  To study diffusion processes on the $p$-Wasserstein space $\scr P_p $ for
$p\in [1,\infty)$ over a separable, reflexive Banach space $X$, we present a
criterion on the quasi-regularity of Dirichlet forms in $L^2(\scr P_p,\Lambda)$
for a reference probability $\Lambda$ on $\scr P_p$. It is formulated in terms
of an upper bound condition with the uniform norm of the intrinsic derivative.
The condition is easy to check in relevant applications and allows to construct
a type of Ornstein-Uhlenbeck process on $\scr P_p$. We find a versatile class
of quasi-regular local Dirichlet forms on $\scr P_p$ by using images of
Dirichlet forms on the tangent space $L^p(X\to X,\mu_0)$ at a reference point
$\mu_0\in \scr P_p$. The Ornstein-Uhlenbeck type Dirichlet form is an important
example in this class. An $L^2$-estimate for the corresponding heat kernel is
derived, based on the eigenvalues of the covariance operator of the underlying
Gaussian measure.
