An approach to Ito linear equations in Hilbert spaces by approximation of white noise with coloured noise. (English) Zbl 0547.60066

Acquistapace, Paolo; Broglia, Fabrizio

Stochastic Anal. Appl. 2, 131-186 (1984).

Consider on a Hilbert space H the stochastic evolution equation (*) \(du(t)=(A(t)u(t)+{1\over2}B^ 2u(t)+f(t))dt+Bu(t)dW_ t\), \(u(0)=X\), where f and X are prescribed data, \(W_ t\) is a real valued Wiener process

•  Encyclopedia of Mathematics
•  Encyclopedia of Mathematics
•  Wikipedia
•  Wolfram MathWorld

, and A(t), B generate an analytic semigroup and a strongly continuous group, respectively. The domains D(A(t)) may vary with t, and D(A(t))\(\subset D(B)\) is required. By analytic methods the authors obtain a generalized solution of (*) as the pathwise limit (a.s.) of associated deterministic solutions with smooth approximations of the Wiener process as input.

Reviewer: P.Kotelenez

MSC:

60H15

Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

approximation; analytic semigroup; strongly continuous group

References:

[1]	Acquistapace P., J. Math. Anal. Appl (1982)
[2]	Amann H., Periodic solutions of semi-linear parabolic equations, Non-linear analysis (1978) · Zbl 0406.35038
[3]	Balakrishnan, A. 1974. Stochastic bilinear partial differential equations. U.S. Italy Conference on variable structure systems. 1974, Oregon.
[4]	Choinowska Michalik A., Stochastic differential equations in Hilbert spaces and some of their applications (1971)
[5]	Curtain R.F., J. Math. Anal. Appl 60 pp 570– (1977) · Zbl 0367.60067 · doi:10.1016/0022-247X(77)90002-6
[6]	Curtain R.F., Infinite dimensional linear systems theory (1978) · Zbl 0389.93001 · doi:10.1007/BFb0006761
[7]	Da Prato, G. Stochastic differential equations with non-continuous coefficients in Hilbert spaces. conference held at the meeting ”Stochastic problems in Mechanics”. 1981, Torino. Pisa pre-print Sc. Norm. Sup
[8]	Da Prato G., Boll. Un. Mat. Ita1 16 (5) pp 168– (1979)
[9]	Da Prato G., Stochastics 6 (5) pp 105– (1982) · doi:10.1080/17442508208833196
[10]	Dawson D.A., Math Biosciences 15 (5) pp 287– (1972) · Zbl 0251.60040 · doi:10.1016/0025-5564(72)90039-9
[11]	Friedman A., Stochastic differential equations and applications 1 (1975) · Zbl 0323.60056
[12]	Ichikawa A., J. Diff. Eq 28 pp 266– (1978) · Zbl 0348.60087 · doi:10.1016/0022-0396(78)90071-2
[13]	Kato T., Osaka Math. J 14 pp 107– (1962)
[14]	Kotelenez P., Stochastics 8 pp 139– (1982) · Zbl 0495.60066 · doi:10.1080/17442508208833233
[15]	Krylov N.V., Izvestia Ak. Nauk CCCP, Math. Series 41 (6) pp 1329– (1977)
[16]	Krylov N.V., Seryia Sovremennye Problemy Matematiki 14 (6) pp 71– (1979)
[17]	Lipster R.S., Theory and applications (1978)
[18]	Metivier M., Z. Wahrscheinlichkeitstheorie verw. Geb 33 pp 1– (1975) · Zbl 0325.60054 · doi:10.1007/BF00539856
[19]	Pardoux E., Equations aux dérivées partielles stochastiques non linéaires monotones (1975)
[20]	Pardoux E., Stochastic partial differential equations and filtering of diffusion processes Stochastics 3 pp 127– (1979) · Zbl 0424.60067
[21]	Stratonovich R.L., Vestnik Moskov Univ. Sere I 1 pp 3– (1964)
[22]	Sussmann H., Ann. of Prob 6 pp 19– (1978) · Zbl 0391.60056 · doi:10.1214/aop/1176995608
[23]	Tanabe B., Osaka Math. J 12 pp 363– (1960)
[24]	Wong E., Z. Wahrscheinlichkeitstheorie verw. Geb 12 pp 87– (1969) · Zbl 0185.44401 · doi:10.1007/BF00531642
[25]	Yosida K., Functional analysis (1968) · Zbl 0152.32102 · doi:10.1007/978-3-662-11791-0
[26]	Zakai M., Z. Wahrscheinlichkei tstheorie verw. Geb 11 pp 230– (1969) · Zbl 0164.19201 · doi:10.1007/BF00536382