×

Evolution equations driven by a fractional Brownian motion. (English) Zbl 1027.60060

The authors study stochastic evolution equations driven by a fractional Brownian motion (fBm) of the form \[ dX_{t}=(A X_{t}+F(X_{t})) dt + G(X_{t}) dB_{t}^{H}, \quad X_{0}=x_{0}, t\in [0,T], \] in a Hilbert space \(V\), where \(A\) is the infinitesimal generator of an analytic semigroup on \(V\), \(B^{H}\) is a \(V\)-valued fBm with Hurst parameter \(H>1/2\) and with nuclear covariance operator. Existence and uniqueness of mild solution are established under some regularity and growth conditions on the coefficients \(F\) and \(G\) and for some values of \(H\). Moreover, an existence result is proved under less restrictive assumptions on the coefficients and the space dimension \(d\). The proofs of these results are based on the approach developed by D. Nualart and A. Răşcanu [Collect. Math. 53, 55-81 (2002; Zbl 1018.60057)] and they combine techniques of fractional calculus and semigroup estimates.
As application, the authors deal with stochastic parabolic equations driven by a fractional white noise with nuclear covariance: \[ \frac{\partial u}{\partial t}= Lu+f(u)+\Phi(u)\frac{\partial B^{H}}{\partial t}, \] with some boundary conditions, where \(L\) is a uniformly elliptic operator, the drift \(f\) is supposed to be continuous with at most linear growth and \(\Phi\) is Lipschitz continuous. Finally, it is worth quoting from the authors: “It may be surprising that if the coefficient \(\Phi\) is Lipschitz and bounded, we find the restriction \(H>\frac{d}{4}\) for having a function space valued solution, which is the same as in T. E. Duncan, B. Maslowski and B. Pasik-Duncan [Stoch. Dyn. 2, 225-250 (2002; Zbl 1040.60054)] for the case of an additive noise with identity covariance”.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agmon, S. A.: Lectures on elliptic boundary value problems. (1965) · Zbl 0142.37401
[2] Alòs, E.; Mazet, O.; Nualart, D.: Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than 1/2. Stochastic process. Appl. 86, 121-139 (2000) · Zbl 1028.60047
[3] Alòs, E.; Mazet, O.; Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. probab. 29, 766-801 (2001) · Zbl 1015.60047
[4] P. Carmona, L. Coutin, Stochastic integration with respect to fractional Brownian motion, preprint. · Zbl 0951.60042
[5] Coutin, L.; Qian, Z.: Stochastic differential equations for fractional Brownian motions. C. R. Acad. sci. Paris sér. I math. 331, 75-80 (2000) · Zbl 0981.60040
[6] Coutin, L.; Qian, Z.: Stochastic analysis, rough paths analysis and fractional Brownian motions. Probab. theory rel. Fields 122, 108-140 (2002) · Zbl 1047.60029
[7] Da Prato, G.; Zabczyk, J.: Stochastic equations in infinite dimensions. (1992) · Zbl 0761.60052
[8] Dai, W.; Heyde, C. C.: Itô’s formula with respect to fractional Brownian motion and its applications. J. appl. Math. stochastic anal. 9, 439-448 (1996) · Zbl 0867.60029
[9] Decreusefond, L.; Üstünel, A. S.: Stochastic analysis of the fractional Brownian motion. Potential anal. 10, 177-214 (1999) · Zbl 0924.60034
[10] Duncan, T. E.; Hu, Y.; Pasik-Duncan, B.: Stochastic calculus for fractional Brownian motion. Itheory. SIAM J. Control optim. 38, 582-612 (2000) · Zbl 0947.60061
[11] Duncan, T. E.; Maslowski, B.; Pasik-Duncan, B.: Fractional Brownian motion and stochastic equations in Hilbert spaces. Stochastics dyn. 2, 225-250 (2002) · Zbl 1040.60054
[12] Eidel’man, S. D.: Parabolic systems. (1969) · Zbl 0181.37403
[13] Feyel, D.; De La Pradelle, A.: Fractional integrals and Brownian processes. Potential anal. 10, 273-288 (1996) · Zbl 0944.60045
[14] Garroni, M. G.; Menaldi, J. L.: Green functions for second order parabolic integro-differential problems. (1992) · Zbl 0806.45007
[15] 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
[16] Grisvard, P.: Commutativité de deux foncteurs d’interpolation et applications. J. math. Pures appl. 45, 143-290 (1966) · Zbl 0173.15803
[17] Hu, Y.: Heat equation with fractional white noise potentials. Appl. math. Optim. 43, 221-243 (2001) · Zbl 0993.60065
[18] Kolmogorov, A. N.: Wienersche spiralen und einige andere interessante kurven im hilbertschen raum. CR (Doklady) acad. URSS (N.S.) 26, 115-118 (1940) · JFM 66.0552.03
[19] Leland, W.; Taqqu, M.; Willinger, W.; Wilson, D.: On the self-similar nature of Ethernet traffic. IEEE/ACM trans. Networking 2, 1-15 (1994)
[20] Lin, S. J.: Stochastic analysis of fractional Brownian motions. Stochastics stochastics rep. 55, 121-140 (1995) · Zbl 0886.60076
[21] Lyons, T.: Differential equations driven by rough signals (I)an extension of an inequality of L.C. Young. Math. res. Lett. 1, 451-464 (1994) · Zbl 0835.34004
[22] Lyons, T.: Differential equations driven by rough signals. Rev. mat. Iberoamericana 14, 215-310 (1998) · Zbl 0923.34056
[23] Mandelbrot, B. B.: The variation of certain speculative prices. J. business 36, 394-419 (1963)
[24] Manthey, R.; Zausinger, T.: Stochastic evolution equations in \(L{\rho}2{\nu}\). Stochastics stochastics rep. 66, 37-85 (1999) · Zbl 0926.60051
[25] Mandelbrot, B. B.; Van Ness, J. W.: Fractional Brownian motions, fractional noises and applications. SIAM rev. 10, No. 4, 422-437 (1968) · Zbl 0179.47801
[26] Nualart, D.; Rascanu, A.: Differential equations driven by fractional Brownian motion. Collectanea math. 53, 55-81 (2002) · Zbl 1018.60057
[27] Ovespian, I.; Pelczynski, A.: On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in L2. Studia math. 54, 149-159 (1975)
[28] Peszat, S.; Zabczyk, J.: Stochastic evolution equations with spatially homogeneous Wiener process. Stochastic process. Appl. 72, 187-204 (1997) · Zbl 0943.60048
[29] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives. Theory and applications. (1993) · Zbl 0818.26003
[30] Seidler, J.: Da prato- zabczyk’s maximal inequality revisited. Math. bohemica 118, 67-106 (1993) · Zbl 0785.35115
[31] Young, L. C.: An inequality of the hölder type connected with Stieltjes integration. Acta math. 67, 251-282 (1936) · Zbl 0016.10404
[32] Zähle, M.: Integration with respect to fractal functions and stochastic calculus, I. Probab. theory related fields 111, 333-374 (1998) · Zbl 0918.60037
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.