arXiv subjects

Esmée Theewis

Publications and source records attributed to Esmée Theewis.

7 recordsLinked to original sources

arXiv2026

Large deviations for stochastic evolution equations beyond the coercive case

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.

math.PR↗

arXiv2026

The large deviation principle for the stochastic 3D primitive equations with transport noise

We prove the small-noise large deviation principle for the three-dimensional primitive equations with transport noise and turbulent pressure. Transport noise is important for geophysical fluid dynamics applications, as it takes into account the effect of small scales on the large scale dynamics. The main mathematical challenge is that we allow for the transport noise to act on the full horizontal velocity, therefore leading to a non-trivial turbulent pressure, which requires an involved analysis to obtain the necessary energy bounds. Both Stratonovich and Itô noise are treated.

math.PR↗

arXiv2026

Maximal regularity for evolution equations with critical singular perturbations

Assuming $A$ has maximal $L^p$-regularity, this paper investigates perturbations of $A$ by time-dependent operators $B$ that are unbounded and satisfy a critical $L^q$-integrability condition in time. We establish two main results. The first proves maximal $L^p$-regularity for the critical endpoint case, generalizing previous work by Prüss and Schnaubelt (2001). The second develops a weighted maximal regularity theory for mixed-scale perturbations, motivated by the linearized skeleton equations appearing in large deviations theory for stochastic PDEs.

math.FA↗

arXiv2026

Uniform large deviation principles for the stochastic heat equation over unbounded sets of initial data

We study small-noise large deviations for the stochastic heat equation (SHE) on the torus with unbounded, multiplicative space-time white noise. We establish uniform large deviation principles (ULDPs) over unbounded sets of initial data, alongside novel well-posedness and regularity results. The ULDPs are obtained for both continuous-in-space and $L^p$ initial data. For these two classes, the ULDP holds uniformly over $L^q$-bounded subsets, with $q<\infty$ and $q<p$ allowed respectively, so these sets are highly unbounded in the state space. The admissible range of $q$ is dictated by the growth of the noise coefficient in the SHE. Crucially, our methods allow us to reach down to $q=1$. This yields ULDPs over $L^1$-bounded subsets, enabling the study of exit times in physical systems with mass conservation.

math.PR↗

arXiv2026

Strong existence and uniqueness for a class of quasilinear stochastic evolution equations

We establish existence of probabilistically strong solutions and pathwise uniqueness for a class of quasilinear stochastic evolution equations on bounded domains. Our results combine recent weak existence results for quasilinear stochastic evolution equations in an $L^p$-setting (with $p > 2$) with Yamada--Watanabe theory. To establish pathwise uniqueness, we rely on an $L^1$-contraction argument.

math.PR↗

arXiv2026

The Yamada-Watanabe-Engelbert theorem for SPDEs in Banach spaces

We give a unified proof of the Yamada-Watanabe-Engelbert theorem for various notions of solutions for SPDEs in Banach spaces with cylindrical Wiener noise. We use Kurtz' generalization of the theorems of Yamada, Watanabe and Engelbert. In addition, we deduce the classical Yamada-Watanabe theorem for SPDEs, with a slightly different notion of `unique strong solution' than that corresponding to the result of Kurtz. Our setting includes analytically strong solutions, analytically weak solutions and mild solutions. Moreover, our approach offers flexibility with regard to the function spaces and integrability conditions that are chosen in the solution notion (and affect the meaning of existence and uniqueness). All results hold in Banach spaces which are either martingale type 2 or UMD. For analytically weak solutions, the results hold in arbitrary Banach spaces. In particular, our results extend the Yamada-Watanabe theorems of Ondreját for mild solutions in 2-smooth Banach spaces, of Röckner et al. for the variational framework and of Kunze for analytically weak solutions, and cover many new settings. As a tool, and of interest itself, we construct a measurable representation $I$ of the stochastic integral in a martingale type 2 or UMD Banach space, in the sense that for any stochastically integrable process $f$ and cylindrical Brownian motion $W$, we have $I(f(ω),W(ω),\mathrm{Law}(f,W)) = (\int_0^{\cdot} f\, \mathrm{d}W)(ω)$ for almost every $ω$.

math.PR↗

arXiv2026

Large Deviations for Stochastic Evolution Equations in the Critical Variational Setting

Using the weak convergence approach, we prove the large deviation principle (LDP) for solutions to quasilinear stochastic evolution equations with small Gaussian noise in the critical variational setting, a recently developed general variational framework. No additional assumptions are made apart from those required for well-posedness. In particular, no monotonicity is required, nor a compact embedding in the Gelfand triple. Moreover, we allow for flexible growth of the diffusion coefficient, including gradient noise. This leads to numerous applications for which the LDP was not established yet, in particular equations on unbounded domains with gradient noise. Since our framework includes the 2D Navier-Stokes and Boussinesq equations with gradient noise and unbounded domains, our results resolve an open problem that has remained unsolved for over 15 years.

math.PR↗