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Generation results for elliptic operators with unbounded diffusion coefficients in $$L^p$$-and $$C_b$$-spaces. (English) Zbl 1152.47031
In this paper, it is proven that for $$N$$-dimensional second order elliptic operators with diffusion coefficients growing at most quadratically and drifts growing at most linearly at infinity, there are realizations generating analytic semigroups in $$L^p$$ (for any $$p$$ in the interval $$[1,+ \infty]$$) and in $$C_b$$ (in $$\mathbb R^N$$). The domain of such generators is explicitly characterized and corresponding results on smooth interior domain with quite general boundary conditions are given. For bounded coefficients, this problem is rather classical. Even for unbounded coefficients, there are various ways of constructing the Markov semigroups.

##### MSC:
 47D07 Markov semigroups and applications to diffusion processes 35B50 Maximum principles in context of PDEs 35J25 Boundary value problems for second-order elliptic equations 35J70 Degenerate elliptic equations
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