The Annals of Probability

Gibbs Measures Relative to Brownian Motion

Hirofumi Osada and Herbert Spohn

Full-text: Open access


We consider Brownian motion perturbed by the exponential of an action. The action is the sum of an external, one-body potential and a two-body interaction potential which depends only on the increments. Under suitable conditions on these potentials, we establish existence and uniqueness of the corresponding Gibbs measure. We also provide an example where uniqueness fails because of a slow decay in the interaction potential.

Article information

Ann. Probab., Volume 27, Number 3 (1999), 1183-1207.

First available in Project Euclid: 29 May 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B21

Gibbs measure Brownian motion pair potential


Osada, Hirofumi; Spohn, Herbert. Gibbs Measures Relative to Brownian Motion. Ann. Probab. 27 (1999), no. 3, 1183--1207. doi:10.1214/aop/1022677444.

Export citation


  • [1] Brascamp, H. J. and Lieb, E. H. (1976). On extensions of the Brunn-Minkowski and Pr´ekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 366-389.
  • [2] Bricmont, J., Lebowitz, J. L. and Pfister, C. (1981). Periodic Gibbs states of ferromagnetic spin systems. J. Statist. Phys. 24 269-278.
  • [3] Dyson, F. J. (1971). Existence of a phase-transition in a one-dimensional Ising ferromagnet. Comm. Math. Phys. 12 91-107.
  • [4] Fr ¨ohlich, 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.
  • [5] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
  • [6] Glimm, J. and Jaffe, A. (1981). Quantum Physics: A Functional Integral Point of View. Springer, Berlin.
  • [7] Lebowitz, J. L. and Presutti, E. (1976). Statistical mechanics of systems of unbounded spins. Comm. Math. Phys. 50 195-218.
  • [8] Papangelou, F. (1984). On the absence of phase transition in one-dimensional random fields I. Sufficient conditions.Wahrsch. Verw. Gebiete 67 255-263.
  • [9] Preston, C. J. (1974). A generalization of the FKG inequality. Comm. Math. Phys. 36 223- 241.
  • [10] Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18 127-159.
  • [11] Simon, B. (1979). Functional Integration and Quantum Physics. Academic Press, London.