Strict Decomposition of Diffusions Associated to Degenerate (Sub)-Elliptic Forms

#### Jiyong Shin

2018, v.24, Issue 4, 695-714

ABSTRACT

For a given strongly local Dirichlet form with possibly degenerate symmetric (sub)-elliptic matrix, we identify a Hunt process (associated to the Dirichlet form) with a weak solution to the corresponding stochastic differential equation starting from all points in $\R^d$. More precisely, using heat kernel estimates, stochastic calculus, and Dirichlet form theory, we obtain the pointwise existence of a weak solution to the stochastic differential equation which has possibly unbounded and discontinuous drift. We also present some conditions that the weak solution becomes a pathwise unique strong solution.

Keywords: subelliptic operators, intrinsic metric, strong existence, Fukushima decomposition, degenerate forms