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## Binary jumps in continuum. II. Non-equilibrium process and a Vlasov-type scaling limit (2011)

### Citations

796 | One–Parameter Semigroups for Linear Evolution Equations - Engel, Nagel - 2000 |

529 |
D.: Statistical Mechanics: Rigorous Results
- Ruelle
- 1999
(Show Context)
Citation Context ...n ∈ N, (2.7) k(n)µ (x1, . . . , xn) := { kµ({x1, . . . , xn}), if (x1, . . . , xn) ∈fl(Rd)n 0, otherwise are called correlation functions of µ, and they are well known in statistical physics, see e.g =-=[26]-=-, [27]. In view of Remark 2.1 and (2.5), the mapping (L̂G)(η) := (K−1LKG)(η), η ∈ Γ0, is well defined for any G ∈ Bbs(Γ0), where K−1 is understood in the sense of Remark 2.2. Let k be a measurable fun... |

112 |
Superstable interactions in classical statistical mechanics,
- Ruelle
- 1970
(Show Context)
Citation Context ... (2.7) k(n)µ (x1, . . . , xn) := { kµ({x1, . . . , xn}), if (x1, . . . , xn) ∈fl(Rd)n 0, otherwise are called correlation functions of µ, and they are well known in statistical physics, see e.g [26], =-=[27]-=-. In view of Remark 2.1 and (2.5), the mapping (L̂G)(η) := (K−1LKG)(η), η ∈ Γ0, is well defined for any G ∈ Bbs(Γ0), where K−1 is understood in the sense of Remark 2.2. Let k be a measurable function ... |

93 | Analysis and geometry on configuration spaces,
- Kondratiev
- 1998
(Show Context)
Citation Context ... with compact support. The corresponding Borel σ-algebra B(Γ) coincides with the smallest σ-algebra on Γ for which all mappings Γ 3 γ 7→ |γΛ| ∈ N0 := N∪{0} are measurable for any Λ ∈ Bb(Rd), see e.g. =-=[1]-=-. It is worth noting that Γ is a Polish space (see e.g. [16]). Let Fcyl(Γ) denote the set of all measurable cylinder functions on Γ. Each F ∈ Fcyl(Γ) is characterized by the following property: F (γ) ... |

66 |
States of classical statistical mechanical systems of infinitely many particles.
- Lenard
- 1975
(Show Context)
Citation Context ...and Γ0 will be called observables and quasi-observables, respectively. We consider the following mapping from L0ls(Γ0) into Fcyl(Γ): (KG)(γ) := ∑ ηbγ G(η), γ ∈ Γ, (2.4) where G ∈ L0ls(Γ0), see, e.g., =-=[15,23,24]-=-. The summation in (2.4) is taken over all finite subconfigurations η ∈ Γ0 of the (infinite) configuration γ ∈ Γ; we denote this by the symbol η b γ. The mapping K is linear, positivity preserving, an... |

59 |
Harmonic analysis on configuration space
- Kondratiev, Kuna
- 2002
(Show Context)
Citation Context ...and Γ0 will be called observables and quasi-observables, respectively. We consider the following mapping from L0ls(Γ0) into Fcyl(Γ): (KG)(γ) := ∑ ηbγ G(η), γ ∈ Γ, (2.4) where G ∈ L0ls(Γ0), see, e.g., =-=[15,23,24]-=-. The summation in (2.4) is taken over all finite subconfigurations η ∈ Γ0 of the (infinite) configuration γ ∈ Γ; we denote this by the symbol η b γ. The mapping K is linear, positivity preserving, an... |

48 |
Correlation functions and the uniqueness of the state in classical statistical mechanics,
- Lenard
- 1973
(Show Context)
Citation Context ...h that KG ≥ 0. It was shown in [24] that any such k is the correlation functional of some probability measure on Γ. If, additionally, k ∈ KC for some C > 0, then this measure is uniquely defined (cf. =-=[22]-=-) and belongs to M1fm(Γ) (cf. [15]). Therefore, to prove the theorem, it is enough to show that kt is Lenard positive definite for any t ∈ (0, T ). Since the measure µ ∈M1fm(Γ) has the correlation fun... |

