  We develop a stochastic model for Lagrangian velocity as it is observed in
experimental and numerical fully developed turbulent flows. We define it as the
unique statistically stationary solution of a causal dynamics, given by a
stochastic differential equation. In comparison to previously proposed
stochastic models, the obtained process is infinitely differentiable at a given
finite Reynolds number, and its second-order statistical properties converge to
those of an Ornstein-Uhlenbeck process in the infinite Reynolds number limit.
In this limit, it exhibits furthermore intermittent scaling properties, as they
can be quantified using higher-order statistics. To achieve this, we begin with
generalizing the two-layered embedded stochastic process of Sawford (1991) by
considering an infinite number of layers. We then study, both theoretically and
numerically, the convergence towards a smooth (i.e. infinitely differentiable)
Gaussian process. To include intermittent corrections, we follow similar
considerations as for the multifractal random walk of Bacry et al. (2001). We
derive in an exact manner the statistical properties of this process, and
compare them to those estimated from Lagrangian trajectories extracted from
numerically simulated turbulent flows. Key predictions of the multifractal
formalism regarding acceleration correlation function and high-order structure
functions are also derived. Through these predictions, we understand
phenomenologically peculiar behaviours of the fluctuations in the dissipative
range, that are not reproduced by our stochastic process. The proposed
theoretical method regarding the modelling of infinitely differentiability
opens the route to the full stochastic modelling of velocity, including the
peculiar action of viscosity on the very fine scales.
