  We discuss Hilbert space-valued stochastic differential equations associated
with the heat semi-groups of the standard model of non-relativistic quantum
electrodynamics and of corresponding fiber Hamiltonians for translation
invariant systems. In particular, we prove the existence of a stochastic flow
satisfying the strong Markov property and the Feller property. To this end we
employ an explicit solution ansatz. In the matrix-valued case, i.e., if the
electron spin is taken into account, it is given by a series of operator-valued
time-ordered integrals, whose integrands are factorized into annihilation,
preservation, creation, and scalar parts. The Feynman-Kac formula implied by
these results is new in the matrix-valued case. Furthermore, we discuss
stochastic differential equations and Feynman-Kac representations for an
operator-valued integral kernel of the semi-group. As a byproduct we obtain
analogous results for Nelson's model.