37 |
Phillips: Dissipative operators in a Banach space
- Lumer, S
- 1961
(Show Context)
Citation Context ...irectly from (3.9), (3.10) and (3.11)– (3.14). To prove that the bounded operator L (n) 0 is the generator of a contraction semigroup in Xn, it is enough to show that L (n) 0 is dissipative (see e.g. =-=[25]-=-). For any κ > 0 and G(n) ∈ Xn, ‖L(n)0 G(n) − κG(n)‖Xn = ∫ (Rd)n ∣∣∣∣∣ n∑ i=1 n∑ j=i+1 ∫ Rd ∫ Rd c (xi, xj , y1, y2, )G (n) ( x1, . . . , y1 ∧ i , . . . , y2 ∧ j , . . . , xn ) dy1dy2 − n∑ i=1 n∑ j=i+... |

30 | Markov evolutions and hierarchical equations in the continuum I. One-component systems, - Kondratiev, Oliveira - 2009 |

23 |
On the metrical properties of the configuration space
- Kondratiev, Kutoviy
(Show Context)
Citation Context ...Γ) coincides with the smallest σ-algebra on Γ for which all mappings Γ 3 γ 7→ |γΛ| ∈ N0 := N∪{0} are measurable for any Λ ∈ Bb(Rd), see e.g. [1]. It is worth noting that Γ is a Polish space (see e.g. =-=[16]-=-). Let Fcyl(Γ) denote the set of all measurable cylinder functions on Γ. Each F ∈ Fcyl(Γ) is characterized by the following property: F (γ) = F (γΛ) for some Λ ∈ Bb(Rd) and for any γ ∈ Γ. A stochastic... |

22 | Equilibrium Kawasaki dynamics of continuous particle systems. - Kondratiev, Lytvynov, et al. - 2007 |

18 | Nonequilibrium stochastic dynamics in continuum: the free case
- Kondratiev, Lytvynov, et al.
(Show Context)
Citation Context ...onsider a free dynamics. In the free Kawasaki dynamics, in the course of a random evolution, each particle of the configuration randomly hops over Rd without any interaction with other particles (see =-=[19]-=- for details). In the dynamics of binary jumps, at each jump time two points of the (infinite) configuration change their positions in Rd. A randomness for choosing a pair of points provides a random ... |

16 |
Time reversible and Gibbsian point processes. II. Markovian particle jump processes on a general phase space.
- Glotzl
- 1982
(Show Context)
Citation Context ... 17, 18, 20, 21] and the references therein. Under the socalled balance condition on the jump rate, a proper Gibbs distribution is an invariant, and even symmetrizing measure for such a dynamics, see =-=[13]-=-. To obtain a simpler measure, e.g. Poissonian, as a symmetrizing measure for a Kawasaki-type dynamics, we should either suppose quite unnatural conditions on the jump rate, or consider a free dynamic... |

11 | Vlasov scaling for the Glauber dynamics in continuum
- Finkelshtein, Kondratiev, et al.
- 1002
(Show Context)
Citation Context ...tart with explaining the idea of the Vlasovtype scaling. A general scheme for both birth-and-death and conservative dynamics may be found in [7]. Certain realizations of this approach were studied in =-=[4, 6, 8, 9]-=-. We would like to construct a scaling of the generator L, say Lε, ε > 0, such that the following requirements are satisfied. Assume we have an evolution Vε(t) corresponding to the equation ∂∂tGt,ε = ... |

11 | Equilibrium Glauber dynamics of continuous particle systems as a scaling limit of Kawasaki dynamics, Random Oper - Kondratiev, Lytvynov |

8 | A note on equilibrium Glauber and Kawasaki dynamics for fermion point processes, Methods Funct. Anal. Topology 14 - Lytvynov, Ohlerich - 2008 |

7 | Operator approach to Vlasov scaling for some models of spatial ecology, ArXiv
- Finkelshtein, Kondratiev, et al.
(Show Context)
Citation Context ...tart with explaining the idea of the Vlasovtype scaling. A general scheme for both birth-and-death and conservative dynamics may be found in [7]. Certain realizations of this approach were studied in =-=[4, 6, 8, 9]-=-. We would like to construct a scaling of the generator L, say Lε, ε > 0, such that the following requirements are satisfied. Assume we have an evolution Vε(t) corresponding to the equation ∂∂tGt,ε = ... |

