On the small time asymptotics of quasilinear parabolic stochastic partial differential equations (Q6115023)

The paper is concerned with small time large deviation principles for quasilinear parabolic stochastic partial differential equations with multiplicative noise, more precisely, \[du + \nabla\cdot (B(u))\, dt = \nabla\cdot (A(u)\nabla u) \, dt +\sigma(t,u)\, dW(t), \quad t\in [0,1] \] on the torus \(\mathbb T^d\) with initial condition \(u(0)=f\in L^m(\mathbb T^d)\) for all \(m\ge 1\). Here, \(B:\mathbb R \to \mathbb R^d\) and \(A:\mathbb R \to \mathbb R^{d\times d}\) are of class \(C^1_{Lip}\), and the matrix-valued function \(A\) is bounded and uniformly positive definite; \(W(t)\) is a cylindrical Brownian motion on some Hilbert space \(U\) and for each \(t\in [0,1]\), \(u\in H=L^2(\mathbb T^d)\), \(\sigma(t,u)\) is a mapping from \(U\) to \(H\), satisfying the usual linear growth and Lipschitz condition. The global well-posedness of the above equation has been established by \textit{M. Hofmanová} and \textit{T. Zhang} [Stochastic Processes Appl. 127, No. 10, 3354--3371 (2017; Zbl 1372.60091)]. The purpose of the present paper is, under some additional conditions on the diffusion coefficient \(\sigma\), to prove a large deviation principle for the laws \(\mu^\varepsilon\) on \(C([0,T], H)\) of processes \(u^\varepsilon(t)= u(\varepsilon t),\, t\in [0,1]\). The key step is to prove that \(\mu^\varepsilon\) is exponentially equivalent to the law of the solutions to \[ v^\varepsilon(t)= f + \sqrt{\varepsilon} \int_0^t \sigma(\varepsilon s, v^\varepsilon(s))\, dW(s). \]