Small time asymptotics for SPDEs with locally monotone coefficients

Abstract

This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous medium equations, stochastic p-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new in the case of general quasilinear SPDEs with multiplicative noise.

Numerical approximations of one-point large deviations rate functions of stochastic differential equations with small noise

“…The small time asymptotics of stochastic systems has received extensive attention in recent years (see e.g., [13,17,20] and references therein). This property characterizes the asymptotical behavior of the underlying stochastic systems with the time tending to zero.…”

On the small time asymptotics of stochastic Ladyzhenskaya-Smagorinsky equations with damping perturbed by multiplicative noise

“…For the small-time LDP for infinite dimensional diffusion processes, the interested readers are referred to see [1,2,12,29,67,70], etc. The small time LDP for stochastic 2D Navier-Stokes equations, stochastic 3D tamed Navier-Stokes equations, stochastic quasi-geostrophic equations in the sub-critical case, stochastic two-dimensional non-Newtonian fluids, 3D stochastic primitive equations, SPDEs with locally monotone coefficients, stochastic convective Brinkman-Forchheimer equations, scalar stochastic conservation laws, are established in [68,56,40,36,18,38,50,19], respectively. Even though the work [38] covers the case of SPDEs with locally monotone coefficients like stochastic power law fluid equations, it won't cover the system under our consideration for arbitrary values of r (for example r > pd d−p ).…”

Freidlin-Wentzell Type Large Deviation Principle for Multi-Scale Locally Monotone SPDEs

“…Later on, such framework has been substantially generalized by the third named author and Röckner in [24,25,26] to more general hypothesises only fulfilling the local monotonicity and generalized coercivity, which covers several SPDEs such as the stochastic porous media equations, stochastic fast-diffusion equations, stochastic 2D Navier-Stokes equations and other hydrodynamical type models, stochastic p-Laplace equations, stochastic power law fluid equations, stochastic Ladyzhenskaya models, etc. We also refer the interested readers to [6,14,22,23,27,32,39,41,42] and reference therein for the properties of solutions associated with such framework.…”