## Infinite interaction diffusion particles I: Equilibrium process and its scaling limit

Venue: | Forum Math |

Citations: | 4 - 3 self |

### BibTeX

@ARTICLE{Kondratiev_infiniteinteraction,

author = {Yuri Kondratiev and Eugene Lytvynov and Michael Röckner},

title = {Infinite interaction diffusion particles I: Equilibrium process and its scaling limit},

journal = {Forum Math},

year = {},

pages = {9--43}

}

### OpenURL

### Abstract

A stochastic dynamics (X(t))t≥0 of a classical continuous system is a stochastic process which takes values in the space Γ of all locally finite subsets (configurations) in Rd and which has a Gibbs measure µ as an invariant measure. We assume that µ corresponds to a symmetric pair potential φ(x − y). An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics—the so-called gradient stochastic dynamics, or interacting Brownian particles—has been investigated. By using the theory of Dirichlet forms from [27], we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form EΓ µ on L2 (Γ; µ), and under general conditions on the potential φ, prove its closability. For a potential φ having a “weak ” singularity at zero, we also write down an explicit form of the generator of EΓ µ on the set of smooth cylinder functions. We then show that, for any Dirichlet form EΓ µ, there exists a diffusion process that is properly associated with it. Finally, in a way parallel to [17], we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in C([0, ∞), D ′), where D ′ is the dual space of D:=C ∞ 0 (Rd).

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Citation Context ...B(Γ)) belongs to G(z,φ) if and only if µ satisfies (2.5), cf. [31, Theorem. 2]. Let us now describe two classes of Gibbs measures which appear in classical statistical mechanics of continuous systems =-=[40, 41]-=-. For every r = (r 1 ,...,r d ) ∈ Z d , we define the cube Qr := { x ∈ R d | r i − 1 2 ≤ xi < r i + 1 2 These cubes form a partition of R d . For any γ ∈ Γ, we set γr := γQr, r ∈ Z d . For N ∈ N let Λ... |

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Citation Context ...y diffusions. Till now, only one type of such dynamics—the so-called gradient stochastic dynamics, or interacting Brownian particles—has been investigated. By using the theory of Dirichlet forms from =-=[27]-=-, we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form EΓ µ on L2 (Γ; µ), and under general conditions ... |

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Citation Context ... 4.2] now carries over to our case. In particular, V 16is continuously embedded into L1 (Γ;µ) and SΓ extends uniquely to a bilinear map from (V, | · |Γ) × (V, | · |Γ) into L1 ( .. Γ;µ). Lemma 4.3 in =-=[28]-=- now reads as follows: Let f ∈ D. Then, 〈f, ·〉 ∈ V and S Γ ∫ (〈f, ·〉)(γ) = A(γ,x)S(f)(x)γ(dx) for µ-a.e. γ ∈ .. Γ. Here, S(f):=S(f,f) and S(f,g)(x):=〈∇f(x), ∇g(x)〉, f,g ∈ D, x ∈ Rd . The proof is agai... |

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Citation Context ...ions (SS), (LR), and (I). Furthermore, the set G(z,φ) is not empty for potentials satisfying (S) and (UI), or equivalently, for stable potentials in the low activity-high temperature regime, see e.g. =-=[29, 30]-=-. A measure µ ∈ G(z,φ) in the latter case is constructed as a limit of finite volume Gibbs measures corresponding to empty boundary conditions. Let us now recall the so-called Ruelle bound (cf. [41]).... |

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Citation Context ...ocess onto ΩT:=C([0,T], D ′ ) without expressing this explicitly. We define the time reversal rT(ω):=ω(T − ·). Let f ∈ D. It is easy to show that 〈f, ·〉 ∈ D(Eǫ). By the Lyons–Zheng decomposition, cf. =-=[25, 15, 26]-=-, we have, for all 0 ≤ t ≤ T: 〈f,X(t)〉 − 〈f,X(0)〉 = 1 2 Mt(ǫ,f) + 1 ( MT −t(ǫ,f)(rT) − MT(ǫ,f)(rT) 2 ) , Pǫ-a.e., where (Mt(ǫ,f))0≤t≤T is a continuous (Pǫ,(Ft/ǫ 2)0≤t≤T)-martingale and (Mt(ǫ,f)(rT))0≤... |

