Global well-posedness of the dynamic \(\Phi^{4}\) model in the plane.

*(English)*Zbl 1381.60098The paper deals with the non linear stochastic partial differential equation (SPDE)
\[
\partial_t X=\triangle X-X^{:3:}+aX+\xi \quad\qquad X(t,\cdot)=X_0
\]
on \(\mathbb R_+\times\mathbb R^2\), where \(\xi\) is a white noise on \(\mathbb R\times\mathbb R^2\), \(a\) is a real parameter and \(X^{:3:}\) denotes the Wick (or normally ordered) power of \(X\). The main result is that – with the due qualifications – the solution exists and is unique (well-posedness) globally in time and on the full space \(\mathbb R^2\) in the probabilistically strong sense, when the initial condition takes values in a suitable weighted Besov space. As recalled by the authors, the relevance of this SPDE naturally arises from its original role in the dynamics of the Euclidean \(\Phi_2^4\) quantum field theory, and in the stochastic quantization. As such it has been extensively discussed in the literature, but its complete understanding is still elusive, in particular for its three-dimensional version.

An acknowledged breakthrough has been achieved in [ibid. 31, No. 4, 1900–1916 (2003; Zbl 1071.81070)] by G. Da Prato and A. Debussche on the two-dimensional torus where they showed the “short time existence and uniqueness in the probabilistically strong sense, via a fixed-point argument in a suitable Besov space.” The present paper builds on this result by extending its methods: the authors first construct periodic solutions for a short time on a torus of arbitrary size, but then the technically new part of the proof is that they are now able to derive an entire panoply of “a priori estimates that are strong enough to imply nonexplosion on the torus for an arbitrary initial condition in a natural Besov space.” Finally, “as the torus grows larger, the family of solutions remains in a compact subset of a suitable Besov space with polynomial weights. This implies the existence of solutions by extracting a converging subsequence.”

The paper is hence eminently technical and innovates on the previous proof methods, even with a few avowed limitations: it is not hoped, for instance, “to be able to construct local solutions of [our SPDE] directly on the full space via a Picard iteration.” The authors however still “hope that the present article will serve as a first step towards proving global-in-time well-posedness for the three-dimensional version of” the given SPDE.

The proposed procedure for solving the SPDE breaks down into two steps: first we consider the solution \(Z\) of the stochastic heat equation on \(\mathbb R_+\times\mathbb R^2\) \[ \partial_t Z=\triangle Z+\xi\qquad\quad Z(0,\cdot)=Z_0 \] then we define – by approximation – its Wick powers, and finally we find that \(X=Y+Z\) solves our SPDE when \(Y\) is a solution of \[ \partial_tY=\triangle Y-Y^3-3Y^2Z-3YZ^{:2:}-Z^{:3:}+a(Y+Z) \] with \(Y(0,\cdot)=0\). The main result (see Theorem 1.1, also restated as Theorem 8.1) is then that, for \(\beta>2\), \(\sigma<2\), \(\alpha>0\), when \(X_0\) is taken in the Besov space \(\widehat{\mathcal B}^{-\alpha,\sigma}_{3p,\infty}\) with polynomial weights \(\widehat{w}_\sigma(x)=\left(1+| x|^2\right)^{-\alpha/2}\), with probability \(1\) there exists a unique \(Y\) solution of the previous SPDE taking values in another Besov space \(\widehat{\mathcal B}^{-\alpha,\sigma}_{3p,\infty}\) with the same weights.

In order to make the paper self contained, the authors go a long way in defining and describing the particular Besov spaces they are working within: nearly half of this extended paper is devoted to this preparation. Then, after analyzing the properties of the solutions \(Z\) of the stochastic heat equation, they show the existence and uniqueness of the global solutions \(Y\) of the second SPDE on the torus, namely in the periodic case. They proceed further to derive strong a priori estimates on the solutions \(Y\) in order to show that all the “periodised” solutions belong to suitable compact subsets of polynomially weighted Besov spaces. And now, the stage is finally set to state the main result about the existence and uniqueness of the global solutions \(Y\), and hence \(X\) of out initial SPDE. Non senza fatiga si giunge al fine (Not without toil we reach the end; G. Frescobaldi, 1627).

An acknowledged breakthrough has been achieved in [ibid. 31, No. 4, 1900–1916 (2003; Zbl 1071.81070)] by G. Da Prato and A. Debussche on the two-dimensional torus where they showed the “short time existence and uniqueness in the probabilistically strong sense, via a fixed-point argument in a suitable Besov space.” The present paper builds on this result by extending its methods: the authors first construct periodic solutions for a short time on a torus of arbitrary size, but then the technically new part of the proof is that they are now able to derive an entire panoply of “a priori estimates that are strong enough to imply nonexplosion on the torus for an arbitrary initial condition in a natural Besov space.” Finally, “as the torus grows larger, the family of solutions remains in a compact subset of a suitable Besov space with polynomial weights. This implies the existence of solutions by extracting a converging subsequence.”

The paper is hence eminently technical and innovates on the previous proof methods, even with a few avowed limitations: it is not hoped, for instance, “to be able to construct local solutions of [our SPDE] directly on the full space via a Picard iteration.” The authors however still “hope that the present article will serve as a first step towards proving global-in-time well-posedness for the three-dimensional version of” the given SPDE.

The proposed procedure for solving the SPDE breaks down into two steps: first we consider the solution \(Z\) of the stochastic heat equation on \(\mathbb R_+\times\mathbb R^2\) \[ \partial_t Z=\triangle Z+\xi\qquad\quad Z(0,\cdot)=Z_0 \] then we define – by approximation – its Wick powers, and finally we find that \(X=Y+Z\) solves our SPDE when \(Y\) is a solution of \[ \partial_tY=\triangle Y-Y^3-3Y^2Z-3YZ^{:2:}-Z^{:3:}+a(Y+Z) \] with \(Y(0,\cdot)=0\). The main result (see Theorem 1.1, also restated as Theorem 8.1) is then that, for \(\beta>2\), \(\sigma<2\), \(\alpha>0\), when \(X_0\) is taken in the Besov space \(\widehat{\mathcal B}^{-\alpha,\sigma}_{3p,\infty}\) with polynomial weights \(\widehat{w}_\sigma(x)=\left(1+| x|^2\right)^{-\alpha/2}\), with probability \(1\) there exists a unique \(Y\) solution of the previous SPDE taking values in another Besov space \(\widehat{\mathcal B}^{-\alpha,\sigma}_{3p,\infty}\) with the same weights.

In order to make the paper self contained, the authors go a long way in defining and describing the particular Besov spaces they are working within: nearly half of this extended paper is devoted to this preparation. Then, after analyzing the properties of the solutions \(Z\) of the stochastic heat equation, they show the existence and uniqueness of the global solutions \(Y\) of the second SPDE on the torus, namely in the periodic case. They proceed further to derive strong a priori estimates on the solutions \(Y\) in order to show that all the “periodised” solutions belong to suitable compact subsets of polynomially weighted Besov spaces. And now, the stage is finally set to state the main result about the existence and uniqueness of the global solutions \(Y\), and hence \(X\) of out initial SPDE. Non senza fatiga si giunge al fine (Not without toil we reach the end; G. Frescobaldi, 1627).

Reviewer: Nicola Cufaro Petroni (Bari)

##### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

81S20 | Stochastic quantization |

81T27 | Continuum limits in quantum field theory |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

35K55 | Nonlinear parabolic equations |

30H25 | Besov spaces and \(Q_p\)-spaces |