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Large deviation principle for a class of SPDE with locally monotone coefficients. (English) Zbl 07242640
The authors investigate stochastic partial differential equations (SPDEs) of the following form \[ dX_t = A(t,X_t)\,dt + B(t,X_t)\,dW_t \] where \(W_t\) is a cylindrical Wiener process on a separable Hilbert space \(U\) and \(A\) resp. \(B\) are coefficients on \([0,T]\times V\) taking values in \(V^*\) resp. the Hilbert-Schmidt operators from \(U\to H\). As usual, \(V\hookrightarrow H \simeq H^* \hookrightarrow V^*\) is a Gelfand triplet. Moreover, the coefficients are assumed to have the following four properties
\((A_1)\)
Hemicountinuity: \(\mathbb R\ni s\mapsto \mbox{}_{V^*}\langle A(t,v_1+sv_2),v \rangle_V\) is continuous
\((A_2)\)
Local monotonicity: \(2\mbox{}_{V^*}\langle A(t,v_1)-A(t,v_2),v_1-v_2 \rangle_V + \|B(t,v_1)-B(t,v_2)\|_{HS}^2 \leq -\delta\|v_1-v_2\|_V^\alpha + (K+\rho(v_2))\|v_1-v_2\|_H^2\) for a suitable fixed \(\alpha>0\) and a function \(\rho\) on \(V\) which is bounded by \(C+\|u\|_V^\alpha\cdot \|v\|_H^\beta\) for some \(\beta\geq 0\)
\((A_3)\)
(Linear) growth: \(v\mapsto B(t,v)\) is bounded by \(C+\|v\|_H\) and is Lipschitz in \(v\) and \[ \|A(t,v)\|_{V^*}^{\alpha/(\alpha-1)} \leq K(1+\|v\|_V^\alpha)(1+\|v\|_H^\beta) \]
\((A_4)\)
Time regularity: \(\|B(t_1,v)-B(t_2,v)\|_{HS}^2\leq L(1+\|v\|_V)|t_1-t_2|^\gamma\)
Under these assumptions, the authors show that the \(\epsilon\)-dependent solutions of the SPDEs \[ dX_t^\epsilon = A(t,X_t^\epsilon)\,dt + \epsilon B(t,X_t^\epsilon)\,dW_t,\quad X_0^\epsilon = x\in H \] satisfy a large deviation principle on \(C([0,T],H)\cap L^\alpha([0,T],V))\) with a good rate function of the form \[ I(z) = \inf\left\{\frac 12\int_0^T \|\phi_s\|_U^2\,ds \mid z=z^\phi\in C([0,T],H)\cap L^\alpha([0,T],V)\; \phi\in L^2([0,T],U)\right\} \] The proof relies on the weak convergence method and Laplace’s principle.
MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60F10 Large deviations
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