The Pluripotential Cauchy-Dirichlet problem for the Complex Monge-Ampère flow with a general measure on the right-hand side

We show that the pluripotential Cauchy-Dirichlet problem for the complex Monge-Ampère flow is solvable for the right-hand side of the form $dt\wedge d\mu$ where $d\mu$ is dominated by a Monge-Ampère measure of a bounded plurisubharmonic function. In particular, we remove the strict positivity assumption on $d\mu$. We use this result to prove the parabolic version of the bounded subsolution theorem due to Kolodziej in pluripotential theory.