#### DMCA

## Binary jumps in continuum. I. Equilibrium processes and their scaling limits

Citations: | 1 - 1 self |

### Citations

369 |
Introduction to the Theory of (Non-symmetric) Dirichlet Forms,
- Ma, Rockner
- 1992
(Show Context)
Citation Context ...is special case. Throughout the paper, we have added a series of statements and remarks regarding such a dynamics. The paper is organized as follows. In Section 2, using the theory of Dirichlet forms =-=[8, 17]-=-, we construct a rather general dynamics on the configuration space, whose generator has the form (1) on a set of test cylinder functions on Γ. In Section 3, we show that the generator (1) with domain... |

183 | One-Parameter Semigroups - Davies - 1980 |

136 |
White Noise. An Infinite Dimensional Calculus, volume 253
- Hida, Kuo, et al.
- 1993
(Show Context)
Citation Context ...− ∂†x1∂†x2 + ∂†x1 + ∂†x2) × (∂x1+h1∂x2+h2 − ∂x1+h1 − ∂x2+h2 − ∂x1∂x2 + ∂x1 + ∂x2), (3) where ∂†x denotes a creation operator at point x ∈ Rd (∂†x being rather an operatorvalued distribution, see e.g. =-=[10]-=-) Noting that the operators ∂x, x ∈ Rd, commute, we get from (3) and (11), (12): −L̃0 = J+ + J0 + J−, where J+ := ∫ Rd z dx1 ∫ Rd z dx2 ∫ Rd dh1 ∫ Rd dh2 q(x2 − x1, h1, h2) (− ∂†x1∂†x2∂x1 + ∂†x1∂†x2∂x... |

79 |
Anticipative calculus for the Poisson process based on the Fock space,
- Nualart, Vives
- 1990
(Show Context)
Citation Context ...absence of fi. We will preserve the notation ∂x for the operator I∂xI −1 :P →P. This operator admits the following explicit representation ∂xF (γ) = F (γ ∪ {x})− F (γ) 11 for piz-a.a. γ ∈ Γ, see e.g. =-=[1, 11, 19]-=-. By the Mecke formula, for any F ∈P, E(F, F ) = 1 4 ∫ Rd z dx1 ∫ Rd z dx2 ∫ Rd dh1 ∫ Rd dh2 q(x2 − x1, h1, h2) × ∫ Γ piz(dγ) ( F (γ ∪ {x1 + h1, x2 + h2})− F (γ ∪ {x1, x2}) )2 . Noting that F (γ ∪ {x1... |

44 |
Analysis and geometry on configuration spaces: The Gibbsian case,
- Kondratiev
- 1998
(Show Context)
Citation Context ...e, in a diffusive scaling limit, to a dynamics of interacting Brownian particles. The form of the generator of the limiting diffusion resembles the generator of the gradient stochastic dynamics (e.g. =-=[1, 20, 24, 27]-=-), while staying symmetric with respect to the Poisson measure, rather than with respect to a Gibbs measure (as it is the case for the gradient stochastic dynamics). We prove the convergence of proces... |

40 | Glauber dynamics of continuous particle systems.
- Kondratiev, Lytvynov
- 2005
(Show Context)
Citation Context ...ss on Γ in which particles hop over Rd so that, at each jump time, only one particle changes its position. For a study of equilibrium Kawasaki dynamics in continuum, we refer the reader to the papers =-=[5, 7, 9, 12, 14, 16]-=- and the references therein. In this paper, we will study an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing (and hence invariant) measure. In several cases, an equi... |

36 | Construction of diffusions on configuration spaces,
- Ma, Rockner
- 2000
(Show Context)
Citation Context ...FC∞b (C∞0 (Rd),Γ)) in L2(Γ, piz). Hence, this form is closable in L2(Γ, piz), and so statement i) is proven. Statements ii) and iii) can be shown analogously to Theorems 6.1 and 6.3 in [13], see also =-=[18]-=- and [23]. A result similar to Theorem 3 and Proposition 4 can be obtained for the stochastic dynamics from Proposition 3, ii). Let us briefly outline it. The scaled q function is given by qε(x, h) :=... |

