Results 1  10
of
63
Entropic Repulsion of the Lattice Free Field
"... Consider the massless free field on the ddimensional lattice Z d ; d 3; that is the centered Gaussian field on R Z d with covariances given by the Green function of the simple random walk on Z d . We show that the probability, that all the spins are positive in a box of volume N d , decay ..."
Abstract

Cited by 20 (4 self)
 Add to MetaCart
Consider the massless free field on the ddimensional lattice Z d ; d 3; that is the centered Gaussian field on R Z d with covariances given by the Green function of the simple random walk on Z d . We show that the probability, that all the spins are positive in a box of volume N d , decays exponentially at a rate of order N d\Gamma2 log N and compute explicitly the corresponding constant in terms of the capacity of the unit cube. The result is extended to a class of transient random walks with transition functions in the domain of the normal and ff stable law.
A handbook of Γconvergence
 in “Handbook of Differential Equations – Stationary Partial Differential Equations
"... The notion of Γconvergence has become, over the more than thirty years after its introduction by Ennio De Giorgi, the commonlyrecognized notion of convergence for variational problems, and it would be difficult nowadays to think of any other ‘limit ’ than a Γlimit when talking about asymptotic an ..."
Abstract

Cited by 15 (9 self)
 Add to MetaCart
The notion of Γconvergence has become, over the more than thirty years after its introduction by Ennio De Giorgi, the commonlyrecognized notion of convergence for variational problems, and it would be difficult nowadays to think of any other ‘limit ’ than a Γlimit when talking about asymptotic analysis in a general variational setting (even though special convergences may fit better specific problems, as Moscoconvergence, twoscale convergence, G and Hconvergence, etc.). This short presentation is meant as an introduction to the many applications of this theory to problems in Partial Differential Equations, both as an effective method for solving asymptotic and approximation issues and as a means of expressing results that are derived by other techniques. A complete introduction to the general theory of Γconvergence is the bynowclassical book by Gianni Dal Maso [85], while a userfriendly introduction can be found in my book ‘for beginners ’ [46], where also simplified onedimensional versions of many of the problems in this article are treated. These notes are addressed to an audience of experienced mathematicians, with some background and interest in Partial Differential Equations, and are meant to direct the reader to what I regard as the most interesting features of this theory. The style of the exposition is how I would present the subject to a colleague in a neighbouring field or to an interested PhD student: the issues that I think
Quasiregular Dirichlet forms: Examples and counterexamples
, 1993
"... We prove some new results on quasiregular Dirichlet forms. These include results on perturbations of Dirichlet forms, change of speed measure, and tightness. The tightness implies the existence of an associated right continuous strong Markov process. We also discuss applications to a number of exam ..."
Abstract

Cited by 14 (7 self)
 Add to MetaCart
We prove some new results on quasiregular Dirichlet forms. These include results on perturbations of Dirichlet forms, change of speed measure, and tightness. The tightness implies the existence of an associated right continuous strong Markov process. We also discuss applications to a number of examples including cases with possibly degenerate (sub)elliptic part, diffusions on loops spaces, and certain FlemingViot processes.
Turbulent diffusion in Markovian flows
 Ann. Appl. Probab
, 1999
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
Abstract

