# Kolmogorov equations in *Rd* with unbounded coefficients

Chapter

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## Abstract

In this chapter we are concerned with the following class of second order elliptic operators
The vector field

$$
\mathcal{L}_0 (x,D) = \sum\limits_{i,j = 1}^d {a_{ij} (x)D_{ij} + \sum\limits_{i = 1}^d {b_i (x)D_{i, } x\varepsilon \mathbb{R}^d } }
$$

*b*= (*b*,...,*b*_{d}) : ℝ^{d}→ ℝ^{d}is of class*C*^{3}and the matrix*a*(*x*) = [*a*_{ij}(*x*)] is symmetric, strictly positive and of class*C*^{3}, so that it can be written as*a(x)*= ½σ*x*σ^{*}*x*,*x*∈ℝ^{d}, for some function σ : ℝ^{d}→ ℒ(ℝ^{d}) of class*C*^{2}(in fact, we can take σ = √a). Both b and a are assumed to have polynomial growth and 6 enjoys some dissipativity conditions which will be described in more details later on. Our aim is to prove the existence and the uniqueness of solutions for the elliptic and the parabolic problems associated with the operator ℒ_{0}. Moreover, we want to give optimal regularity results in the space of bounded Holder continuous functions (*Schauder type estimates*).## Keywords

Variation Equation Parabolic Problem Previous Proposition KOLMOGOROV Equation Gronwall Lemma
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2001