Using the weak convergence approach, we prove the small-noise large deviation principle (LDP) for stochastic evolution equations in an L^2 -setting. As the coefficients are allowed to be non-coercive, our framework encompasses a much broader scope than variational settings. In place of coercivity, we require only well-posedness of the stochastic evolution equation and two concrete, verifiable a priori estimates. Furthermore, we accommodate drift nonlinearities satisfying a modified criticality condition and we allow for vanishing drift perturbations. The latter permits the inclusion of Itô–Stratonovich correction terms, enabling the treatment of both noise interpretations. We apply our results to a reaction-diffusion system that lacks coercivity. In separate joint work, the present framework has been used to establish an LDP for the 3D primitive equations with full transport noise, further demonstrating its versatility. Finally, we show that even in the coercive case, our framework yields new LDP results for equations with critical nonlinearities that rely on our modified criticality condition, including the stochastic 2D Allen–Cahn equation in the weak setting.
In recent work, Al-Maskari and Karaa [2] considered error estimates for a fully discrete scheme to solve a stochastic time-fractional Allen–Cahn equation driven by additive integrated noise, assuming the existence of a solution u ∈ L^2p (Ω ; Ḣ_0^1 (𝒟)), p ≥ 1 and that the noise satisfies ‖ A^β -1/2‖ _ℒ_2^0 < ∞ for β∈ [0, 2] . In this paper, we extend the results of [2]. We first apply the Yosida approximation to show that the stochastic time-fractional Allen–Cahn equation driven by additive integrated noise admits a unique solution u ∈ L^2p(Ω ; L^6(𝒟)), p ≥ 1 . Under the more natural assumptions on the noise, ‖ A^σ -κ/2‖ _ℒ_2^0 < ∞ , where σ∈ [d/3, 2] , d = 1, 2, 3 , and κ∈ [0, 2] depends on the Caputo time-fractional order α∈ (0, 1) and the Riemann–Liouville fractional integral order γ∈ [0, 1] , we establish the spatial and temporal regularities of the solution. We then approximate the spatial derivative using the finite element method, while the Caputo time-fractional derivative and the Riemann–Liouville time-fractional integral are approximated using convolution quadrature formulas. The additive noise is discretized using the L^2 projection method and the backward Euler method. Based on the smoothing properties of the Mittag-Leffler functions, we derive optimal error estimates with respect to nonsmooth initial data, as well as both smooth and nonsmooth integrated noise. Numerical experiments are conducted to verify the consistency of our theoretical findings with the numerical results.
The paper deals with linear passive scalar transport-diffusion equation subject to a velocity field which is white noise in time and is mainly active at small scales in space. The purpose is investigating the enhancement of dissipation and decay given by the small-scale transport. We modify and improve estimates from the previous work by Flandoli et al. (J. Differ. Equations 394, 237–277, 2024), in order to investigate a different regime, namely small molecular diffusion and small noise intensity – corresponding to small turbulent kinetic energy. The noise specification is carefully tuned to Kolmogorov theory of turbulent fluids.
The purpose of this paper is to investigate the problem of autocorrelated moving boundaries for a one-dimensional heat equation, where the moving boundary is a fractional Brownian motion B^H with Hurst parameter H >1/2 and the dependent Volterra-type Dirichlet boundary term ξ is verified to lie in the exponential Orlicz space 𝒮_Ψ^(v)(Ω× (0,T]) with v∈ (0,1] . This problem is interpreted as a single-layer potential involving stochastic heat kernel G( t-s,x-B^H(s)) = 1/√(4π (t-s))exp( -( x-B^H(s)) ^2/4(t-s)) , with 0z ) , z> E [ sup _(s,t)∈𝒬 B^H(t)-B^H(s)/(t-s)^1/2+γ] , where 𝒬={(s,t): 0≤ s ≤ t≤ T } and γ∈ [0,H-1/2) . By potential theoretic arguments, we establish the unique weak solution to the fractional Brownian moving boundary problem in L^p(Ω ; C_(v)((0,T];C_b(-∞ ,B^H(· )))) with p>1 and v∈ (0,1] .
