  We use tools from $n$-dimensional Brownian motion in conjunction with the
Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact
Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that
any nodal domain $\Omega_\lambda$ almost fully contains a ball of radius $\sim
\frac{1}{\sqrt{\lambda}}$. This also gives a slight refinement of a result by
Mangoubi, concerning the inradius of nodal domains (\cite{Man2}). On the other
hand, we also prove that no nodal domain can be contained in a reasonably thin
tubular neighbourhood of unions of finitely many surfaces inside $M$.
