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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 \(n\nu\).

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