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].

An Explicit Large Deviation Analysis of the Spatial Cycle Huang–Yang–Luttinger Model

We introduce a family of ‘spatial’ random cycle Huang–Yang–Luttinger (HYL)-type models in which the counter-term only affects cycles longer than some cut-off that diverges in the thermodynamic limit. Here, spatial refers to the Poisson reference process of random cycle weights. We derive large deviation principles and explicit pressure expressions for these models, and use the zeroes of the rate functions to study Bose–Einstein condensation. The main focus is a large deviation analysis for the diverging counter term where we identify three different regimes depending on the scale of divergence with respect to the main large deviation scale. Our analysis derives explicit bounds in critical regimes using the Poisson nature of the random cycle distributions.

Large deviations for a class of tempered subordinators and their inverse processes

We consider a class of tempered subordinators, namely a class of subordinators with one-dimensional marginal tempered distributions which belong to a family studied in [3]. The main contribution in this paper is a non-central moderate deviations result. More precisely we mean a class of large deviation principles that fill the gap between the (trivial) weak convergence of some non-Gaussian identically distributed random variables to their common law, and the convergence of some other related random variables to a constant. Some other minor results concern large deviations for the inverse of the tempered subordinators considered in this paper; actually, in some results, these inverse processes appear as random time-changes of other independent processes.

Large deviations for longest runs in Markov chains

We continue our investigation on general large deviation principles (LDPs) for longest runs. Previously, a general LDP for the longest success run in a sequence of independent Bernoulli trails was derived in [Z. Liu and X. Yang, A general large deviation principle for longest runs, Statist. Probab. Lett. 110 2016, 128–132]. In the present note, we establish a general LDP for the longest success run in a two-state (success or failure) Markov chain which recovers the previous result in the aforementioned paper. The main new ingredient is to implement suitable estimates of the distribution function of the longest success run recently established in [Z. Liu and X. Yang, On the longest runs in Markov chains, Probab. Math. Statist. 38 2018, 2, 407–428].