Large Deviation Principles of Obstacle Problems for Quasilinear Stochastic PDEs

Abstract

In this paper, we present a sufficient condition for the large deviation criteria of Budhiraja, Dupuis and Maroulas for functionals of Brownian motions. We then establish a large deviation principle for obstacle problems of quasi-linear stochastic partial differential equations. It turns out that the backward stochastic differential equations will play an important role.

Large deviation principles for first-order scalar conservation laws with stochastic forcing

“…Recently, a new sufficient condition (Condition B below) to verify the assumptions in condition A (hence the large deviation principle) is proposed by Matoussi, Sabagh and Zhang in [22]. It turns out this new sufficient condition is suitable for establishing the large deviation principle for the scalar conservation laws.…”

“…In particular, for Brownian functionals, an elegant variational representation formula has been established by Boué and Dupuis in [2] and by Budhiraja and Dupuis in [3]. Recently, a sufficient condition to verify the large deviation criteria of Budhiraja, Dupuis and Maroulas for functionals of Brownian motions is proposed by Matoussi, Sabbagh and Zhang in [22], which turns out to be more suitable for SPDEs arising from fluid mechanics. Thus, in the present paper, we adopt this new sufficient condition.…”

Large deviation principle for stochastic Burgers type equation with reflection

“…The main difficulties come from the highly non-linear coefficient and the singularity caused by the reflection. Here, we adopt a new sufficient condition for the weak convergence criteria, which is proposed by Matoussi, Sabbagh and Zhang [14].…”

“…Based on a viscosity solution approach, an LDP for the diffusions with Lipschitz continuous oblique reflections on regular domains was obtained in [13]. Using the weak convergence approach introduced in [1,3,4], LDPs for obstacle problems of linear/quasi-linear SPDEs were established in [22] and [14], respectively. There are also several papers studying the large deviations for multivalued stochastic differential equations (SDEs), which in particular contain a class of SDEs with reflection in a convex domain, see, e.g., [16,17,18,20].…”

“…In this paper, we will adopt a new sufficient condition for the weak convergence criteria given by Matoussi et al [14]. This new condition seems to be more suitable for stochastic dynamics generated by SPDEs with reflections.…”