zbMATH — the first resource for mathematics

Strong solutions to the stochastic quantization equations. (English) Zbl 1071.81070
Let $$G=[0, 2 \pi ]^2$$ 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.

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
Full Text:
References:
 [1] Albeverio, S. and Röckner, M. (1991). Stochastic differential equations in infinite dimensions: Solutions via Dirichlet forms. Probab. Theory Related Fields 89 347–386. · Zbl 0725.60055 [2] Albeverio, S., Haba, Z. and Russo, F. (2001). A two-space dimensional semilinear heat equation perturbed by (Gaussian) white noise. Probab. Theory Related Fields 121 319–366. · Zbl 0993.60062 [3] Borkar, V. S., Chari, R. T. and Mitter, S. K. (1988). Stochastic quantization of field theory in finite and infinite volume. J. Funct. Anal. 81 184–206. · Zbl 0657.60084 [4] Chemin, J.-Y. (1995). Fluides parfaits incompressibles . Astérisque 230 . · Zbl 0829.76003 [5] Chemin, J.-Y. (1996). About Navier–Stokes system. Prépublication R96023, Laboratoire d’Analyse Numérique de l’Université Paris 6. [6] Da Prato, G. and Debussche, A. (2002). Two-dimensional-Navier–Stokes equations driven by a space–time white noise. J. Functional Anal. 196 180–210. · Zbl 1013.60051 [7] Da Prato, G. and Tubaro, L. (1996). Introduction to stochastic quantization. Pubblicazione UTM 505, Dipartimento di Matematica dell’Università di Trento. · Zbl 0853.60052 [8] Da Prato, G. and Tubaro, L. (2000). A new method to prove self-adjointness of some infinite dimensional Dirichlet operator. Probab. Theory Related Fields 118 131–145. · Zbl 0971.47019 [9] Da Prato, G. and Zabczyk, J. (1992). Stochastic equations in infinite dimensions. Encyclopedia Math. Appl. 44 . Cambridge Univ. Press. · Zbl 0761.60052 [10] Gatarek, D. and Goldys, B. (1996). Existence, uniqueness and ergodicity for the stochastic quantization equation. Studia Math. 119 179–193. · Zbl 0858.60058 [11] Jona Lasinio, G. and Mitter, P. K. (1985). On the stochastic quantization of field theory. Comm. Math. Phys. 101 409–436. · Zbl 0588.60054 [12] Liskevich, V. and Röckner, M. (1998). Strong uniqueness for a class of infinite dimensional Dirichlet operators and application to stochastic quantization. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 69–91. · Zbl 0953.60056 [13] Mikulevicius, R. and Rozovskii, B. (1998). Martingale problems for stochastic PDE’s. Math. Surveys Monogr. 64 243–326. · Zbl 0938.60047 [14] Parisi, G. and Wu, Y. S. (1981). Perturbation theory without gauge fixing. Sci. China Ser. A 24 483–490. [15] Simon, B. (1974). The $$P(\phi)_2$$ Euclidean (Quantum) Field Theory . Princeton Univ. Press. · Zbl 1175.81146
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.