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Citation Context ..., they could not write down the corresponding generator explicitly, hence could not prove that their processes actually solve (1.1) weakly. This, however, was proved in [5] (see also the survey paper =-=[36]-=-) by showing an integration by parts formula for the respective Gibbs measures. But the gradient stochastic dynamics is, of course, not the unique diffusion process which has µ as an invariant measure... |

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Citation Context ... our forthcoming research. There also exists another type of stochastic dynamics of a classical continuous system— the so-called Glauber-type dynamics, which is a spatial birth-and-death process, see =-=[22]-=-. Let us briefly describe the contents of the paper. After some preliminary information about Gibbs measures in Section 2, we construct a bilinear form E Γ µ on L 2 (Γ;µ) in Section 3. This form is de... |

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Citation Context ... x (t), x(t) ∈ X(t), (1.1) X(0) = γ ∈ Γ, y(t)∈X(t), y(t)̸=x(t) where (B x )x∈γ is a sequence of independent Brownian motions. The study of such diffusions has been initiated by R. Lang [24] (see also =-=[42, 13]-=-), who considered the case φ ∈ C 3 0 (Rd ) using finite-dimensional approximations of stochastic differential equations. More singular φ, which are of particular interest from the point of view of sta... |

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Citation Context ...hich is properly associated with the (closed) Dirichlet form EΓ µ. This process lives, in general, in the bigger space .. Γ of all locally finite multiple configurations, but we prove, analogously to =-=[38]-=-, that the process indeed lives in Γ in case d ≥ 2. (If d = 1, one cannot, of course, exclude collisions of particles.) According to (1.5), the constructed diffusion process informally solves (1.2). S... |

14 |
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Citation Context ...proximations of stochastic differential equations. More singular φ, which are of particular interest from the point of view of statistical mechanics, have been treated by H. Osada [32] and M. Yoshida =-=[46]-=-. These authors were the 1first to use the Dirichlet form approach from [27] for the construction of such processes. However, they could not write down the corresponding generator explicitly, hence c... |

13 |
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(Show Context)
Citation Context ...e scaling limit of the gradient stochastic dynamics. Then, H. Rost gave some heuristic arguments for the existence of a limiting generalized Ornstein–Uhlenbeck 3process [39]. In the celebrated paper =-=[44]-=-, H. Spohn described a proof of convergence of the scaled processes in the case where the underlying potential φ is smooth, compactly supported, and positive, and d ≤ 3. In [18], M. Z. Guo and G. Papa... |

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Citation Context ...y(t))dt + √ 2dB x (t), x(t) ∈ X(t), (1.1) X(0) = γ ∈ Γ, y(t)∈X(t), y(t)̸=x(t) where (B x )x∈γ is a sequence of independent Brownian motions. The study of such diffusions has been initiated by R. Lang =-=[24]-=- (see also [42, 13]), who considered the case φ ∈ C 3 0 (Rd ) using finite-dimensional approximations of stochastic differential equations. More singular φ, which are of particular interest from the p... |

9 | Canonical Gibbs measures - Georgii - 1980 |

9 |
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Citation Context ...ocess onto ΩT:=C([0,T], D ′ ) without expressing this explicitly. We define the time reversal rT(ω):=ω(T − ·). Let f ∈ D. It is easy to show that 〈f, ·〉 ∈ D(Eǫ). By the Lyons–Zheng decomposition, cf. =-=[25, 15, 26]-=-, we have, for all 0 ≤ t ≤ T: 〈f,X(t)〉 − 〈f,X(0)〉 = 1 2 Mt(ǫ,f) + 1 ( MT −t(ǫ,f)(rT) − MT(ǫ,f)(rT) 2 ) , Pǫ-a.e., where (Mt(ǫ,f))0≤t≤T is a continuous (Pǫ,(Ft/ǫ 2)0≤t≤T)-martingale and (Mt(ǫ,f)(rT))0≤... |

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Citation Context ...the Boltzmann–Gibbs principle in our situation, which remains an open problem. It is also possible to study an invariance principle (scaling limit) of a tagged particle of interacting diffusions (cf. =-=[33, 34]-=-). This will be the subject of future research. Finally, we would like to mention that, though some proofs of the results of this paper use the ideas and techniques developed for the gradient dynamics... |

