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The dynamic \({\Phi^4_3}\) model comes down from infinity. (English) Zbl 1384.81068

This paper deals with dynamic \(\Phi^4_3\) model of quantum field theory. The authors prove an a priori bound for the dynamic \(\Phi^4_3\) model on the torus, which is given by the stochastic partial differential equation \(\partial_{t} X=\Delta X-X^3+mX+\xi\), on \(R_{+}\times[-1,1]^3\), \(X(0,\cdot)=X_0\), where \(\xi\) is a white noise over \(R_{+}\times[-1,1]^3\), \(m\in R\). The main result implies that for every \(p<\infty\) and \(\varepsilon>0\) sufficiently small, we have \(\text{E}\left[\sup_{0<t\leq1}\sup_{X_0\in{\mathcal B}^{-(1/2)-\varepsilon}_{\infty}}\left(\sqrt{t}\| X(t)\|_{{\mathcal B}^ {-(1/2)-\varepsilon}_{\infty}}\right)^{p}\right]<\infty\). Here, for \(\alpha>0\), \({\mathcal B}^{-\alpha}_{\infty}\) is the Besov-Hölder space of negative regularity \(-\alpha\). Using this bound it is proved the global existence of solutions to the considered equation. Also it can be used to construct invariant measures via the Krylov-Bogoliubov method.

MSC:

81T10 Model quantum field theories
83C47 Methods of quantum field theory in general relativity and gravitational theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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