Cited by 14 (10 self)
 Add to MetaCart
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
The Art of Random Walks
 Lecture Notes in Mathematics 1885
, 2006
"... 1.1 Basic definitions and preliminaries................ 8 ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
1.1 Basic definitions and preliminaries................ 8
Reversible Markov Chains
, 1994
"... ly, call f : [0; 1) ! [0; 1) completely monotone (CM) if there is a nonnegative measure on [0; 1) such that f(t) = Z 1 0 e \Gamma`t (d`); 0 t ! 1: (41) Our applications will use only the special case of a finite sum f(t) = X m am e \Gamma` m t ; for some am ; ` m 0: (42) but finiten ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
ly, call f : [0; 1) ! [0; 1) completely monotone (CM) if there is a nonnegative measure on [0; 1) such that f(t) = Z 1 0 e \Gamma`t (d`); 0 t ! 1: (41) Our applications will use only the special case of a finite sum f(t) = X m am e \Gamma` m t ; for some am ; ` m 0: (42) but finiteness plays no essential role. If f is CM then (provided they exist) so are \Gammaf 0 (t) F (t) j Z 1 t f(s)ds (43) A probability distribution on [0; 1) is called CM if its tail distribution function F (t) = (t; 1) is CM; equivalently, if its density function f is CM (except here we must in the general case allow the possibility f(0) = 1). In more probabilistic language, is CM iff it can be expressed as the distribution of =, where and are independent random variables such that has exponential(1) distribution; ? 0: (44) 19 Given a CM function or distribution, the spectral gap 0 can be defined consistently by = infft ? 0 : [0; t] ? 0g in setting (41) = minf`m g in setting...
Generalized Mehler semigroups and applications
, 1994
"... We construct and study generalized Mehler semigroups (p t ) t#0 and their associated Markov processes M. The construction methods for (p t ) t#0 are based on some new purely functional analytic results implying, in particular, that any strongly continuous semigroup on a Hilbert space H can be extend ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
We construct and study generalized Mehler semigroups (p t ) t#0 and their associated Markov processes M. The construction methods for (p t ) t#0 are based on some new purely functional analytic results implying, in particular, that any strongly continuous semigroup on a Hilbert space H can be extended to some larger Hilbert space E, with the embedding H # E being HilbertSchmidt. The same analytic extension results are applied to construct strong solutions to stochastic differential equations of type dX t = CdW t + AX t dt (with possibly unbounded linear operators A and C on H) on a suitably chosen larger space E. For Gaussian generalized Mehler semigroups (p t ) t#0 with corresponding Markov process M, the associated (nonsymmetric) Dirichlet forms (E , D(E)) are explicitly calculated and a necessary and sufficient condition for path regularity of M in terms of (E , D(E)) is proved. Then, using Dirichlet form methods it is shown that M weakly solves the above stochastic differential ...
Sch’nol’s theorem for strongly local forms
, 2009
"... We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions. ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions.
Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets
 J. Noncommut. Geom
"... Abstract. An analogue of the Riemannian Geometry for an ultrametric Cantor set (C, d) is described using the tools of Noncommutative Geometry. Associated with (C, d) is a weighted rooted tree, its Michon tree [28]. This tree allows to define a family of spectral triples (CLip(C), H, D) using the ℓ 2 ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
Abstract. An analogue of the Riemannian Geometry for an ultrametric Cantor set (C, d) is described using the tools of Noncommutative Geometry. Associated with (C, d) is a weighted rooted tree, its Michon tree [28]. This tree allows to define a family of spectral triples (CLip(C), H, D) using the ℓ 2space of its vertices, giving the Cantor set the structure of a noncommutative Riemannian manifold. Here CLip(C) denotes the space of Lipschitz continuous functions on (C, d). The family of spectral triples is indexed by the space of choice functions which is shown to be the analogue of the sphere bundle of a Riemannian manifold. The Connes metric coming from the Dirac operator D then allows to recover the metric on C. The corresponding ζfunction is shown to have abscissa of convergence, s0, equal to the upper box dimension of (C, d). Taking the residue at this singularity leads to the definition of a canonical probability measure on C which in certain cases coincides with the Hausdorff measure at dimension s0. This measure in turns induces a measure on the space of choices. Given a choice, the commutator of D with
On the Local Property for Positivity Preserving Coercive Forms
, 1995
"... . We show that, under mild conditions, two wellknown definitions for the local property of a Dirichlet form are equivalent. We also show that forms that come from di#erential operators are local. 1991 AMS Subject Classification: 31C25 The purpose of this paper is to clarify the relationship between ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
. We show that, under mild conditions, two wellknown definitions for the local property of a Dirichlet form are equivalent. We also show that forms that come from di#erential operators are local. 1991 AMS Subject Classification: 31C25 The purpose of this paper is to clarify the relationship between two di#erent notions of locality that have appeared in the literature of Dirichlet forms. The first is a slightly modified version of the definition of locality found in the book of Bouleau and Hirsch [BH 91; Chapter I, Corollary 5.1.4], while the second comes from the book of Ma and Rockner [MR 92; Chapter V, Proposition 1.2]. But here we do not assume that the form satisfies any normal contraction property, but only that it is positivity preserving (see Definition 0.1 below). Let (E, F , m) be a measure space, and suppose (E , D(E)) is a densely defined, closed, bilinear form on L 2 (E, F , m). Following [MR 92], we call such a form (E , D(E)) coercive if E(u, u) # 0 for all u ...