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 dμ d t ∧ d μ where dμ d μ is dominated by a Monge-Ampère measure of a bounded plurisubharmonic function. In particular, we remove the strict positivity assumption on dμ d μ . We use this result to prove the parabolic version of the bounded subsolution theorem due to Kołodziej in pluripotential theory.