  We study an index of a transversal Dirac operator on an odd-dimensional
manifold $X$ with locally free $\mathbb{S}^1$-action. One difficulty of using
heat kernel method lies in the understanding of the asymptotic expansion as
$t\to 0^+$. By a probabilistic approach via the Feynman-Kac formula, the
transversal heat kernel on $X$ can be linked to the ordinary heat kernel for
functions on the orbifold $M=X/\mathbb{S}^1$ which is more tractable. After
some technical results for a uniform bound estimate as $t\to 0^+$, we are
reduced from the transversal, orbifold situation to the classical situation
particularly at points of the principal stratum. One application asserts that
for a certain class of spin orbifolds $M$, to the classical index problem of
Kawasaki in the Riemannian setting the net contributions arising from the
lower-dimensional strata beyond the principal one vanish identically.
