## 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)].

### 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

### Keywords:

invariant measure; coupling; kick-noise

Zbl 0974.60046
Full Text: