Gibbs measures relative to Brownian motion. (English) Zbl 0965.60095

Osada, Hirofumi; Spohn, Herbert

Ann. Probab. 27, No. 3, 1183-1207 (1999).

In relation with the physical problem of a quantum particle subject to a potential and coupled to a free bosonic field, the authors consider Gibbs measures \(\mu\) on \(C(R,R)\) with external potential \(\varphi : R \rightarrow R \cup \{\infty \}\) and interaction potential \(w : R^2\rightarrow R\) defined by \[ \mu_{T,\xi} = Z_{T,\xi}^{-1} e^{-H_T(X,\xi)} dW_{T,\xi}, \qquad T\in R_+, \quad \xi \in C(R,R), \] where \(\mu_{T,\xi}\) is the law \(\mu\) of \((X_t)_{t\in R_+}\) given \(X_t=\xi_t\), \(\forall |t|\geq T\), \(W_{T,\xi}\) is the Brownian bridge measure on \(C([-T,T],R)\) conditioned by \(\{X_T = \xi_T, X_{-T} = \xi_{-T}\}\), \(Z_{T,\xi}^{-1}\) is a normalization constant, and \[ \begin{split} H_T (X,\xi) = \int_{|t |\leq T} \varphi (X_t) dt + {1 \over 2} \int_{|t |, |u |\leq T} w(t-u,X_t-X_u) dt du\\ + \int_{|t|\leq T < |u |} w(t-u,X_t-\xi_u) dt du.\end{split} \] An existence result is obtained for such measures via a finite volume construction, and uniqueness is proved among translation invariant measures satisfying a finite moment condition. Examples of potentials which do not define a unique translation invariant Gibbs measure are given as \(\varphi (x) = \beta (x^4-x^2)\) and \(w(t,x) = \alpha (1+|t|)^{-\gamma } x^2\), for some values of \(1<\gamma \leq 2\) and \(\alpha, \beta >0\).

Reviewer: Nicolas Privault (La Rochelle)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B21	Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics

Keywords:

Gibbs measures; Brownian motion; pair potential

References:

[1]	Brascamp, H. J. and Lieb, E. H. (1976). On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 366-389. · Zbl 0334.26009 · doi:10.1016/0022-1236(76)90004-5
[2]	Bricmont, J., Lebowitz, J. L. and Pfister, C. (1981). Periodic Gibbs states of ferromagnetic spin systems. J. Statist. Phys. 24 269-278. · doi:10.1007/BF01007648
[3]	Dyson, F. J. (1971). Existence of a phase-transition in a one-dimensional Ising ferromagnet. Comm. Math. Phys. 12 91-107. · Zbl 1306.47082 · doi:10.1007/BF01645907
[4]	Fr öhlich, J. and Spencer, T. (1982). The phase transition in the one-dimensional Ising model with 1/r2 interaction energy. Comm. Math. Phys. 84 87-102. · Zbl 1110.82302 · doi:10.1007/BF01208373
[5]	Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin. · Zbl 0657.60122 · doi:10.1515/9783110850147
[6]	Glimm, J. and Jaffe, A. (1981). Quantum Physics: A Functional Integral Point of View. Springer, Berlin. · Zbl 0461.46051
[7]	Lebowitz, J. L. and Presutti, E. (1976). Statistical mechanics of systems of unbounded spins. Comm. Math. Phys. 50 195-218. · doi:10.1007/BF01609401
[8]	Papangelou, F. (1984). On the absence of phase transition in one-dimensional random fields I. Sufficient conditions.Wahrsch. Verw. Gebiete 67 255-263. · Zbl 0545.60098 · doi:10.1007/BF00535002
[9]	Preston, C. J. (1974). A generalization of the FKG inequality. Comm. Math. Phys. 36 223- 241. · doi:10.1007/BF01645981
[10]	Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18 127-159. · Zbl 0198.31101 · doi:10.1007/BF01646091
[11]	Simon, B. (1979). Functional Integration and Quantum Physics. Academic Press, London. · Zbl 0434.28013