The authors prove the existence of non-random invariant sets and establish comparison results for a class of parabolic systems of \(m\) stochastic partial differential equations in \(m\) unknowns, where \(m\in\mathbb{N}^+\) is arbitrary. The problem is the fact that the systems are defined on bounded domains of \(\mathbb{R}^n\) where \(n\in\mathbb{N}^+\) is arbitrary, and the authors consider the following system of nonlinear Itô parabolic SPDE’s \[ du^\ell(x,t)=(-A^\ell(x,t,D)u^\ell(x,t)+f^\ell(x,t,u(x,t))Du(x,t)dt+ \sum^\infty_{j=1}q_jg^\ell_j(x,t,u(x,t))dW(t),\tag{1} \] \(\ell=1,2,\dots,m\), \(x\in {\mathcal O}\), \(t>0\), in a bounded smooth domain \({\mathcal O}\in\mathbb{R}^n\) with the boundary and initial conditions \[ B^\ell(x,D)u^\ell(x,t)=0,\;x\in\partial{\mathcal O},\;t>0;\quad u^\ell(0,x)=u^\ell_0(x),\;x\in\overline{\mathcal O},\;\ell=1,2,\dots,m.\tag{2} \] The key of the proof is a Wong-Zakai type approximation theorem for the stochastic problem (1) and (2).
Section 2 is entitled “Main hypotheses and preliminaries”. In Section 3 the authors prove the existence and uniqueness theorem for the stochastic problem (1) and (2). In the first part of Section 4 they study properties of smooth approimation of the problem (1) and (2) and introduce the concept of \(\varepsilon\)-residual approximate solutions to the problem (1) and (2) and prove their existence. The authors invoke some ideas developed by Ya. I. Belopolskaya and Yu. L. Daletskij [“Stochastic equations and differential geometry” (1990; Zbl 0696.60053)] in the abstract setting of bounded operators. Using these considerations they conclude the proof of the approximation theorem.
Section 5 contains the main results. The authors give conditions on \(f^i(x,t,u,\eta)\) and \(g^i_j(x,t,u)\) under which problem (1) and (2) possesses solutions \(u(x,t,\omega)\) that belong to some deterministic set \(D\subset\mathbb{R}^m\) for all \((x,t)\in\overline{\mathcal O}\times\mathbb{R}_+\) and for almost all \(\omega \in \Omega\) (see Theorem 5.6). The result on the existence of invariant sets for (1) and (2) extends, to stochastic systems, the theorem proved by H. Amann (1978) in the deterministic case. The authors invoke these results to obtain the corresponding result for the smooth regularizations of the system (1) and (2). They also prove a comparison principle (see Theorem 5.8) for the problem considered in the case of cooperative \(f\) and \(g_j\). Section 6 treats several examples concerning certain stochastic Lotka-Volterra or Ginzburg-Landau equations.