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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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[1] de Acosta, A.: A genegral non-convex large deviation result with applications to stochastic equations. Probab. Theory Related Fields, 483–521 (2000) · Zbl 0993.60028
[2] de Acosta, A.: Large deviations for vector valued Lévy processes. Stochastic Process. Appl. 51, 75–115 (1994) · Zbl 0805.60020 · doi:10.1016/0304-4149(94)90020-5
[3] Albeverio, S., Wu, J.L., Zhang, T.S.: Parabolic SPDEs driven by Poisson White Noise. Stochastic Process. Appl. 74, 21–36 (1998) · Zbl 0934.60055 · doi:10.1016/S0304-4149(97)00112-9
[4] Chow, P.: Large deviation problem for some parabolic Itô equations. Comm. Pure Appl. Math. XLV, 97–120 (1992) · Zbl 0739.60055 · doi:10.1002/cpa.3160450105
[5] Chojnowska–Michalik, A.: On processes of Ornstein–Uhlenbeck type in Hilbert space. Stochastics 21, 251–286 (1987) · Zbl 0622.60072
[6] Chenal, F., Millet, A.: Uniform large deviations for parabolic SPDEs and applications. Stochastic Process. Appl. 72, 161–186 (1997) · Zbl 0942.60056 · doi:10.1016/S0304-4149(97)00091-4
[7] Cerrai, S., Röckner, M.: Large deviations for stochastic reaction–diffusion systems with multiplicative noise and non–Lipschtiz reaction term. Ann. Probab. 32(1), 1100–1139 (2004) · Zbl 1054.60065 · doi:10.1214/aop/1079021473
[8] Cardon-Weber, C.: Large deviations for a Burgers’-type SPDE. Stochastic Process. Appl. 84, 53–70 (1999) · Zbl 0996.60073 · doi:10.1016/S0304-4149(99)00047-2
[9] Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Jones and Bartlett Publishers, Boston, London (1992) · Zbl 0793.60030
[10] Fournier, N.: Malliavin calculus for parabolic SPDEs with jumps. Stochastic Process. Appl. 87(1), 115–147 (2000) · Zbl 1045.60067 · doi:10.1016/S0304-4149(99)00107-6
[11] Fuhrmann, M., Röckner, M.: Generalized Mehler semigroups: the non-Gaussian case. Potential Anal. 12, 1–47 (2000) · Zbl 0957.47028 · doi:10.1023/A:1008644017078
[12] Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin, New York (1994) · Zbl 0838.31001
[13] Gyöngy, I.: On stochastic equations with respect to semimartingales. III. Stochastics 7, 231–254 (1982) · Zbl 0495.60067
[14] Gyöngy, I., Krylov, N.V.: On stochastic equations with respect to semimartingales. I. Stochastics 4, 1–21 (1980/81) · Zbl 0439.60061
[15] Gyöngy, I., Krylov, N.V.: On stochastic equations with respect to semimartingales. II. Itô formula in Banach spaces. Stochastics 6, 153–173 (1981/82)
[16] Ikeda, N., Watanable, S.: Stochastic Differential Equations and Diffusion Processes. North–Holland/Kodansha, Amsterdam, Oxford, New York (1989)
[17] Krylov, N.V., Rosowskii, B.L.: Stochastic Evolution Equations. Current Problems in Mathematics, vol. 14 (Russian), pp. 71–147, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow (1979)
[18] Lions, J.L.: Equations Differentielles Operationnelles Et Problemes Aux Limites. Springer, Berlin Heidelberg New York (1961)
[19] Mueller, C.: The heat equation with Levy noise. Stochastic Process. Appl. 74(1), 67–82 (1998) · Zbl 0934.60056 · doi:10.1016/S0304-4149(97)00120-8
[20] Mytnik, L.: Stochastic partial differential equations driven by stable noise. Probab. Theory Related Fields 123(2), 157–201 (2002) · Zbl 1009.60053 · doi:10.1007/s004400100180
[21] Pardoux, E.: Stochastic partial differential equations and filtering of diffussion processes. Stochastics 3, 127–167 (1979) · Zbl 0424.60067
[22] Prato, G.D., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press (1992) · Zbl 0761.60052
[23] Protter, P.: Stochastic Integration and Differential Equations. Springer, Berlin Heidelberg New York (1990) · Zbl 0694.60047
[24] Zhang, T.S.: On small time asymptotics of diffusions on Hilbert spaces. Ann. Probab. 28(2), 537–557 (2000) · Zbl 1044.60071 · doi:10.1214/aop/1019160252
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