Large deviation principles for a 2D stochastic Allen–Cahn–Navier–Stokes driven by jump noise

In this paper, we derive a large deviation principle for a stochastic 2D Allen–Cahn–Navier–Stokes system with a multiplicative noise of Lévy type. The model consists of the Navier–Stokes equations for the velocity, coupled with a Allen–Cahn system for the order (phase) parameter. The proof is based on the weak convergence method introduced in [A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincarà ⓒ Probab. Stat. 47(3) (2011) 725–747].

Modeling dynamic processes with panel data: An application of continuous time models to prevention research

Many interventions are characterized by repeated observations on the same individuals (e.g., baseline, mid-intervention, two to three post-intervention observations), which offer the opportunity to consider differences in how individuals vary over time. Effective interventions may not be limited to changing means, but instead may also include changes to how variables affect each other over time. Continuous time models offer the opportunity to specify differing underlying processes for how individuals change from one time to the next, such as whether it is the level or change in a variable that is related to changes in an outcome of interest. After introducing continuous time models, we show how different processes can produce different expected covariance matrices. Thus, models representing differing underlying processes can be compared, even with a relatively small number of repeated observations. A substantive example comparing models that imply different underlying continuous time processes will be fit using panel data, with parameters reflecting differences in dynamics between control and intervention groups.

Embedded Markov chain approximations in Skorokhod topologies

We prove a J1-tightness condition for embedded Markov chains and discuss four Skorokhod topologies in a unified manner. To approximate a continuous time stochastic process by discrete time Markov chains, one has several options to embed the Markov chains into continuous time processes. On the one hand, there is a Markov embedding which uses exponential waiting times. On the other hand, each Skorokhod topology naturally suggests a certain embedding. These are the step function embedding for J1, the linear interpolation embedding forM1, the multistep embedding for J2 and a more general embedding for M2. We show that the convergence of the step function embedding in J1 implies the convergence of the other embeddings in the corresponding topologies. For the converse statement, a J1-tightness condition for embedded time-homogeneous Markov chains is given.Additionally, it is shown that J1 convergence is equivalent to the joint convergence in M1 and J2.