Geodesic
Stochastic evolution equations driven by Liouville fractional Brownian motion
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 1-27
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Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of ${\cal L}(H,E)$-valued functions with respect to $H$-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0\beta 1$. For $0\beta \frac 12$ we show that a function $\Phi \colon (0,T)\to {\cal L}(H,E)$ is stochastically integrable with respect to an $H$-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an $H$-cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations $$ {\rm d}U(t) = AU(t) {\rm d}t + B {\rm d}W_H^\beta (t) $$ driven by an $H$-cylindrical Liouville fractional Brownian motion, and prove existence, uniqueness and space-time regularity of mild solutions under various assumptions on the Banach space $E$, the operators $A\colon \scr D(A)\to E$ and $B\colon H\to E$, and the Hurst parameter $\beta $. As an application it is shown that second-order parabolic SPDEs on bounded domains in $\mathbb R^d$, driven by space-time noise which is white in space and Liouville fractional in time, admit a mild solution if $\frac {1}{4}d\beta 1$.
Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of ${\cal L}(H,E)$-valued functions with respect to $H$-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0\beta 1$. For $0\beta \frac 12$ we show that a function $\Phi \colon (0,T)\to {\cal L}(H,E)$ is stochastically integrable with respect to an $H$-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an $H$-cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations $$ {\rm d}U(t) = AU(t) {\rm d}t + B {\rm d}W_H^\beta (t) $$ driven by an $H$-cylindrical Liouville fractional Brownian motion, and prove existence, uniqueness and space-time regularity of mild solutions under various assumptions on the Banach space $E$, the operators $A\colon \scr D(A)\to E$ and $B\colon H\to E$, and the Hurst parameter $\beta $. As an application it is shown that second-order parabolic SPDEs on bounded domains in $\mathbb R^d$, driven by space-time noise which is white in space and Liouville fractional in time, admit a mild solution if $\frac {1}{4}d\beta 1$.
DOI : 10.1007/s10587-012-0011-z
Classification : 35R60, 47D06, 60G18, 60H05
Keywords: (Liouville) fractional Brownian motion; fractional integration; stochastic evolution equations 
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Brzeźniak, Zdzisław; van Neerven, Jan; Salopek, Donna. Stochastic evolution equations driven by Liouville fractional Brownian motion. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 1-27. doi: 10.1007/s10587-012-0011-z

[1] Anh, V. V., Grecksch, W. A.: A fractional stochastic evolution equation driven by fractional Brownian motion. Monte Carlo Methods Appl. 9 (2003), 189-199. | MR | Zbl

[2] Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001), 766-801. | DOI | MR

[3] Biagini, F., Hu, Y., Øksendal, B., Zhang, T.: Stochastic Calculus for Fractional Brownian Motion and Applications. Probability and its Applications. Springer London (2008). | MR

[4] Brze'zniak, Z.: On stochastic convolutions in Banach spaces and applications. Stochastics Stochastics Rep. 61 (1997), 245-295. | DOI | MR

[5] Brze'zniak, Z., Neerven, J. M. A. M. van: Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise. J. Math. Kyoto Univ. 43 (2003), 261-303. | DOI | MR

[6] Brze'zniak, Z., Zabczyk, J.: Regularity of Ornstein-Uhlenbeck processes driven by a Lévy white noise. Potential Anal. 32 (2010), 153-188. | DOI | MR

[7] Caithamer, P.: The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise. Stoch. Dyn. 5 (2005), 45-64. | DOI | MR | Zbl

[8] Carmona, R., Fouque, J.-P., Vestal, D.: Interacting particle systems for the computation of rare credit portfolio losses. Finance Stoch. 13 (2009), 613-633. | DOI | MR

[9] Prato, G. Da, Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, Vol. 44. Cambridge University Press Cambridge (2008). | MR

[10] Decreusefond, L., Üstünel, A. S.: Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1999), 177-214. | DOI | MR | Zbl

[11] Dettweiler, J., Weis, L., Neerven, J. M. A. M. van: Space-time regularity of solutions of the parabolic stochastic Cauchy problem. Stochastic Anal. Appl. 24 (2006), 843-869. | DOI | MR

[12] Duncan, T. E., Pasik-Duncan, B., Maslowski, B.: Fractional Brownian motion and stochastic equations in Hilbert spaces. Stoch. Dyn. 2 (2002), 225-250. | DOI | MR | Zbl

[13] Fernique, X.: Intégrabilité des vecteurs gaussiens. C. R. Acad. Sci. Paris, Sér. A 270 (1970), 1698-1699 French. | MR | Zbl

[14] Feyel, D., Pradelle, A. de La: On fractional Brownian processes. Potential Anal. 10 (1999), 273-288. | DOI | MR | Zbl

[15] Goldys, B., Neerven, J. M. A. M. van: Transition semigroups of Banach space valued Ornstein-Uhlenbeck processes. Acta Appl. Math. 76 (2003), 283-330 Revised version: arXiv:math/0606785. | DOI | MR

[16] Grecksch, W., Roth, C., Anh, V. V.: $Q$-fractional Brownian motion in infinite dimensions with application to fractional Black-Scholes market. Stochastic Anal. Appl. 27 (2009), 149-175. | DOI | MR | Zbl

[17] Guasoni, P.: No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Finance 16 (2006), 569-582. | DOI | MR | Zbl