7 | operators on differential forms over configuration spaces
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Citation Context ... that there exists a sequence (un) ∞ n=1 ⊂ D(EΓ µ) such that each un, n ∈ N, is a continuous function on .. Γ, un → 1N pointwise as n → ∞, and sup n∈N E Γ µ (un) < ∞. Let f ∈ C ∞ 0 (R) be such that 1 =-=[0,1]-=- ≤ f ≤ 1 [−1/2,3/2) and |f ′ | ≤ 3 × 1 [−1/2,3/2). For any n ∈ N and i = (i1,... ,id) ∈ Z d , define a function f (n) i f (n) i (x):= ∈ D by d∏ f(nxk − ik), x ∈ R d . k=1 Let also I (n) i (x):= ∏d k=1... |

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Citation Context ... x (t), x(t) ∈ X(t), (1.1) X(0) = γ ∈ Γ, y(t)∈X(t), y(t)̸=x(t) where (B x )x∈γ is a sequence of independent Brownian motions. The study of such diffusions has been initiated by R. Lang [24] (see also =-=[42, 13]-=-), who considered the case φ ∈ C 3 0 (Rd ) using finite-dimensional approximations of stochastic differential equations. More singular φ, which are of particular interest from the point of view of sta... |

7 |
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Citation Context ...sson measure on (Γ, B(Γ)) with intensity measure zm(dx). This measure can be characterized by its Laplace transform ∫ (∫ exp[〈f,γ〉]πz(dγ) = exp Rd (e f(x) − 1)zm(dx) Γ x∈γ ) , f ∈ D. We refer e.g. to =-=[45, 4]-=- for a detailed discussion of the construction of the Poisson measure on the configuration space. Now, we proceed to consider Gibbs measures. A pair potential is a Borel measurable function φ: Rd → R∪... |

6 | Röckner: On a relation between intrinsic and extrinsic Dirichlet forms for interacting particle systems - Silva, Kondratiev, et al. - 1998 |

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(Show Context)
Citation Context ...or of EΓ µ on the set of smooth cylinder functions. We then show that, for any Dirichlet form EΓ µ, there exists a diffusion process that is properly associated with it. Finally, in a way parallel to =-=[17]-=-, we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in C([0, ∞), D ′), where D ′ i... |

5 |
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(Show Context)
Citation Context ...starts from a given configuration, or a given distribution, is still open. Actually, this problem may be studied by using the ideas developed for the Hamiltonian and gradient stochastic dynamics, see =-=[21]-=-, that is, by obtaining equations for the time evolution of correlation functions, or corresponding generating (Bogoliubov) functionals. This will be the subject of our forthcoming research. There als... |

5 |
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(Show Context)
Citation Context ...ions (SS), (LR), and (I). Furthermore, the set G(z,φ) is not empty for potentials satisfying (S) and (UI), or equivalently, for stable potentials in the low activity-high temperature regime, see e.g. =-=[29, 30]-=-. A measure µ ∈ G(z,φ) in the latter case is constructed as a limit of finite volume Gibbs measures corresponding to empty boundary conditions. Let us now recall the so-called Ruelle bound (cf. [41]).... |

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3 |
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Citation Context ...limiting Markov process for the scaling limit of the gradient stochastic dynamics. Then, H. Rost gave some heuristic arguments for the existence of a limiting generalized Ornstein–Uhlenbeck 3process =-=[39]-=-. In the celebrated paper [44], H. Spohn described a proof of convergence of the scaled processes in the case where the underlying potential φ is smooth, compactly supported, and positive, and d ≤ 3. ... |

2 |
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Citation Context ... ρdx ∈ D ′ , where we identify the configuration with the corresponding sum of Dirac measures, ρ is the first correlation function of µ, and D ′ is the dual space of D:=C ∞ 0 (Rd ). T. Brox showed in =-=[9]-=- that, in the low activity-high temperature regime, the Gibbs measure µ converges under the scaling Sout, ǫSin, ǫ to a corresponding white noise measure νc with covariance operator cId, where the cons... |

2 | W.: The law of large numbers and the law of the iterated logarithm for infinite dimensional interacting processes - Schmuland, Sun |