Scale-invariant diffusions: Transience and non-polar points
Consider a diffusion \(X\) on \(\mathbb{R}^d\) whose generator \(\sum_{i,j} a_{ij}(x) \frac{\partial^2} {\partial x_i\partial x_j}+ \sum_i \frac{b_i(x)} {|x|} \frac{\partial} {\partial x_i}\) has diffusion and drift components depending not on the distance to the origin 0. The author introduces a parameter \(\alpha\), explicitly given in terms of the invariant measure of the diffusion on the sphere, and proves the following results: (i) If \(\alpha>1\), then \(X\) is transient to \(\alpha\) and almost surely does not hit 0 \((\tau_0= \infty\) a.e.). (ii) If \(\alpha< 1\), then \(X_t\to 0\) on \(\tau_0= \infty\) a.e. (iii) If \(\alpha< 1\) and \(\inf_{x\neq 0} (\operatorname {tr} a(x)+ \frac{2}{|x|}< x\), \(b(x)> 0)> 0\), then \(\tau_0< \infty\) a.e. Some examples for \(\alpha= 1\) are provided. The behavior of the coefficients at the origin is of special interest, compare the author [Stochastics Stochastics Rep. 65, No. 1/2, 1-11 (1998; Zbl 0916.60066)].
scientific article; zbMATH DE number 1348245