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

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 $$\alpha, \beta >0$$.

### 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
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### References:

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