Stochastic quantization of the two-dimensional polymer measure
Denote by \(X\) the space of all continuous paths \(\omega: [0,1]\to\mathbb R^{2}\) with \(\omega(0) = 0\) and by \(\nu_{0}\) the Wiener measure on \(X\). Let \(\alpha\) be the self-intersection local time of the planar Brownian measure, defined formally as \(\alpha = \int^{1}_{0}\int^{1}_{t} \delta (\omega_{t}-\omega_{s}) ds dt\). A rigorous construction of the renormalized version \(\xi\) of \(\alpha\) may be found e.g. in \textit{J. F. Le Gall}'s paper [in: Séminaire de probabilités XIX. Lect. Notes Math. 1123, 314-331 (1985; Zbl 0563.60072)]. There exists a \(g_{0} > 0\) such that \(\mathbf E \exp(-g\xi)<\infty\) for each \(g\in\left]-g_{0},+\infty \right[\). If \(g>g_{0}\), then the two-dimensional polymer measure \(\nu_{g}\) is a Borel probability measure on \(X\) with density \([\mathbf E\exp(-g\xi)]^{-1}\exp(-g\xi)\) with respect to \(\nu_0\). Using the theory of Dirichlet forms, the authors construct a \(\nu_{g}\)-symmetric \(X\)-valued diffusion process \(M\). In particular, \(\nu_g\) is an invariant measure for \(M\), and the process \(M\) is ergodic under time translations: if \(P_{t}\) denotes the transition semigroup of \(M\), then \(P_{t}f\to \int_{X} f d\nu_{g}\) as \(t\to \infty\) quasi-everywhere for every bounded real Borel function \(f\) on \(X\).
scientific article; zbMATH DE number 1369641