Small time large deviation principle for Burgers type stochastic equation with reflection

In this paper, we establish the small time large deviation principle for Burgers type stochastic equation with reflection. The method we adopt is a new sufficient condition for the weak convergence method proposed by Matoussi, Sabbagh, and Zhang [14]. We show in particular the existence and uniqueness of a random equation with reflection and without Laplacian operator. The main difficulties come from the high nonlinearity, the singularity of the reflection wall, and the existence of small perturbation on the Laplacian term. By introducing a related penalized equation, the delicate estimations of the penalized term are shown. The results of this paper can be generalized to Burgers type stochastic equations with two reflections.