30 | Markov evolutions and hierarchical equations in the continuum I. One-component systems,
- Kondratiev, Oliveira
- 2009
(Show Context)
Citation Context ...ss on Γ in which particles hop over Rd so that, at each jump time, only one particle changes its position. For a study of equilibrium Kawasaki dynamics in continuum, we refer the reader to the papers =-=[5, 7, 9, 12, 14, 16]-=- and the references therein. In this paper, we will study an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing (and hence invariant) measure. In several cases, an equi... |

30 |
Calculus on Gaussian and Poisson white noises
- Itô, Kubo
- 1988
(Show Context)
Citation Context ...absence of fi. We will preserve the notation ∂x for the operator I∂xI −1 :P →P. This operator admits the following explicit representation ∂xF (γ) = F (γ ∪ {x})− F (γ) 11 for piz-a.a. γ ∈ Γ, see e.g. =-=[1, 11, 19]-=-. By the Mecke formula, for any F ∈P, E(F, F ) = 1 4 ∫ Rd z dx1 ∫ Rd z dx2 ∫ Rd dh1 ∫ Rd dh2 q(x2 − x1, h1, h2) × ∫ Γ piz(dγ) ( F (γ ∪ {x1 + h1, x2 + h2})− F (γ ∪ {x1, x2}) )2 . Noting that F (γ ∪ {x1... |

22 | Equilibrium Kawasaki dynamics of continuous particle systems.
- Kondratiev, Lytvynov, et al.
- 2007
(Show Context)
Citation Context ...ss on Γ in which particles hop over Rd so that, at each jump time, only one particle changes its position. For a study of equilibrium Kawasaki dynamics in continuum, we refer the reader to the papers =-=[5, 7, 9, 12, 14, 16]-=- and the references therein. In this paper, we will study an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing (and hence invariant) measure. In several cases, an equi... |

18 | Nonequilibrium stochastic dynamics in continuum: the free case
- Kondratiev, Lytvynov, et al.
(Show Context)
Citation Context ...ure is a free dynamics, i.e., there is no interaction between particles. For example, this is true for a Surgailis process (in particular, the Glauber dynamics without interaction) [25, 26] (see also =-=[15]-=-). Note that a Surgailis generator, in the symmetric Fock space realization of the L2-space of Poisson measure, is the second quantization of the generator of a one-particle dynamics. Another example ... |

16 |
Time reversible and Gibbsian point processes. II. Markovian particle jump processes on a general phase space.
- Glotzl
- 1982
(Show Context)
Citation Context |

10 | Infinite interacting diffusion particles I: Equilibrium process and its scaling limit. Forum Math.,
- Kondratiev, Lytvynov, et al.
- 2006
(Show Context)
Citation Context ...etric form (E0,FC∞b (C∞0 (Rd),Γ)) in L2(Γ, piz). Hence, this form is closable in L2(Γ, piz), and so statement i) is proven. Statements ii) and iii) can be shown analogously to Theorems 6.1 and 6.3 in =-=[13]-=-, see also [18] and [23]. A result similar to Theorem 3 and Proposition 4 can be obtained for the stochastic dynamics from Proposition 3, ii). Let us briefly outline it. The scaled q function is given... |

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

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

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 ...e 2 particle process. In fact, in the mentioned Fock space realization, the generator of this dynamics has a Jacobi matrix (three-diagonal) form. When this paper was nearing completion, the reference =-=[2]-=- came to our attention. There, at a rather heuristic level, the authors discuss a special kind of a stochastic dynamics of binary jumps, and derive a Boltzmann-type equation through a Vlasovtype scali... |

2 |
Binary jumps in continuum. II. Non-equilibrium process and Vlasov-type scaling limit,” in preparation
- Kutoviy, V, et al.
(Show Context)
Citation Context ... the generator of the initial dynamics of binary jumps. We also note that the result of the scaling essentially depends on the initial distribution of the dynamics. 3 In the second part of this paper =-=[6]-=- we discuss non-equilibrium dynamics of binary jumps. In particular, we show that a Vlasov-type mesoscopic scaling for such a dynamics leads to a generalized Boltzmann non-linear equation for the part... |