Abstract
Models of quantum mechanical anharmonic lattice systems ("anharmonic crystals") are described. Temperature quantum Gibbs states are represented by classical Gibbs measures for lattice systems of loop-valued spin variables. These Gibbs measures are also obtained as invariant (equilibrium) measures of a system of stochastic differential equations ("stochastic dynamics", "stochastic quantization"). Existence and uniqueness results for these equations are established and a construction of the solution via a finite volume approximation is given. The Markov property of this solution is also exhibited and properties of the Gibbs distributions (existence, a prioiri estimates, regularity of support) are characterized in terms of the stochastic dynamics. Ergodicity and uniqueness of the Gibbs distributions are also discussed.
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