  The paper identifies families of quasi-stationary initial conditions for
infinite Brownian particle systems within a large class and provides a
construction of the particle systems themselves started from such initial
conditions. Examples of particle systems falling into our framework include
Brownian versions of TASEP-like processes such as the diffusive scaling limit
of the q-TASEP process. In this context the spacings between consecutive
particles form infinite-dimensional versions of the softly reflected Brownian
motions recently introduced in the finite-dimensional setting by O'Connell and
Ortmann and are of independent interest. The proof of the main result is based
on intertwining relations satisfied by the particle systems involved which can
be regarded as infinite-dimensional analogues of the suitably generalized
Burke's Theorem.
