Remarks on non-linear noise excitability of some stochastic heat equations

We consider nonlinear parabolic SPDEs of the form $\partial_t u=\Delta u + \lambda \sigma(u)\dot w$ on the interval $(0, L)$, where $\dot w$ denotes space-time white noise, $\sigma$ is Lipschitz continuous. Under Dirichlet boundary conditions and a linear growth condition on $\sigma$, we show that the expected $L^2$-energy is of order $\exp[\text{const}\times\lambda^4]$ as $\lambda\rightarrow \infty$. This significantly improves a recent result of Khoshnevisan and Kim. Our method is very different from theirs and it allows us to arrive at the same conclusion for the same equation but with Neumann boundary condition. This improves over another result of Khoshnevisan and Kim

Large Deviations for a Class of Semilinear Stochastic Partial Differential Equations

We prove the large deviations principle (LDP) for the law of the solutions to a class of semilinear stochastic partial differential equations driven by multiplicative noise. Our proof is based on the weak convergence approach and significantly improves earlier methods