On the small‐time asymptotics of 3D stochastic primitive equations

Dong, Zhao; Zhang, Rangrang

doi:10.1002/mma.5142

Cited by 17 publications

(11 citation statements)

References 28 publications

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“…martingale solutions whose regularity in space and time is the one of a weak solution, by an implicit Euler scheme is given in [24]. Large deviation principles [13] and moderate deviation principles [55] are known to hold for small multiplicative noise and short times [15]. The existence of a Markov selection is proven in [14] for additive noise.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

Saal¹,

Slavík²

2021

Preprint

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We establish the existence and uniqueness of pathwise strong solutions to the stochastic 3D primitive equations with only horizontal viscosity and diffusivity driven by transport noise on a cylindrical domain M = (−h, 0) × G, G ⊂ R 2 bounded and smooth, with the physical Dirichlet boundary conditions on the lateral part of the boundary. Compared to the deterministic case where the uniqueness of z-weak solutions holds in L 2 , more regular initial data are necessary to establish uniqueness in the anisotropic space H 1 z L 2 xy so that the existence of local pathwise solutions can be deduced from the Gyöngy-Krylov theorem. Global existence is established using the logarithmic Sobolev embedding, the stochastic Gronwall lemma and an iterated stopping time argument.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

Saal¹,

Slavík²

2021

Preprint

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“…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.…”

Section: Application To the Small Time Ldp Of Sdesmentioning

confidence: 99%

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

Hong,

Jin,

Sheng

2021

Preprint

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In this paper, we study the numerical approximation of the one-point large deviations rate functions of nonlinear stochastic differential equations (SDEs) with small noise. We show that the stochastic θ-method satisfies the one-point large deviations principle with a discrete rate function for sufficiently small step-size, and present a uniform error estimate between the discrete rate function and the continuous one on bounded sets in terms of step-size. It is proved that the convergence orders in the cases of multiplicative noises and additive noises are 1/2 and 1 respectively. Based on the above results, we obtain an effective approach to numerically approximating the large deviations rate functions of nonlinear SDEs with small time. To the best of our knowledge, this is the first result on the convergence rate of discrete rate functions for approximating the one-point large deviations rate functions associated with nonlinear SDEs with small noise.

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“…Moreover, the second named author with Röckner and Zhu [44] also obtained the small time LDP for stochastic 2D quasigeostrophic equations in the sub-critical case. The small time LDP of stochastic 3D primitive equations was investigated by Dong and Zhang [19]. Recently, the small time LDP of scalar stochastic conservation laws was also studied in [61].…”

Section: Shihu LI Wei Liu and Yingchao Xiementioning

confidence: 99%

“…The proof of the main result here mainly follows the idea in Zhang's work [62] by using exponential equivalence arguments, which is a very powerful method used by many scholars to study the small time LDP for SPDEs, see, e.g. [12,19,37,44,50,60,61]. More precisely, consider a zero drift stochastic differential equation with the same initial data (see (4) below), where the small noise and small time asymptotics problems are equivalent.…”

Section: Shihu LI Wei Liu and Yingchao Xiementioning

confidence: 99%

Small time asymptotics for SPDEs with locally monotone coefficients

Li¹,

Liu²,

Xie³

2020

Discrete &Amp; Continuous Dynamical Systems - B

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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.

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On the small‐time asymptotics of 3D stochastic primitive equations

Abstract: In this paper, we establish a small-time large deviation principle for the strong solution of three-dimensional stochastic primitive equations driven by multiplicative noise, which involves not only the study of small noise but also the challenging nonlinear drift terms.

Cited by 17 publications

References 28 publications

Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

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

Small time asymptotics for SPDEs with locally monotone coefficients

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