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The Ornstein-Uhlenbeck semigroup in exterior domains. (English) Zbl 1114.47043
Let $$K$$ be a compact in $${\mathbb R}^n$$ with $$C^{1,1}$$-boundary, $$\Omega ={\mathbb R}^n\setminus K$$ and matrices $$M\in M_n({\mathbb R})\setminus\{0\}$$ be given. Then: (i) in $$L^p(\Omega )$$, $$1<p<\infty$$, operators of the form $$Lu(x):=\Delta u(x)+Mx\cdot \nabla u(x)$$, $$x\in\Omega$$, with $$D(L)=\{u\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$$: $$Mx\cdot \nabla u\in L^p(\Omega )\}$$, generate $$C_0$$-semigroups and $$\|e^{tL}\|\leq e^{-(\operatorname{tr} M)t/p}$$, $$t\geq0$$; (ii) for $$q\in[p,\infty)$$, upper bounds of $$\|e^{tL}f\|_q$$ and $$\|\nabla e^{tL}f\|_q$$ obtained ($$t>0$$, $$\|f\|_p=1$$) are similar to those in the case of an “interior” bounded domain $$\Omega$$.
If $$\partial \Omega$$ is only Lipschitz, (i) holds for $$p$$ close to 2 and, given also the so-called uniform outer ball condition, (i) and (ii) follow for $$1<p\leq q\leq 2$$.

##### MSC:
 47D07 Markov semigroups and applications to diffusion processes 47D06 One-parameter semigroups and linear evolution equations
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