Quasi regular Dirichlet forms and the stochastic quantization problem

Abstract

After recalling basic features of the theory of symmetric quasi regular Dirichlet forms we show how by applying it to the stochastic quantization equation, with Gaussian space-time noise, one obtains weak solutions in a large invariant set. Subsequently, we discuss non symmetric quasi regular Dirichlet forms and show in particular by two simple examples in infinite dimensions that infinitesimal invariance, does not imply global invariance. We also present a simple example of non-Markov uniqueness in infinite dimensions.

Stochastic Partial Differential Equations: An Introduction

“…N ; N F x ; N Q x ; . N F x t // where N F x WD B.H/˝B.B/˝B.W 0 / .B.H/˝B tC" .B/˝B tC" .W 0 /; N N x /; where N N x WD fN 2 N F x j N Q x .N/ D 0g.As in the proof of Lemma E.0.12 one shows that for x outside a -zero set N1 2 B.H/, .…; … 3 / on . N ; N F x ; N Q x ; .…”

“…Since k1 OE0;t n OE .s/.t n s/˛1 S.t n s/'.s/ 1 OE0;tOE .s/.t s/˛ 1 S.t s/'.s/k 6 F t n .s/ C F t .s/ for all s 2 OE0; T the assertion follows. u t Thus we have found a tool to check whether the process S.t s/B.X.…”

“…For a detailed analysis of Example 1.2.1 and for the non-linear much harder stochastic quantization equations for interacting Euclidean quantum fields we refer to [2] and the recent survey [1] (including the references therein). All other examples above under the respective appropriate conditions are covered by the theory presented in this course and all of them are analyzed in detail here.…”

“…So, already in two dimensions one can only expect Schwarz distribution-valued solutions to the SPDE, hence the simplest non-linear parts of the drift, as e.g. a power of the solution, have to be defined by renormalization techniques (see [2] and also [1]). Recently, a breakthrough has been achieved in this direction by Martin Hairer in [44] (for which, together with his work on the KPZ-equation [43] and other beautiful results in the field, he was awarded the Fields Medal in 2014).…”

Non-local Markovian Symmetric Forms on Infinite Dimensional Spaces I. The closability and quasi-regularity

“…The natural setting is the one where the Hilbert space, where the Dirichlet forms are defined (as quadratic forms), is L 2 (E; m), with E a general locally compact separable metric space (when E is a topological vector space, the dimension of the space is thus finite) and m a positive Radon measure on it (cf., e.g., [46,48,49,60,70,84,88], [4], [34,35] and references therein). Also, many results have been developed on the theory of general (non-symmetric) local Dirichlet forms defined on L 2 (E; m), with general topological spaces including the case of some infinite dimensional topological vector spaces, and m some Radon measures on them (cf., e.g., [3,[14][15][16][17][25][26][27][29][30][31][32]68,69,85,87] and references therein). In the general abstract framework, in particular [3,69], the Dirichlet forms need not be local ones, however all examples except those considered through the framework of subordination given in [A,Rüdiger] (cf.…”