Stochastic quantization of field theory in finite and infinite volume (Q1110915)

The program of stochastic quantization proposed by \textit{G. Parisi} and \textit{Y. S. Wu} [Sci. Sin. 24, No. 4, 483--496 (1981; Zbl 1480.81051)] suggests the construction of a nonlinear stochastic differential equation taking values in the space of distributions on a finite rectangle in \(R^ 2\), such that the resultant process is ergodic Markov with its unique invariant measure coincident with the finite volume Euclidean \((\phi^ 4)_ 2\) measure. This paper constructs a class of such processes using the theory of Dirichlet forms. Ergodicity of the process and its uniqueness as a weak solution of the corresponding stochastic differential equation are proved using change of measure arguments. As the rectangle increases to \(R^ 2\), an infinite volume limit of these processes is shown to exist along a subsequence, in an appropriate sense. This uses recent results on weak convergence of distribution-valued processes.