Strong solutions to the stochastic quantization equations.

*(English)*Zbl 1071.81070Let \(G=[0, 2 \pi ]^2\) and \(H=L^2(G)\). The authors consider stochastic quantization equations in space dimension 2 with periodic boundary conditions:
\[
d X = ( AX + : p(X):) \,dt + d W(t), \quad X(0) = x \tag{1}
\]
where \(A : D(A) \subset H \to H\) is the linear operator \(A x = \varDelta x - x\), \(D(A) = H_{\#}^2(G)\). \(H_{\#}^2(G)\) is the subspace of \(H^2(G)\) of all functions which are periodic together with their first derivatives, \(p(\xi)\) \(=\) \(\sum_{k=0}^n a_k \xi^k\) is a polynomial of odd degree \(n \geq 3\) with \(a_n < 0\), and \(: p(x) :\) means the renormalization of \(p(x)\). \(W\) is a cylindrical Wiener process defined in a probability space \(( \Omega, {\mathcal F}, {\mathbb P})\) and taking values on \(L^2(G)\). Formally, (1) is a gradient system with an invariant Gibbs measure \(\nu\) defined as
\[
\nu(dx) = c \cdot e^{( : q(x):, 1)} \mu(dx), \tag{2}
\]
where \(q\) is a primitive of \(p\), \(c\) is a normalization constant, and \(\mu\) is Gaussian invariant measure of the free field. The goal of this paper is to construct a unique strong (in the probability sense) solution for the problem (1) for \(\mu\)-almost every initial data \(x\).

One of the main difficulties when dealing with renormalized products is that they do not depend continuously on their arguments, so that a fixed point theorem is not applicable any more to the case. The authors’ idea to overcome this is to split the unknown into two parts: \(X = Y + z\), where \(z(t)\) is the stochastic convolution \[ z(t) = \int_{-\infty}^t e^{(t-s)A} \,d W(s) \tag{3} \] and is also a stationary solution to the linear version of (1). Moreover, the problem (1) becomes \[ \frac{dY}{d t} = A Y + \sum_{k=0}^n a_k \sum_{l=0}^k C_k^l Y^l : z^{k-l}:, \quad Y(0) = x - z(0). \tag{4} \] Since the law of \(z(t)\) is equal to \(\mu\) for any \(t \in {\mathbb R}\), the term \(: z^n :\) can be defined in the classical way through the formula \({\mathbb E}[ g( : z^n :)]\) \(=\) \(\int_{{\mathcal H}} g(: x^n :) \mu(dx)\) for any Borel bounded real function \(g\). The main advantage of considering (4) is that now the nonlinear term is a continuous function with respect to the unknown \(Y\). However, the price to pay is to work with Besov spaces \({\mathcal B}_{p,r}^s\). Fixed point results on Besov spaces can be applied to solve (4). Note that \(X\) is a solution of (1) if and only if \(X = Y + z\) and \(Y\) is a mild solution of (4).

Here is the principal result (Theorem 4.2) in this paper: “Let \(\alpha = 2/p + 2 s\), \(p > n\), \(r \geq 1\) and \[ 0 > s > \max \left\{ - \frac{2}{p(2n + 1)}, \frac{-1}{n-1} \left( 1 - \frac{n}{r} \right) \right\}. \] Then for all \(\nu\)-almost every \(x\) there exists for any \(T \geq 0\) a unique solution of (1) such that \[ Y \in C( [0, T]; {\mathcal B}_{p,r}^s(G)) \cap L^p(0,T; {\mathcal B}_{p,r}^{\alpha}(G)). \text{''} \] The method is similar to what is used in the recent work done by the same authors [J. Funct. Anal. 196, No. 1, 180–210 (2002; Zbl 1013.60051)] for the two-dimensional Navier-Stokes equations.

One of the main difficulties when dealing with renormalized products is that they do not depend continuously on their arguments, so that a fixed point theorem is not applicable any more to the case. The authors’ idea to overcome this is to split the unknown into two parts: \(X = Y + z\), where \(z(t)\) is the stochastic convolution \[ z(t) = \int_{-\infty}^t e^{(t-s)A} \,d W(s) \tag{3} \] and is also a stationary solution to the linear version of (1). Moreover, the problem (1) becomes \[ \frac{dY}{d t} = A Y + \sum_{k=0}^n a_k \sum_{l=0}^k C_k^l Y^l : z^{k-l}:, \quad Y(0) = x - z(0). \tag{4} \] Since the law of \(z(t)\) is equal to \(\mu\) for any \(t \in {\mathbb R}\), the term \(: z^n :\) can be defined in the classical way through the formula \({\mathbb E}[ g( : z^n :)]\) \(=\) \(\int_{{\mathcal H}} g(: x^n :) \mu(dx)\) for any Borel bounded real function \(g\). The main advantage of considering (4) is that now the nonlinear term is a continuous function with respect to the unknown \(Y\). However, the price to pay is to work with Besov spaces \({\mathcal B}_{p,r}^s\). Fixed point results on Besov spaces can be applied to solve (4). Note that \(X\) is a solution of (1) if and only if \(X = Y + z\) and \(Y\) is a mild solution of (4).

Here is the principal result (Theorem 4.2) in this paper: “Let \(\alpha = 2/p + 2 s\), \(p > n\), \(r \geq 1\) and \[ 0 > s > \max \left\{ - \frac{2}{p(2n + 1)}, \frac{-1}{n-1} \left( 1 - \frac{n}{r} \right) \right\}. \] Then for all \(\nu\)-almost every \(x\) there exists for any \(T \geq 0\) a unique solution of (1) such that \[ Y \in C( [0, T]; {\mathcal B}_{p,r}^s(G)) \cap L^p(0,T; {\mathcal B}_{p,r}^{\alpha}(G)). \text{''} \] The method is similar to what is used in the recent work done by the same authors [J. Funct. Anal. 196, No. 1, 180–210 (2002; Zbl 1013.60051)] for the two-dimensional Navier-Stokes equations.

Reviewer: Isamu Dôku (Saitama)

##### MSC:

81S20 | Stochastic quantization |

28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |

42B99 | Harmonic analysis in several variables |

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\textit{G. Da Prato} and \textit{A. Debussche}, Ann. Probab. 31, No. 4, 1900--1916 (2003; Zbl 1071.81070)

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