[18] Gubinelli, M., Lejay, A., Tindel, S.: Young integrals and SPDEs. Potential Anal. 25 (2006), 307-326. | DOI | MR | Zbl

[19] Hairer, M., Ohashi, A.: Ergodic theory for SDEs with extrinsic memory. Ann. Probab. 35 (2007), 1950-1977. | DOI | MR | Zbl

[20] Hu, Y.: Heat equations with fractional white noise potentials. Appl. Math. Optimization 43 (2001), 221-243. | DOI | MR | Zbl

[21] Hu, Y., Øksendal, B., Zhang, T.: General fractional multiparameter white noise theory and stochastic partial differential equations. Commun. Partial. Diff. Equations 29 (2004), 1-23. | DOI | MR

[22] Jumarie, G.: Would dynamic systems involving human factors be necessarily of fractal nature?. Kybernetes 31 (2002), 1050-1058. | DOI | Zbl

[23] Jumarie, G.: New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations. Math. Comput. Modelling 44 (2006), 231-254. | DOI | MR | Zbl

[24] Kalton, N. J., Neerven, J. M. A. M. van, Veraar, M. C., Weis, L.: Embedding vector-valued Besov spaces into spaces of gamma-radonifying operators. Math. Nachr. 281 (2008), 238-252. | DOI | MR

[25] Kalton, N. J., Weis, L.: The {$H^\infty$}-calculus and square function estimates. In preparation. | Zbl

[26] Leland, W., Taqqu, M., Willinger, W., Wilson, D.: On the self-similar nature of ethernet traffic. IEEE/ACM Trans. Networking 2 (1994), 1-15. | DOI

[27] Mandelbrot, B. B., Ness, J. W. Van: Fractional Brownian motion, fractional noises, and applications. SIAM Rev. 10 (1968), 422-437. | DOI | MR

[28] Maslowski, B., Nualart, D.: Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202 (2003), 277-305. | DOI | MR | Zbl

[29] Maslowski, B., Schmalfuss, B.: Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion. Stochastic Anal. Appl. 22 (2004), 1577-1607. | DOI | MR | Zbl

[30] Neerven, J. M. A. M. van: $\gamma$-radonifying operators---a survey. Proceedings CMA 44 (2010), 1-61. | MR

[31] Neerven, J. M. A. M. van: Stochastic Evolution Equations. Lecture Notes of the Internet Seminar 2007/08, OpenCourseWare, TU Delft, http://ocw.tudelft.nl</b>

[32] Neerven, J. M. A. M. van, Veraar, M. C., Weis, L.: Stochastic integration in UMD Banach spaces. Ann. Probab. 35 (2007), 1438-1478. | DOI | MR

[33] Neerven, J. M. A. M. van, Veraar, M. C., Weis, L.: Conditions for stochastic integrability in UMD Banach spaces. Inn: Banach Spaces and their Applications in Analysis: In Honor of Nigel Kalton's 60th Birthday. De Gruyter Proceedings in Mathematics Walter De Gruyter Berlin (2007), 127-146. | MR

[34] Neerven, J. M. A. M. van, Veraar, M. C., Weis, L.: Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal. 255 (2008), 940-993. | DOI | MR

[35] Neerven, J. M. A. M. van, Weis, L.: Stochastic integration of functions with values in a Banach space. Stud. Math. 166 (2005), 131-170. | DOI | MR

[36] Nualart, D.: Fractional Brownian motion: stochastic calculus and applications. In: Proceedings of the International Congress of Mathematicians. Volume III: Invited Lectures, Madrid, Spain, August 22-30, 2006 European Mathematical Society Zürich (2006), 1541-1562. | MR | Zbl

[37] Ohashi, A.: Fractional term structure models: no-arbitrage and consistency. Ann. Appl. Probab. 19 (2009), 1553-1580. | DOI | MR | Zbl

[38] Palmer, T. N., Shutts, G. J., Hagedorn, R., Doblas-Reyes, F. J., Jung, T., Leutbecher, M.: Representing model uncertainty in weather and climate prediction. Annu. Rev. Earth Planet. Sci. 33 (2005), 163-193. | DOI | MR

[39] Palmer, T. N.: A nonlinear dynamical perspective on model error: A proposal for non-local stochastic-dynamic parametrization in weather and climate prediction models. Q. J. Meteorological Soc. 127 (2001) B 279-304.

[40] Pasik-Duncan, B., Duncan, T. E., Maslowski, B.: Linear stochastic equations in a Hilbert space with a fractional Brownian motion. In: Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems. International Series in Operations Research & Management Science Springer New York (2006), 201-221. | MR | Zbl

[41] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44. Springer New York (1983). | MR

[42] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach New York (1993). | MR | Zbl

[43] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Math. Library, vol. 18. North-Holland Amsterdam (1978). | MR

[44] Tindel, S., Tudor, C. A., Viens, F.: Stochastic evolution equations with fractional Brownian motion. Probab. Theory Relat. Fields 127 (2003), 186-204. | DOI | MR | Zbl

[45] Weis, L.: Operator-valued Fourier multiplier theorems and maximal $L\sb p$-regularity. Math. Ann. 319 (2001), 735-758. | DOI | MR

[46] Willinger, W., Taqqu, M., Leland, W. E., Wilson, D. V.: Self-similarity in high speed packet traffic: analysis and modeling of Ethernet traffic measurements. Stat. Sci. 10 (1995), 67-85. | DOI | Zbl

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