  ODE solvers with randomly sampled timestep sizes appear in the context of
chaotic dynamical systems, differential equations with low regularity, and,
implicitly, in stochastic optimisation. In this work, we propose and study the
stochastic Euler dynamics - a continuous-time Markov process that is equivalent
to a linear spline interpolation of a random timestep (forward) Euler method.
We understand the stochastic Euler dynamics as a path-valued ansatz for the ODE
solution that shall be approximated. We first obtain qualitative insights by
studying deterministic Euler dynamics which we derive through a first order
approximation to the infinitesimal generator of the stochastic Euler dynamics.
Then we show convergence of the stochastic Euler dynamics to the ODE solution
by studying the associated infinitesimal generators and by a novel local
truncation error analysis. Next we prove stability by an immediate analysis of
the random timestep Euler method and by deriving Foster-Lyapunov criteria for
the stochastic Euler dynamics; the latter also yield bounds on the speed of
convergence to stationarity. The paper ends with a discussion of second-order
stochastic Euler dynamics and a series of numerical experiments that appear to
verify our analytical results.
