Gibbs measures relative to Brownian motion
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 \[ \begin{multlined} H_T (X,\xi) = \int_{|t |\leq T} \varphi (X_t) dt + {1 \over 2} \int_{|t |, |u |\leq T} w(t-u,X_t-X_u) dt du\\ + \int_{|t|\leq T < |u |} w(t-u,X_t-\xi_u) dt du.\end{multlined} \] 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 \(\alpha, \beta >0\).
scientific article; zbMATH DE number 1462548