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**A coupling approach to randomly forced nonlinear PDE’s. I.**
*(English)*
Zbl 0991.60056

Let \(H\) be a separable Hilbert space, \(\{e_{k}\}\) its orthonormal basis, and \(S: H\to H\) a mapping Lipschitz continuous on bounded sets. Let \(\eta_{k}\), \(k\geq 1\), be independent identically distributed \(H\)-valued random variables of the form \(\eta_k = \sum^\infty_{j=1} b_{j}\xi_{jk}e_{j}\), where \(b_{j}\geq 0\) satisfy \(\sum^\infty_{j=1} b^2_{j} <\infty\), and for each \(j\geq 1\) the law of the real-valued random variable \(\xi_{jk}\) has a density \(p_{j}\) with respect to Lebesgue measure, \(p_{j}\) being a function of bounded variation, zero outside \([-1,1]\), and such that \(\int_{[-\varepsilon,\varepsilon]} p_{j}(r)dr >0\) for all \(\varepsilon>0\). Long-time behaviour of a Markov chain \((u_{n},\mathbb P_{x})\), defined recursively by \(u_{k} = S(u_{k-1}) + \eta_{k}\), \(k\geq 1\), \(\mathbb P_{x}\{u_0 = x\} =1\), is studied.

Suppose that for any \(R>r>0\) there exist \(a<1\) and \(n_0\in\mathbb N\) such that \(\|S^{n}(y)\|\leq \max(a\|y\|, r)\) for all \(y\in H\), \(\|y\|\leq R\), and all \(n\geq n_0\). Denote by \(K\) the set \(\{\sum^\infty_{j=1} f_{j}e_{j}\), \(|f_{j} |\leq b_{j}\) for all \(j\geq 1\}\), and for any bounded set \(B\) define \(\mathcal A_{0}(B) = B\), \(\mathcal A_{k}(B) = S(\mathcal A_{k-1} (B)) + K\), \(k\geq 1\). Assume that there exists \(\rho >0\) such that for every bounded set \(B\) a \(k_0\in \mathbb N\) may be found such that \(\mathcal A_{k}(B)\) is contained in a ball with radius \(\rho\) centered at \(0\) for all \(k\geq k_0\). Let \(Q_{N}\) be the orthogonal projection onto the orthogonal complement of the space spanned by \(e_1,\ldots,e_{N}\). For each \(R>0\), let there exist \(\gamma_{N}(R)\), \(\gamma_{N}(R)\searrow 0\) as \(N\to\infty\), such that \[ \|Q_{N}S(y_1)-Q_{N}S(y_2)\|\leq \gamma_{N}(R)\|y_1 - y_2\| \] for all \(y_1,y_2\in H\), \(\|y_{i}\|\leq R\). Under the above hypotheses it is proven by using the coupling method that there exists an \(N\in\mathbb N\) such that the chain \(\{u_{n}\}\) has a unique invariant measure \(\mu\) whenever \(b_{j}\neq 0\), \(j=1,\ldots, N\). Moreover, \[ |\mathbb E_{x}f(u_{k}) - \mu(f)|\leq C_{R} \exp(-ck^{1/2}) \Bigl(\sup_{H}|f|+ \text{Lip}(f)\Bigr) \] for some constant \(c>0\), all bounded Lipschitz continuous functions \(f\), all \(R>0\) and \(x\in H\), \(\|x\|\leq R\); with a constant \(C_{R}\) dependent only on \(R\). Markov chains of the above type have arisen in the authors’ investigations concerning two-dimensional Navier-Stokes equations perturbed by a random kick-force [Commun. Math. Phys. 213, No. 2, 291-330 (2000; Zbl 0974.60046)].

Suppose that for any \(R>r>0\) there exist \(a<1\) and \(n_0\in\mathbb N\) such that \(\|S^{n}(y)\|\leq \max(a\|y\|, r)\) for all \(y\in H\), \(\|y\|\leq R\), and all \(n\geq n_0\). Denote by \(K\) the set \(\{\sum^\infty_{j=1} f_{j}e_{j}\), \(|f_{j} |\leq b_{j}\) for all \(j\geq 1\}\), and for any bounded set \(B\) define \(\mathcal A_{0}(B) = B\), \(\mathcal A_{k}(B) = S(\mathcal A_{k-1} (B)) + K\), \(k\geq 1\). Assume that there exists \(\rho >0\) such that for every bounded set \(B\) a \(k_0\in \mathbb N\) may be found such that \(\mathcal A_{k}(B)\) is contained in a ball with radius \(\rho\) centered at \(0\) for all \(k\geq k_0\). Let \(Q_{N}\) be the orthogonal projection onto the orthogonal complement of the space spanned by \(e_1,\ldots,e_{N}\). For each \(R>0\), let there exist \(\gamma_{N}(R)\), \(\gamma_{N}(R)\searrow 0\) as \(N\to\infty\), such that \[ \|Q_{N}S(y_1)-Q_{N}S(y_2)\|\leq \gamma_{N}(R)\|y_1 - y_2\| \] for all \(y_1,y_2\in H\), \(\|y_{i}\|\leq R\). Under the above hypotheses it is proven by using the coupling method that there exists an \(N\in\mathbb N\) such that the chain \(\{u_{n}\}\) has a unique invariant measure \(\mu\) whenever \(b_{j}\neq 0\), \(j=1,\ldots, N\). Moreover, \[ |\mathbb E_{x}f(u_{k}) - \mu(f)|\leq C_{R} \exp(-ck^{1/2}) \Bigl(\sup_{H}|f|+ \text{Lip}(f)\Bigr) \] for some constant \(c>0\), all bounded Lipschitz continuous functions \(f\), all \(R>0\) and \(x\in H\), \(\|x\|\leq R\); with a constant \(C_{R}\) dependent only on \(R\). Markov chains of the above type have arisen in the authors’ investigations concerning two-dimensional Navier-Stokes equations perturbed by a random kick-force [Commun. Math. Phys. 213, No. 2, 291-330 (2000; Zbl 0974.60046)].

Reviewer: Jan Seidler (Praha)

### MSC:

60J05 | Discrete-time Markov processes on general state spaces |

35R60 | PDEs with randomness, stochastic partial differential equations |

37H10 | Generation, random and stochastic difference and differential equations |

76F55 | Statistical turbulence modeling |