  Heat-invariants are a class of spectral invariants of Laplace-type operators
on compact Riemannian manifolds that contain information about the geometry of
the manifold, e.g., the metric and connection. Since Brownian motion solves the
heat equation, these invariants can be obtained studying Brownian motion on
manifolds. In this article, we consider Brownian motion on the Toeplitz
algebra, discrete Heisenberg group algebras, and non-commutative tori to define
Laplace-type operators and heat-semigroups on these C*-bialgebras. We show that
their traces can be $\zeta$-regularized and compute "heat-traces" on these
algebras, giving us a notion of dimension and volume. Furthermore, we consider
$SU_q(2)$ which does not have a Brownian motion but a class of driftless
Gaussians which still recover the dimension of $SU_q(2)$.
