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Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles. (English) Zbl 1119.60057
In the framework of a Gelfand triple $$V\subset H=H^*\subset V^*$$ of Hilbert spaces, the authors investigate an evolution equation
$dY_t= -AY_tdt+b(Y_t)dt+\sigma(Y_t)dB_t+\int_Xf(Y_{t-},x)\,{\widetilde N}(dt,dx),\quad Y_0= h\in H,$ where $$A:V\to V^*$$ is linear bounded and coercive, $$B$$ is a real-valued Brownian motion, $$N(dt,dx)$$ is a Poisson counting measure on $${\mathbb R}_+\times X$$ ($$X$$ is a measurable space) with intensity $$dt\,\nu(dx)$$, $${\tilde N}(dt,dx)=N(dt,dx)-dt\,\nu(dx)$$, and $$b,\sigma: H\to H$$, $$f: H\times X\to H$$ are measurable mappings. It is proved, via succesive approximations, that under natural assumptions on coefficients (linear growth and Lipschitz) the equation (understood in the integral form) has a unique solution $$Y$$ which is progressively measurable, càdlàg in $$H$$ and $$E\int_0^T\| Y_t\| ^2_V\,dt<\infty$$, $$T>0$$. This result is then extended to the Poisson part of the equation of the form $\int_0^t\int_Ug(Y_{t-},x)\,\widetilde N(dt,dx)+ \int_0^t\int_{X \setminus U}g(Y_{t-},x)\,N(dt,dx),$ where $$\nu(X\setminus U)<\infty$$. It is also proved that if the coercivity hypothesis has the form $$\langle Au,u\rangle\geq\alpha\| u\| ^2_V$$, $$u\in V$$, for some $$\alpha>0$$, and if $$f$$ is exponentially integrable, then both $$\sup_{0\leq t\leq 1}| Y_t| _H$$ and $$\| Y\| _{L^2([0,1]\to V)}$$ are exponentially integrable. In the last part of the paper a large deviation principle is derived for the solutions of the following Ornstein-Uhlenbeck equations driven by Lévy noise: $Y_t^n=x-\int_0^tAY^n_s\,ds+bt+ {1\over n^{1/2}}W_t+{1\over n}\int_0^t\int_Xf(x)\,\widetilde N_n(ds,dx),$ where $$W$$ is an $$H$$-valued Wiener process and $$\tilde N$$ is a compensated Poisson measure with charactersistic measure $$n\nu$$.

MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations 60F10 Large deviations
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