Weak convergence of infinite-dimensional diffusions. (English) Zbl 0886.60005
The author considers a sequence of stochastic evolution equations
\[
dX_n=\big( AX_n+f_n(t,X_n) \big) dt+\sigma_n(t,X_n) dW_n, \qquad n=0,1,2,\ldots,
\]
in an infinite-dimensional Hilbert space \(H\). Here \(W_n\) are (possibly cylindrical) Wiener processes in a Hilbert space \(Y\) with covariance operators \(Q_n\in L(Y)\), \(A\) is the infinitesimal generator of a compact semigroup \(S\) on \(H\), \(f_n : [0,T]\times H\to H\), and \(\sigma : [0,T]\times H\to L(Y,H)\). The author obtains sufficient conditions for the weak convergence of the martingale solutions \(X_n\) to \(X_0\) in \(C([0,T],H)\) as \(n\to\infty\). The conditions are given in terms of convergence of the corresponding initial distributions, coefficients \(f_n\) and \(\sigma_n\), and covariance operators \(Q_n\). Particular results are obtained in the cases, where \(S\) is an analytical semigroup and where all \(W_n\) are standard cylindrical Wiener processes. The results obtained are applied to equations with rapidly oscillating coefficients and to a particular stochastic parabolic equation.
Reviewer: V.Mackevičius (Vilnius)
MSC:
60B10 | Convergence of probability measures |
60J60 | Diffusion processes |
60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
References:
[1] | DOI: 10.2969/jmsj/03910093 · Zbl 0616.47032 · doi:10.2969/jmsj/03910093 |
[2] | DOI: 10.1007/BF01292676 · Zbl 0794.60074 · doi:10.1007/BF01292676 |
[3] | Chojnowska–Michalik A., Existence, uniqueness and invariant. Rneasures for stochast.ic sernilinear equations on Hilbert spaces (1994) |
[4] | DOI: 10.1017/CBO9780511666223 · doi:10.1017/CBO9780511666223 |
[5] | DOI: 10.1016/0167-7152(93)90259-L · Zbl 0786.60089 · doi:10.1016/0167-7152(93)90259-L |
[6] | Gaarek D., Stochastics Stochastics Rep 46 pp 41– (1994) |
[7] | DOI: 10.1080/07362999408809346 · Zbl 0798.60064 · doi:10.1080/07362999408809346 |
[8] | Gikhman I.I., Ukrain. Mat. Zh 4 pp 215– (1952) |
[9] | Goldys, B. On weak solutions of stochastic evolution equations with unbounded coefficients@Miniconference on probability and analysis. Proc. Centre Math. Appl. Austral. Nat. Univ. 1991, Sydney. Vol. 29, pp.387–394. Canberra: Austral. Nat. Univ. |
[10] | Heunis A.J., Stochastics 16 pp 157– (1986) |
[11] | Krasnosel’Ski M.A., Uspekhi Mat. Nauk 10 pp 147– (1955) |
[12] | Makhno S. Ya, Theory of Randoui Processes 126 pp 113– |
[13] | DOI: 10.1007/BF00994917 · Zbl 0810.60054 · doi:10.1007/BF00994917 |
[14] | Tanabe H., Equations of evolution (1979) · Zbl 0417.35003 |
[15] | Triebel H., Interpolation theory, function spaces, differential operators (1978) · Zbl 0387.46033 |
[16] | Vrko[cddot] I., Math. Bohem 120 pp 91– (1995) |
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