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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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