Volume growth and parabolicity for non-symmetric operators

The interplay between a weighted volume measure and its naturally associated weighted Laplace operator is fundamental to the geometric analysis of weighted structures governed by gradient vector fields. Beyond the gradient setting, however, the theory has remained considerably less developed, largely because this natural correspondence is lost. In this paper, we identify invariant measures as the appropriate framework for extending the theory to general drifted Laplace operators. This approach yields a systematic treatment of parabolicity in the non-gradient setting, reveals its connection with volume growth, and leads to new volume finiteness results for non-gradient geometric structures.