenerate two dimensional Monge-Ampère equations

We derive a monotonicity formula for smooth solutions u of degenerate two dimensional Monge-Ampère equations, and use this to obtain a local H?lder gradient estimate, depending on for some . Received August 9, 1999; in final form December 8, 1999/ Published online December 8, 2000     

Dependent percolation in two dimensions

For a natural number k, define an oriented site percolation on ℤ² as follows. Let x _i , y _j be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ² closed if x _i = y _j , and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i ₀, j ₀)(i ₁, j ₁) · · · such that 0 = i ₀≤i ₁≤· · ·, 0 = j ₀≤j ₁≤· · ·, and each site (i _n , j _n ) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x _i ) and (y _j ). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods. Received: 4 March 1999 / Revised version: 27 September 1999 / Published online: 21 June 2000     

Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group

Denoting Δ_? the Laplacian operator on the (2N+1)-dimensional Heisenberg group ? ^N , we prove some nonexistence results for solutions of inequalities of the three types

A family of diffusion processes on Sierpinski carpets

We construct a family of diffusions P ^α = {P ^x} on the d-dimensional Sierpinski carpet F^. The parameter α ranges over d _H < α < ∞, where d _H = log(3 ^d − 1)/log 3 is the Hausdorff dimension of the d-dimensional Sierpinski carpet F^. These diffusions P ^α are reversible with invariant measures μ = μ^[α]. Here, μ are Radon measures whose topological supports are equal to F^ and satisfy self-similarity in the sense that μ(3A) = 3^α·μ(A) for all A∈ℬ(F^). In addition, the diffusion is self-similar and invariant under local weak translations (cell translations) of the Sierpinski carpet. The transition density p = p(t, x, y) is locally uniformly positive and satisfies a global Gaussian upper bound. In spite of these well-behaved properties, the diffusions are different from Barlow-Bass' Brownian motions on the Sierpinski carpet. Received: 30 September 1999 / Revised version: 15 June 2000 / Published online: 24 January 2000