Diffusion equation techniques in stochastic monotonicity and positive correlations

Herbst, Ira; Pitt, Loren D.

doi:10.1007/bf01312211

Cited by 48 publications

(51 citation statements)

References 27 publications

Supporting

Mentioning

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Order By: Relevance

“…Since FKG inequality is used several times, we recall that, as a consequence of Corollary 1.7 in [11], measures of the form …”

Section: Appendix: Proofs Of Some Technical Estimatesmentioning

confidence: 99%

Non-Gaussian surface pinned by a weak potential

Deuschel¹,

Velenik²

2000

Probab Theory Relat Fields

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Abstract. We consider a model of a two-dimensional interface of the (continuous) SOS type, with finite-range, strictly convex interactions. We prove that, under an arbitrarily weak pinning potential, the interface is localized. We consider the cases of both square well and δ potentials. Our results extend and generalize previous results for the case of nearest-neighbours Gaussian interactions in [7] and [1]. We also obtain the tail behaviour of the height distribution, which is not Gaussian.

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“…Since FKG inequality is used several times, we recall that, as a consequence of Corollary 1.7 in [11], measures of the form …”

Section: Appendix: Proofs Of Some Technical Estimatesmentioning

confidence: 99%

Non-Gaussian surface pinned by a weak potential

Deuschel¹,

Velenik²

2000

Probab Theory Relat Fields

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“…We sum-up the obtained results in the following statement: 15) and for every α + > 4G, α − < 4G and > 0, (1.14) holds P(dσ )-a.s.. 16) and for every α + > 4Q, α − < 4Q and > 0…”

Section: Super/sub-gaussian σ + Tailsmentioning

confidence: 99%

Enhanced interface repulsion from quenched hard–wall randomness

Bertacchi¹,

Giacomin²

2002

Probab Theory Relat Fields

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We consider the harmonic crystal, or massless free field, ϕ = {ϕ x } x∈Z d , d ≥ 3, that is the centered Gaussian field with covariance given by the Green function of the simple random walk on Z d . Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition ϕ x to be larger than σ x , σ = {σ x } x∈Z d is an IID field (which is also independent of ϕ), for every x in a large region D N = ND ∩ Z d , with N a positive integer and D a bounded subset of R d . We are mostly motivated by results for given typical realizations of σ (quenched set-up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, living in a (d + 1)-dimensional space, constrained not to go below an inhomogeneous substrate that acts as a hard wall. We consider various types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of σ 0 is heavier than Gaussian, while essentially no effect is observed if the tail is sub-Gaussian. In the critical case, that is the one of approximately Gaussian tail, the interplay of the two sources of randomness, ϕ and σ , leads to an enhanced repulsion effect of additive type. This generalizes work done in the case of a flat wall and also in our case the crucial estimates are optimal Large Deviation type asymptotics as N ∞ of the probability that ϕ lies above σ in D N .

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“…Massey (1987), Herbst and Pitt (1991), Chen and Wang (1993), Chen (2004), and Daduna and Szekli (2006) described stochastic ordering for discrete-state spaces, for diffusions, and for diffusions with jumps in terms of generators. For bounded generators and in the case of discrete-state spaces, Daduna and Szekli (2006) gave a comparison result for stochastic ordering in terms of a comparison of the generators.…”

Section: Introductionmentioning

confidence: 99%

On a Comparison Result for Markov Processes

Rüschendorf

2008

J. Appl. Probab.

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A comparison theorem is stated for Markov processes in Polish state spaces. We consider a general class of stochastic orderings induced by a cone of real functions. The main result states that stochastic monotonicity of one process and comparability of the infinitesimal generators imply ordering of the processes. Several applications to convex type and to dependence orderings are given. In particular, Liggett's theorem on the association of Markov processes is a consequence of this comparison result.

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Diffusion equation techniques in stochastic monotonicity and positive correlations

Cited by 48 publications

References 27 publications

Non-Gaussian surface pinned by a weak potential

Non-Gaussian surface pinned by a weak potential

Enhanced interface repulsion from quenched hard–wall randomness

On a Comparison Result for Markov Processes

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