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

Let Γ denote the space of all locally finite subsets (configurations) in Rd. A stochastic dynamics of binary jumps in continuum is a Markov process on Γ in which pairs of particles simultaneously hop over Rd. In this paper, we study an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing (and hence invariant) measure. The existence and uniqueness of the corresponding stochastic dynamics are shown. We next prove the main result of this paper: a big class of dynamics of binary jumps converge, in a diffusive scaling limit, to a dynamics of interacting Brownian particles. We also study another scaling limit, which leads us to a spatial birth-and-death process in continuum. A remarkable property of the limiting dynamics is that its generator possesses a spectral gap, a property which is hopeless to expect from the initial dynamics of binary jumps.

[1]  M. Reed,et al.  Methods of Modern Mathematical Physics. 2. Fourier Analysis, Self-adjointness , 1975 .

[3]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[4]  Yuri G. Kondratiev,et al.  Equilibrium Glauber dynamics of continuous particle systems as a scaling limit of Kawasaki dynamics , 2006 .

[5]  Time reversible and Gibbsian point processes, II. Markovian particle jump processes on a general phase space , 1982 .

[6]  Diffusion approximation for equilibrium Kawasaki dynamics in continuum , 2007, math/0702178.

[7]  Zhi-Ming Ma,et al.  Construction of diffusions on configuration spaces , 2000 .

[8]  Zhi-Ming Ma,et al.  Introduction to the theory of (non-symmetric) Dirichlet forms , 1992 .

[9]  Non-equilibrium stochastic dynamics in continuum: The free case , 2007, math/0701736.

[10]  A support property for infinite-dimensional interacting diffusion processes , 1998, math/9801143.

[11]  K. Parthasarathy.  PROBABILITY MEASURES IN A METRIC SPACE , 1967 .

[12]  J. Carstensen.  One Component Systems , 2000 .

[13]  D. Surgailis.  On Poisson multiple stochastic integrals and associated equilibrium Markov processes , 1983 .

[14]  V. Belavkin,et al.  Asymptotic dynamics of a system of a large number of particles described by the Kolmogorov-Feller equations , 1981 .

[15]  D. Nualart,et al.  Anticipative calculus for the Poisson process based on the Fock space , 1990 .

[16]  J. Doob.  Stochastic processes , 1953 .

[17]  Dmitri Finkelshtein,et al.  Binary jumps in continuum. II. Non-equilibrium process and a Vlasov-type scaling limit , 2011 .

[18]  D. Surgailis.  On multiple Poisson stochastic integrals and associated Markov semigroups , 1984 .

[19]  Yuri G. Kondratiev,et al.  Markov evolutions and hierarchical equations in the continuum. I: one-component systems , 2007, 0707.0619.

[20]  M. Röckner,et al.  Infinite interacting diffusion particles I: Equilibrium process and its scaling limit , 2003, math/0311444.

[21]  H. Osada.  Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions , 1996 .

[22]  H. Spohn.  Equilibrium fluctuations for interacting Brownian particles , 1986 .

[23]  K. Parthasarathy,et al.  Probability measures on metric spaces , 1967 .

[24]  Minoru W. Yoshida.  Construction of infinite dimensional interacting diffusion processes through Dirichlet forms , 1996 .

[25]  E. Lytvynov,et al.  Equilibrium Kawasaki dynamics of continuous particle systems , 2005 .

[26]  Minoru W. Yoshida,et al.  On the Essential Self-Adjointness of Wick Powers of Relativistic Fields and of Fields Unitary Equivalent to Random Fields , 2004 .

[27]  Yoshifusa Ito,et al.  Calculus on Gaussian and Poisson white noises , 1988, Nagoya Mathematical Journal.

[28]  M. Röckner,et al.  Classical dirichlet forms on topological vector spaces-the construction of the associated diffusion process , 1989 .

[29]  Jürgen Potthoff,et al.  White Noise: An Infinite Dimensional Calculus , 1993 .

[30]  SYSTEMS OF CLASSICAL PARTICLES IN THE GRAND CANONICAL ENSEMBLE, SCALING LIMITS AND QUANTUM FIELD THEORY , 2005, math-ph/0601021.

[31]  M. Fukushima,et al.  Dirichlet forms and symmetric Markov processes , 1994 .

[32]  I︠u︡. M. Berezanskiĭ,et al.  Spectral Methods in Infinite-Dimensional Analysis , 1995 .

[33]  Sergio Albeverio,et al.  Analysis and Geometry on Configuration Spaces: The Gibbsian Case☆ , 1998 .

[34]  E. Davies,et al.  One-parameter semigroups , 1980 .