# zbMATH — the first resource for mathematics

On the Poisson equation and diffusion approximation. I. (English) Zbl 1029.60053
The authors consider the Poisson equation $$Lu=-f$$ in the entire $$d$$-dimensional space. $$L$$ denotes a second-order elliptic differential operator which is the generator of a positive recurrent diffusion process $$X$$. The function $$f$$ is assumed to be centred with respect to the invariant measure of $$X$$. The first part of the paper investigates under what conditions the function $$u(x)=\int_0^\infty E_x[f(X_t)]dt$$ defines a solution of the above Poisson equation. The two main conditions are that the diffusion coefficient is strongly elliptic and that the drift coefficient $$b$$ satisfies the mixing condition $$\langle b(x),x/|x|\rangle\leq -r |x|^\alpha$$ for certain $$r>0$$ and $$\alpha\geq -1$$ outside a compact set. Several properties of the solution are derived by probabilistic methods. The second part applies these results to singularly perturbed random differential equations and establishes their convergence to a stochastic differential equation. In the appendix a version of the Itô-Krylov formula is proved.

##### MSC:
 60H30 Applications of stochastic analysis (to PDEs, etc.) 35J15 Second-order elliptic equations 60J45 Probabilistic potential theory 60J60 Diffusion processes
Full Text:
##### References:
 [1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201 [2] Bouc, R. and Pardoux, E. (1984). Asymptotic analysis of PDEs with wide-band noise disturbance expansion of the moments. Stochastic Anal. Appl. 2 369-422. · Zbl 0574.60066 · doi:10.1080/07362998408809044 [3] Dynkin, E. B. (1965). Markov Processes 2. Springer, Berlin. · Zbl 0132.37901 [4] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence, Wiley, New York. · Zbl 0592.60049 [5] Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer, Berlin. · Zbl 0562.35001 [6] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam. · Zbl 0495.60005 [7] Khasminski, R.(1966). A limit theorem for solutions of differential equations with random right-hand sides. Theory Probab. Appl. 11 390-406. · Zbl 0202.48601 [8] Khasminski, R.(1980). Stochastic Stability of Differential Equations. Sijthoff and Nordhoff, The Netherlands. [9] Krylov, N. V. (1980). Controlled Diffusion Processes (trans. by A. B. Aries). Springer, Berlin. · Zbl 0436.93055 · doi:10.1070/SM1980v037n01ABEH001946 [10] Ladyzenskaja, O., Solonnikov, V. and Ural’ceva, N. (1968). Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI. · Zbl 0174.15403 [11] Papanicolaou, G. C., Stroock, D. W. and Varadhan, S. R. S. (1977). Martingale approach to some limit theorems. In Conference on Statistical Mechanics, Dynamical Systems and Turbulence (M. Reed ed.) Duke Univ. Press. · Zbl 0387.60067 [12] Pardoux, E. and Veretennikov, A. Yu. (1997). Averaging of backward stochastic differential equations, with application to semi-linear PDEs. Stochastics Stochastic. Rep. 60 255- 270. · Zbl 0891.60053 [13] Revuz, D. (1984). Markov Chains, 2nd rev. ed. North-Holland, Amsterdam. · Zbl 0539.60073 [14] Stratonovich, R. L. (1963, 1967). Topics in the Theory of Random Noise 1, 2. Gordon and Breach, New York. · Zbl 0183.22007 [15] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, Berlin. · Zbl 0426.60069 [16] Veretennikov, A. Yu. (1982). Parabolic equations and It o’s stochastic equations with coefficients discontinuous in the time variable. J. Math. Notes 31 278-283 (trans. from Mat. Zametki 31 549-557). · Zbl 0563.60056 · doi:10.1007/BF01138937 [17] Veretennikov, A. Yu. (1987). Bounds for the mixingrates in the theory of stochastic equations. Theory Probab. Appl. 32 273-281. · Zbl 0663.60046 · doi:10.1137/1132036 [18] Veretennikov, A. Yu. (1997). On polynomial mixingbounds for stochastic differential equations. Stochastic Process. Appl. 70 115-127. · Zbl 0911.60042 · doi:10.1016/S0304-4149(97)00056-2 [19] LATP, UMR-CNRS 6632 Centre de Mathématiques et d’Informatique Université de Provence 39, rue F. Joliot Curie 13453 Marseille cedex 13 France E-mail: pardoux@cmi.univ-mrs.fr Institute of Information Transmission Problems 19, Bolshoy Karetnii 101447 Moscow Russia E-mail: veretenn@iitp.ru
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.