Poincaré Inequality on the Path Space of Poisson Point Processes

Wang, Feng‐Yu; Chen, Yuan

doi:10.1007/s10959-009-0232-8

Cited by 7 publications

(19 citation statements)

References 15 publications

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“…holds (see also Wang and Yuan (2010) for a more elementary proof). Moreover, by (1) we have |F (w + x1 [t,∞) )| ≤ ∥F ∥ ∞ for (Λ × ν × dt)-a.e.…”

Section: The Weak Poincaré Inequality For Weighted Dirichlet Formsmentioning

confidence: 93%

“…We shall adopt an induction argument as in Wang and Yuan (2010). By the monotone class theorem and an approximation argument, we shall assume that…”

Section: The Elementary Proof Of Theorem 11mentioning

confidence: 99%

“…Therefore, our proof is more elementary since it avoids the use of the martingale representation on Poisson spaces and the Itô formula for jump processes. We remark that the idea of our proof is essentially due to Wang and Yuan (2010). The remainder of the note is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An elementary proof of the log-Sobolev inequality for Poisson point processes

Deng

Song

2011

Statistics & Probability Letters

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“…holds (see also Wang and Yuan (2010) for a more elementary proof). Moreover, by (1) we have |F (w + x1 [t,∞) )| ≤ ∥F ∥ ∞ for (Λ × ν × dt)-a.e.…”

Section: The Weak Poincaré Inequality For Weighted Dirichlet Formsmentioning

confidence: 93%

“…We shall adopt an induction argument as in Wang and Yuan (2010). By the monotone class theorem and an approximation argument, we shall assume that…”

Section: The Elementary Proof Of Theorem 11mentioning

confidence: 99%

See 1 more Smart Citation

An elementary proof of the log-Sobolev inequality for Poisson point processes

Deng

Song

2011

Statistics & Probability Letters

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“…In the case where instead of a single measure we have a familly of measures as in the case of the semigroup {P t , t ≥ 0}, then the constant C may depend on the time t, i.e. C = C(t), as is the case for the examples studied in [35], [3] and [10]. The aforementioned papers used the so-called semigroup method that will be also followed in the current work.…”

Section: Introductionmentioning

confidence: 99%

Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes

Hodara

Papageorgiou

2019

Mathematics

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We study Talagrand concentration and Poincaré type inequalities for unbounded pure jump Markov processes. In particular we focus on processes with degenerate jumps that depend on the past of the whole system, based on the model introduced by Galves and Löcherbach in [19], in order to describe the activity of a biological neural network. As a result we obtain exponential rates of convergence to equilibrium.2010 Mathematics Subject Classification. 60K35, 26D10, 60G99, Key words and phrases. Talagrand inequality, Poincaré inequality, brain neuron networks.

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“…For which (random) perturbations are the distributions of W S and W S + ξ equivalent? Related results for compound Poisson processes can be found in [17] and [15]. In this note, we assume that ξ t = h(S t ) where h ∈ W M and t ∈ [0, T ].…”

mentioning

confidence: 99%

On a Cameron–Martin Type Quasi-Invariance Theorem and Applications to Subordinate Brownian Motion

Deng

Schilling

2015

Stochastic Analysis and Applications

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We present a Cameron-Martin type quasi-invariance theorem for subordinate Brownian motion. As applications, we establish an integration by parts formula and construct a gradient operator on the path space of subordinate Brownian motion, and we obtain some canonical Dirichlet forms. These findings extend the corresponding classical results for Brownian motion. IntroductionRecently the stability of properties of Markov processes and their semigroups under subordination in the sense of Bochner has attracted great interest. In [6], Wang's dimension free Harnack inequality was established for a class of subordinate semigroups. Nash and Poincaré inequalities are preserved under subordination, cf. [13,5]. In our recent paper [2], we show that shift Harnack inequalities (in the sense of [16]) remain valid under subordination in the sense of Bochner. It is a natural question whether further probabilistic properties, e.g. quasi-invariance, are preserved by subordination.The Cameron-Martin theorem, which was discovered by R.H. Cameron and W.T. Martin [1] (see e.g. [3,7,8] and the references therein for further developments), plays a fundamental role in the analysis on the path space of diffusion processes. It states that the Wiener measure (i.e. the distribution of Brownian motion) is quasi-invariant under a Cameron-Martin shift. In this paper, we shall derive an analogous result for subordinate Brownian motion.Let us recall some basic notations. Throughout this paper, we setand make the convention v u = (u,v) for all 0 ≤ u < v ≤ ∞. By S = (S t ) t∈[0,T ] , where 0 < T ≤ ∞, we denote a non-trivial subordinator, i.e. an increasing Lévy process with S 0 = 0 and Laplace transform Ee −uSt = e −tφ(u) , u > 0, t ∈ [0, T ]. 1 − e −ux ν(dx), u > 0, where b ≥ 0 is the drift parameter and ν is a Lévy measure, i.e. a measure on (0, ∞) satisfying ∞ 0 (x ∧ 1) ν(dx) < ∞; we use [12] as our standard reference for Bernstein functions and subordination. If T = ∞ then S ∞ := lim t↑∞ S t = ∞ a.s. Let M = ess sup S T = sup {r > 0 : P(S T < r) < 1} . Remark 1.1. M can attain the following values: (i) If T = ∞, then S ∞ = ∞ a.s. and M = ∞. (ii) If T < ∞ and S t is deterministic, i.e. S t = ct for some constant c > 0, then M = cT < ∞. (iii) If T < ∞ and S t is non-deterministic, then M = ∞. Indeed: Since ν = 0, there exists some finite interval [u, v] ⊂ (0, ∞) such that η := ν([u, v]) ∈ (0, ∞). The jump times of jumps with size in the interval [u, v] define a Poisson process (N t ) t∈[0,T ] with intensity η. Since S T ≥ uN T , we conclude that ess sup S T = ∞. Let (W t ) t∈[0,M ] be a standard d-dimensional Brownian motion starting from zero. The Wiener measure µ, i.e. the distribution of (W t ) t∈[0,M ] , is a probability measure on the path space W M = w : [0, M] → R d : w is continuous and w(0) = 0 ,which is endowed with the topology of locally uniform convergence. We write

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Poincaré Inequality on the Path Space of Poisson Point Processes

Cited by 7 publications

References 15 publications

An elementary proof of the log-Sobolev inequality for Poisson point processes

An elementary proof of the log-Sobolev inequality for Poisson point processes

Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes

On a Cameron–Martin Type Quasi-Invariance Theorem and Applications to Subordinate Brownian Motion

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