ISSN 1064-5624, Doklady Mathematics, 2011, Vol. 83, No. 3, pp. 309–313. © Pleiades Publishing, Ltd., 2011. Published in Russian in Doklady Akademii Nauk, 2011, Vol. 438, No. 2, pp. 154–159. 309 This paper is a companion to [1]. Suppose we are given an elliptic operator We say that a Borel locally finite measure μ on  d (pos- sibly signed) satisfies the stationary Kolmogorov equa- tion if the coefficient b is locally integrable with respect to the measure |μ| and the following equality is fulfilled: Let b ∈ (dx) with some p > d. As shown in [2], the measure μ has a density u ∈ ( d ) with respect to Lebesgue measure. In particular, one can choose a continuous version of u. In this case the equation can be written as the following divergence form elliptic equation: (1) Throughout we assume that b ∈ (dx) with p > d, so we shall study Eq. (1). A nonnegative and integrable on the whole  d solution u whose integral equals 1 will be called a probability solution. According to Har- nack’s inequality, any nonnegative nonzero solution is strictly positive. Sufficient conditions for the existence of a probability solution are obtained in [3] (see also the survey paper [4]). L ϕ Δϕ b ∇ϕ , ( ) . + = L * μ 0 = L ϕμ d ∫ 0 ϕ ∀ C 0 ∞  d ( ) . ∈ = L loc p W loc 1 p , div ∇u bu – ( ) 0 . = L loc p Suppose that a probability solution to Eq. (1) exists. In this paper we investigate the following two prob- lems. (1) Under what conditions is a given probability solution a unique integrable solution up to multiplica- tion by a constant? (2) Under what conditions is a given probability solution a unique probability solution? The uniqueness in the class of probability solutions and the uniqueness in the class of integrable solutions are different concepts indeed already in dimension d = 1. As shown in [5], in the one-dimensional case the local integrability of the coefficient b implies the uniqueness of a probability solution. However, the following example shows that in this situation the equation can have two linearly independent integrable solutions. Example 1. Set Then is a positive, integrable solution to the second order equation (u' – bu)' = 0. Let us verify that the function u(x) = x(1 + 4x 4 ) –1 . is also a solution. We observe that u = · x . Then Therefore, (u' – bu)' = 0. This example can be easily extended to the case of arbitrary dimension d. The problem of uniqueness of a probability solu- tion to the stationary Kolmogorov equation has been studied in many papers, see [4–9]. Sufficient condi- tions for the uniqueness have been found and exam- ples of nonuniqueness of probability solutions have been constructed. For instance, for the uniqueness of a probability solution it suffices to have one of the following conditions: (1) b ∈L 2 ( d , dx), ζ ρ x () 1 1 4 x 4 + ------------- e x 4 – , bx () ' x () x () --------- . = = ζ ρ ζ ρ ζ ρ ζ ρ e x 4 u' bu – xe x 4 ( )' 1 1 4 x 4 + ------------- e x 4 – 1 4 x 4 + ( ) e x 4 ⋅ 1 . = = = ζ ρ ζ ρ ζ ρ On Probability and Integrable Solutions to the Stationary Kolmogorov Equation 1 V. I. Bogachev a , A. I. Kirillov b , and S. V. Shaposhnikov a Presented by Academician I.A. Ibragimov December 14, 2010 Received January 21, 2011 DOI: 10.1134/S1064562411030112 a Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119992 Russia b Russian Foundation for Basic Research, Leninskii pr. 32a, Moscow, 117334 Russia e-mail: vibogach@mail.ru MATHEMATICS 1 The article was translated by the authors.