### Homogenization of Fractional Kinetic Equations with Random Initial Data

**Gi-Ren Liu**

*(National Taiwan University)*

**Narn-Rueih Shieh**

*(National Taiwan University)*

#### Abstract

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 962-980

Publication Date: April 14, 2011

DOI: 10.1214/EJP.v16-896

#### References

[1] V. V. Anh and N. N. Leonenko, Spectral analysis of fractional kinetic equations with random data, J. Statist. Phys. 104 (2001), 1349-1387. MR1859007

[2] V. V. Anh and N. N. Leonenko, Renormalization and homogenization of fractional diffusion equations with random data, Probab. Theory Rel. Fields 124 (2002), 381-408. MR1939652

[3] V. V. Anh and N. N. Leonenko, Spectral theory of renormalized fractional random fields, Theor.Probability and Math. Statist. 66 (2003), 1-13. MR1931155

[4] V. V. Anh, N. N. Leonenko, and L. M. Sakhno, Higher-order spectral densities of fractionalrandom fields, J. Statist. Phys. 26 (2003), 789-814. MR1972127

[5] O. E. Barndor-Nielsen and N. N. Leonenko, Burgers' turbulence problem with linear orquadratic external potential, J. Appl. Probab. 42 (2005), 550-565. MR2145493

[6] A. Benassi, S. Cohen, and J. Istas, Identification and properties of real harmonizable fractionalLevy motions. Bernoulli 8 (2002), 97-115. MR1884160

[7] A. Benassi, S. Cohen, and J. Istas, On roughness indices for fractional fields. Bernoulli 10 (2004),357-373. MR2046778

[8] M. M. Djrbashian, Harmonic Analysis and Boundary Value Problems in Complex Domain,Birkhauser 1993. MR1249271

[9] P. Doukhan, G. Oppenheim, and M.S. Taqqu, Theory and Applications of Long-Range Dependence, Birkhauser 2003. MR1956041

[10] A. Erdely, W. Magnus, F. Obergettinger, F.G. Tricomi, Higher Transcendental Functions (III).McGraw-Hill 1955. MR0066496

[11] R. L. Dobrushin and P. Major, Non-central limit theorems for nonlinear functionals of Gaussianelds, Z. Wahrsch. verw. Geb. 50 (1979), 1-28. MR550122

[12] I. I. Gikhman and A. V. Skorohod, The Theory of Stochastic Processes (I). Springer 2004. MR2058259

[13] M. Ya. Kelbert, N. N. Leonenko, and M. D. Ruiz-Medina, Fractional random fields associatedwith stochastic fractional heat equations, Adv. Appl. Prob. 37 (2005), 108-133. MR2135156[14] N. N. Leonenko, Limit Theorems for Random Fields with Singular Spectrum. Kluwer Academic1999. MR1687092

[15] N. N. Leonenko and M. D. Ruiz-Medina, Scaling laws for the multi-dimensional Burgers' equationwith quadratic external potential, J. Statist. Phys. 124 (2006), 191-205. MR2256621

[16] N. N. Leonenko and W. A. Woyczynski, Scaling limits of solutions of the heat equation withnon-Gaussian data, J. Statist. Phys. 91 (1998), 423-438. MR1632518

[17] G.-R. Liu and N.-R. Shieh, Scaling limits for some P.D.E. systems with random initial conditions,

Stoch. Anal. Appl. 28 (2010), 505-522. MR2739572

[18] G.-R. Liu and N.-R. Shieh, Scaling limits for time-fractional diffusion-wave systems with randominitial data, Stochastics and Dynamics 10 (2010), 1-35. MR2604676

[19] G.-R. Liu and N.-R. Shieh, Multi-scaling limits for relativistic diffusion equations with random initial data. working project.

[20] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett. 9 (1996), 23-28. MR1419811

[21] F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoustics 9 (2001), 1417-1436. MR1881799

[22] P. Major, Multiple Wiener-Ito Integrals, Lecture Note in Math. 849, Springer 1981. MR0611334

[23] P. Morters and N.-R. Shieh, Small scale limit theorems for the intersection local times of Brownianmotions, Elec. J. Probab. 4 (1999), Paper No. 9 1-23. MR1690313

[24] E.M. Stein, Singular Integrals and Differentiability Properties of Function. Princeton Univ. Press1970. MR0290095

[25] M. S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrsch. verw. Geb. 50 (1979), 53-83. MR0550123

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