Uniform large deviations for parabolic SPDEs and applications
Large deviations for mild solutions to a stochastic parabolic equation \[ \frac {\partial X^\varepsilon }{\partial t} = LX^\varepsilon + b(t,x,X^\varepsilon) + \varepsilon \sigma (t,x,X^\varepsilon) \frac {\partial ^{2}W}{\partial t\partial x}, \quad X^\varepsilon (0) = \varphi , \;0\leq t\leq T, \;0\leq x\leq 1, \] with homogeneous Dirichlet or Neumann boundary data are studied. Here, \(L\) denotes an elliptic second order differential operator of the form \(Lh = D\partial _{xx}h - Ch\), \(D>0\), \(C\geq 0\), \(W\) is a Brownian sheet, the functions \(b(t,x,\cdot)\), \(\sigma (t,x,\cdot)\) are Lipschitz and of a linear growth uniformly in \((t,x)\), and \(\sigma \) is continuous on \([0,T]\times [0,1]\times \mathbb R\) (satisfying, however, no nondegeneracy assumption in general). Let us denote by \(C^\alpha \) the space of all real functions \(F\) on \([0,T]\times [0,1]\) such that \(\sup [F]_\alpha <\infty \), where \[ [F]_\alpha (s,t,x,y) = (|s-t|^{2}+|x-y|^{4})^{-\alpha /2}|F(t,x)-F(s,y)|, \] and by \(C^{\alpha ,0}\) the subspace consisting of functions satisfying \([F]_\alpha \to 0\) as \(|s-t|+|x-y|\to 0\). It is shown that the solutions \((X^\varepsilon)_{\varepsilon >0}\) obey a large deviations principle in the space \(C^\alpha \) for any \(\alpha \in \left ]0,\frac 14\right [\) uniformly in the initial condition \(\varphi \), provided the initial data are taken in a compact subset of \(C^{\alpha ,0}\). This result is then applied to study asymptotics as \(\varepsilon \downarrow 0\) for the exit time of \(X^\varepsilon \) from a bounded subset of \(C^{\alpha ,0}\).
stochastic parabolic partial differential equations
scientific article; zbMATH DE number 1409156
scientific article