# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.