7 | Diffusion approximation for equilibrium Kawasaki dynamics in continuum. Stochastic Process. - Kondratiev, Kutiviy, et al. - 2008 |

6 |
On a general kinetic equation for many-particle systems with interaction, fragmentation and coagulation
- Belavkin, Kolokoltsov
- 2003
(Show Context)
Citation Context ...) , (1.4) then any such measure will even be symmetrizing. Binary jumps in continuum. II. Non-equilibrium process 3 It should be noted that similar dynamics of finite particle systems were studied in =-=[2, 3, 14]-=-. In particular, in [3], the authors studied a non-equilibrium dynamics of velocities of particles, such that the law of conservation of momentum is satisfied for this system. However, the methods app... |

6 |
S.: The asymptotic dynamics of a system with a large number of particles described by Kolmogorov–Feller equations
- Belavkin, Maslov, et al.
- 1981
(Show Context)
Citation Context ...) , (1.4) then any such measure will even be symmetrizing. Binary jumps in continuum. II. Non-equilibrium process 3 It should be noted that similar dynamics of finite particle systems were studied in =-=[2, 3, 14]-=-. In particular, in [3], the authors studied a non-equilibrium dynamics of velocities of particles, such that the law of conservation of momentum is satisfied for this system. However, the methods app... |

6 | Kawasaki dynamics in continuum: micro- and mesoscopic descriptions,
- Kondratiev, Kozitsky, et al.
- 2011
(Show Context)
Citation Context ...tart with explaining the idea of the Vlasovtype scaling. A general scheme for both birth-and-death and conservative dynamics may be found in [7]. Certain realizations of this approach were studied in =-=[4, 6, 8, 9]-=-. We would like to construct a scaling of the generator L, say Lε, ε > 0, such that the following requirements are satisfied. Assume we have an evolution Vε(t) corresponding to the equation ∂∂tGt,ε = ... |

6 | O.: Vlasov scaling for stochastic dynamics of continuous systems
- Finkelshtein, Kondratiev, et al.
- 2010
(Show Context)
Citation Context ...23 4 Vlasov-type scaling For the reader’s convenience, we start with explaining the idea of the Vlasovtype scaling. A general scheme for both birth-and-death and conservative dynamics may be found in =-=[7]-=-. Certain realizations of this approach were studied in [4, 6, 8, 9]. We would like to construct a scaling of the generator L, say Lε, ε > 0, such that the following requirements are satisfied. Assume... |

5 | Glauber dynamics in continuum: a constructive approach to evolution of states, ArXiv
- Finkelshtein, Kondratiev, et al.
(Show Context)
Citation Context |

5 |
Kinetic equations for the pure jump models of k-nary interacting particle systems
- Kolokoltsov
(Show Context)
Citation Context ...) , (1.4) then any such measure will even be symmetrizing. Binary jumps in continuum. II. Non-equilibrium process 3 It should be noted that similar dynamics of finite particle systems were studied in =-=[2, 3, 14]-=-. In particular, in [3], the authors studied a non-equilibrium dynamics of velocities of particles, such that the law of conservation of momentum is satisfied for this system. However, the methods app... |

1 | Binary jumps in continuum. I. Equilibrium processes and their scaling limits,” arXiv:1101.4765v1
- Finkelshtein, Kondratiev, et al.
- 2011
(Show Context)
Citation Context ...the hopping points (and does not depend on the other points of the configuration). A Poisson measure may be invariant, or even symmetrizing for such a dynamics. In the first part of the present paper =-=[10]-=-, we considered such a process with generator (LF )(γ) = ∑ {x1, x2}⊂γ ∫ (Rd)2 Q(x1, x2, dh1 × dh2) × (F (γ \ {x1, x2} ∪ {x1 + h1, x2 + h2})− F (γ)). (1.1) Here, the measure Q(x1, x2, dh1 × dh2) descri... |