Let E be a Hermitian vector bundle over a Riemannian manifold M with metric g, and let â be a metric covariant derivative on E. We study the generalized OrnsteinâUhlenbeck differential expression Pâ=ââ âu+â(dÏ)âŻuââXu+Vu, where ââ is the formal adjoint of â, (dÏ)⯠is the vector field corresponding to dÏ via g, X is a smooth real vector field on M, and V is a self-adjoint locally integrable section of the endomorphsim bundle EndE. In the setting of a geodesically complete M, we establish a sufficient condition for the equality of the maximal and minimal realizations of Pâ in the (weighted) space ÎLÎŒp(E) of LÎŒp-type sections of E, where 1<p<â and dÎŒ=eâÏdÎœg, with Îœg being the usual volume measure. Furthermore, we show that (the negative of) the maximal realization âHp,max generates an analytic quasi-contractive semigroup in ÎLÎŒp(E), 1<p<â. Additionally, in the same context, we establish a coercivity estimate, which leads to the so-called separation property of the covariant Schrödinger operator (that is, Pâ with XâĄ0 and ÏâĄ0) in the (unweighted) space ÎLp(E), 1<p<â. With the generation result at our disposal, we describe a FeynmanâKac representation for the LÎŒp-semigroup generated by Hp,max, 1<p<â. For the OrnsteinâUhlenbeck differential expression acting on functions, that is, Pd=Îu+(dÏ)âŻuâXu+Vu, where Î is the (non-negative) scalar Laplacian on M and V is a locally integrable real-valued function, we consider another way of realizing Pd as an operator in LÎŒp(M) and, by imposing certain geometric conditions on M, we prove another semigroup generation result. The study of the mentioned realization of Pd depends, among other things, on the fulfillment of the so-called Lp-CalderĂłnâZygmund inequality on M.