  This paper deals with the application of probabilistic time integration
methods to semi-explicit partial differential-algebraic equations of parabolic
type and its semi-discrete counterparts, namely semi-explicit
differential-algebraic equations of index 2. The proposed methods iteratively
construct a probability distribution over the solution of deterministic
problems, enhancing the information obtained from the numerical simulation.
Within this paper, we examine the efficacy of the randomized versions of the
implicit Euler method, the midpoint scheme, and exponential integrators of
first and second order. By demonstrating the consistency and convergence
properties of these solvers, we illustrate their utility in capturing the
sensitivity of the solution to numerical errors. Our analysis establishes the
theoretical validity of randomized time integration for constrained systems and
offers insights into the calibration of probabilistic integrators for practical
applications.
