On the existence of diffusions with singular drift coefficient

Let \(L = \frac{1}{2}\sum\limits_{i,j = 1}^d {a^{ij} } (x)\frac{{\partial ^2 }}{{\partial x^i \partial x^j }} + \sum\limits_{i = 1}^d {b^i (x)\frac{\partial }{{\partial x^i }}}\) be an operator inR ^d, where the matrix (a ^ij ) is bounded, Hölder continuous and uniformly positive definite, and (b ⁱ (x)) is Borel measurable. In this paper we prove the existence ofL-diffusion under the hypothesis that $$\mathop {\sup }\limits_x \int_{|y - x| \leqslant \frac{1}{2}} {g_d (x - y)} |b(y)|^2 dy< \infty ,$$ whereg ₁(z)=1,g ₂(z)=?ln|z| andg _d (z)=|z|^2?d ford≥3.