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Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. (English) Zbl 1076.60054
The article is devoted to the stochastic differential equation \[ dX(t)=A(t)X(t)dt+\sum _{k=1}^mB_kX(t)d\beta ^H_k(t).\tag{1} \] Here \(A(t), B_k, k=1,\dots,m\), are unbounded densely defined linear operators in the infinite-dimensional Hilbert space. The processes \(\beta ^H_k, k=1,\dots,m,\) are independent one-dimensional fractional Brownian motions with the Hurst parameter \(H.\) The authors propose the representation for the strong solution of (1). They suppose, in particular, that the operators \(B_k, k=1,\dots,m,\) generate mutually commuting strongly continuous groups \(S_k(s), k=1,\dots,m,\) and the family \(\tilde{A}(t)=A(t)-Ht^{2H-1}\sum _{k=1}^mB_k^2\) generates a strongly continuous evolution family of operators \(\{U(t,s); 0\leq s\leq t\leq T\}.\) Then the solution of (1) can be obtained by the following formula \[ X(t)=\prod _{k=1}^mS_k(\beta ^H_k(t))U(t,0)x_0. \] Such representation allows to study the stability of the solution using the law of the iterated logarithm for fractional Brownian motion and some kind of Lyapunov function despite of the absence of the Markov property of \(X.\) Examples, when \(\beta ^H\) can have the stabilizing influence, are also presented.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
47D06 One-parameter semigroups and linear evolution equations
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[1] Alos, E.; Nualart, D., Stochastic integration with respect to fractional Brownian motion, Stoch. stoch. rep., 75, 129-152, (2003) · Zbl 1028.60048
[2] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge · Zbl 0761.60052
[3] Datko, R., Extending a theorem of A. Lyapunov to Hilbert space, J. math. anal. appl., 32, 610-616, (1970) · Zbl 0211.16802
[4] Duncan, T.E., Some applications of fractional Brownian motion to linear systems, (), 97-105 · Zbl 0974.93061
[5] Duncan, T.E.; Hu, Y.Z.; Pasik-Duncan, B., Stochastic calculus for fractional Brownian motion. I: theory, SIAM J. control optim., 38, 582-612, (2000) · Zbl 0947.60061
[6] T.E. Duncan, J. Jakubowski, B. Pasik-Duncan, Stochastic integration for fractional Brownian motion in a Hilbert space, preprint. · Zbl 1095.60017
[7] Duncan, T.E.; Maslowski, B.; Pasik-Duncan, B., Fractional Brownian motion and linear stochastic equations in Hilbert space, Stochastic dyn., 2, 225-250, (2002) · Zbl 1040.60054
[8] Grecksch, W.; Anh, V.V., A parabolic stochastic differential equation with fractional Brownian motion input, Statist. probab. lett., 41, 337-345, (1999) · Zbl 0937.60064
[9] Hu, Y., Heat equation with fractional noise potentials, Appl. math. optim., 43, 221-243, (2001) · Zbl 0993.60065
[10] Y. Hu, B. Oksendal, T. Zhang, Stochastic partial differential equations driven by multiparameter fractional white noise, in: F. Gesztesy et al. (Eds.), Stochastic Processes, Physics and Geometry: New Interplays, vol. II. (Can. Math. Soc. Proc.) 29 (2000) 327-337.) · Zbl 0982.60054
[11] Hu, Y.; Oksendal, B.; Zhang, T., General fractional multiparameter white noise theory and stochastic partial differential equations, Commun. partial differential equations, 29, 1-23, (2004) · Zbl 1067.35161
[12] Hunt, G.A., Random Fourier series, Trans. amer. math. soc., 71, 38-69, (1951) · Zbl 0043.30601
[13] Hurst, H.E., Long-term storage capacity in reservoirs, Trans. amer. soc. civil eng., 116, 400-410, (1951)
[14] Ichikawa, A., Stability of semilinear stochastic evolution equations, J. math. anal. appl., 90, 12-44, (1982) · Zbl 0497.93055
[15] Kolmogorov, A.N., Wienersche spiralen und einige andere interessante kurven im hilbertschen raum, C. R. (doklady) acad. URSS (N.S.), 26, 115-118, (1940) · JFM 66.0552.03
[16] Leland, W.E.; Taqqu, M.S.; Willinger, W.; Wilson, D.V., On the self-similar nature of Ethernet traffic, IEEE/ACM trans. networking, 2, 1-15, (1994)
[17] Mandelbrot, B.B., The variation of certain speculative prices, J. business, 36, 394-419, (1963), (reprinted in: P.H. Cootner (Ed.), The Random Character of Stock Market Prices, MIT Press, Cambridge, MA, 1964, pp. 297-337).
[18] Mandelbrot, B.B.; Van Ness, J.W., Fractional Brownian motion, fractional noises and applications, SIAM rev., 10, 422-437, (1968) · Zbl 0179.47801
[19] Maslowski, B.; Nualart, D., Evolution driven by fractional Brownian motion, J. funct. anal., 202, 277-305, (2003) · Zbl 1027.60060
[20] B. Maslowski, B. Schmalfuss, Random dynamical systems and stationary solutions of differential equations driven by fractional Brownian motion. Stoc. Anal. Appl. 22 (2004) 1577-1609. · Zbl 1062.60060
[21] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer New York · Zbl 0516.47023
[22] Peszat, S.; Zabczyk, J., Stochastic evolution equations with a spatially homogeneous Wiener process, Stochast. process. appl., 72, 187-204, (1997) · Zbl 0943.60048
[23] Tanabe, H., Equations of evolution, (1979), Pitman London
[24] Tindel, S.; Tudor, C.A.; Viens, F., Stochastic evolution equations with fractional Brownian motion, Probab. th. rel. fields, 127, 186-204, (2003) · Zbl 1036.60056
[25] Zähle, M., Integration with respect to fractal functions and stochastic calculus, Probab. th. rel. fields, 111, 3, 333-374, (1998) · Zbl 0918.60037
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