On the existence of diffusions with singular drift coefficient (Q1119276)

Let \(L=(1/2)\sum^{d}_{i,j=1}a^{ij}(x)\partial^ 2/\partial x^ i\partial x^ j+\sum^{d}_{i=1}b^ i(\) x)\(\partial /\partial xi\) be an operator in \(R^ d\), where the matrix \((a^{ij})\) is bounded, Hölder continuous and uniformly positive definite, and \((b^ i(x))\) is Borel measurable. The author proves the existence of L-diffusion under the hypotheses that \[ \sup_{x}\int_{| y-x| \leq}g_ d(x- y)b(y)<\infty, \] where \(g_ 1(z)=1\), \(g_ 2(z)=-\ln | z|\) and \(g_ d(z)=| z|^{2-d}\) for \(\geq 3\). The proof uses the Girsanov transformation to change the drift coefficient and a condition which characterizes a BMO martingale.