A viscosity solution approach to the large deviation principle for stochastic convective Brinkman-Forchheimer equations
Sagar Gautam, Manil T. Mohan
TL;DR
This work addresses the large deviation principle for the stochastic convective Brinkman-Forchheimer (SCBF) equations on the torus $\mathbb{T}^d$ with small noise $1/\sqrt{n}$. It develops a viscosity-solution framework that identifies the Laplace limit as the solution to an infinite-dimensional HJB equation and then leverages exponential tightness to obtain the LDP in both the Skorohod path space and the continuous-path space. The core novelty lies in exploiting the monotone structure provided by the damping term $\beta|u|^{r-1}u$ (and the diffusion $-\mu\Delta u$) to prove a comparison principle without truncation, allowing a direct Laplace principle-based route instead of the Budhiraja–Dupuis weak convergence approach. The results yield an explicit rate function $I$ via a value-function representation of an associated deterministic control problem, connecting stochastic SCBF dynamics to infinite-dimensional optimal control and PDE techniques for rare-event analysis.
Abstract
This article develops the viscosity solution approach to the large deviation principle for the following two- and three-dimensional stochastic convective Brinkman-Forchheimer equations on the torus $\mathbb{T}^d,\ d\in\{2,3\}$ with small noise intensity: \begin{align*} \mathrm{d}\boldsymbol{u}_n+[-μΔ\boldsymbol{u}_n+ (\boldsymbol{u}_n\cdot\nabla)\boldsymbol{u}_n +α\boldsymbol{u}_n+β|\boldsymbol{u}_n|^{r-1}\boldsymbol{u}_n+\nabla p_n]\mathrm{d} t=\boldsymbol{f}\mathrm{d} t+\frac{1}{\sqrt{n}}\mathrm{Q}^{\frac12}\mathrm{d}\mathrm{W}, \ \nabla\cdot\boldsymbol{u}_n=0, \end{align*} where $μ,α,β>0$, $r\in[1,\infty)$, $\mathrm{Q}$ is a trace class operator and $\mathrm{W}$ is Hilbert-valued calendrical Wiener process. We build our analysis on the framework of Varadhan and Bryc, together with the techniques of [J. Feng et.al., Large Deviations for Stochastic Processes, American Mathematical Society (2006) vol. \textbf{131}]. By employing the techniques from the comparison principle, we identify the Laplace limit as the convergence of the viscosity solution of the associated second-order singularly perturbed Hamilton-Jacobi-Bellman equation. A key advantage of this method is that it establishes a Laplace principle without relying on additional sufficient conditions such as Bryc's theorem, which the literature commonly requires. For $r>3$ and $r=3$ with $2βμ\geq1$, we also derive the exponential moment bounds without imposing the classical orthogonality condition $((\boldsymbol{u}_n\cdot\nabla)\boldsymbol{u}_n,\mathrm{A}\boldsymbol{u}_n)=0$, where $\mathrm{A}=-Δ$, in both two-and three-dimensions. We first establish the large deviation principle in the Skorohod space. Then, by using the $\mathrm{C}-$exponential tightness, we finally establish the large deviation principle in the continuous space.