Space-time fractional diffusions in Gaussian noisy environment

References

• R. Balan and D. Conus, A note on intermittency for the fractional heat equation, Statist. Probab. Lett. 95 (2014), pp. 6–14.
   Web of Science ®Google Scholar
• R. Balan and D. Conus, Intermittency for the wave and heat equations with fractional noise in time, to appear in Ann. Probab. (2015).
   Web of Science ®Google Scholar
• R. Balan and C. Tudor, Stochastic heat equation with multiplicative fractional-colored noise, J. Theor. Probab. 23 (2010), pp. 834–870.
   Web of Science ®Google Scholar
• L. Chen, Nonlinear stochastic time-fractional diffusion equations on ℝ: Moments, Hölder regularity and intermittency, Trans. Amer. Math. Soc. preprint (2014). Available at arXiv:1410.1911.
   Google Scholar
• L. Chen and R.C. Dalang, Moment bounds and asymptotics for the stochastic wave equation, Stoch. Process. Appl. 125(4) (2015), pp. 1605–1628.
   Web of Science ®Google Scholar
• L. Chen and R.C. Dalang, Moments and growth indices for nonlinear stochastic heat equation with rough initial conditions, to appear in Ann. Probab. 43(6) (2015), pp. 3006–3051.
   Web of Science ®Google Scholar
• L. Chen and R.C. Dalang, Moments, intermittency, and growth indices for the nonlinear fractional stochastic heat equation, to appear in Stoch. Partial Differ. Equ. Anal. Comput. 3(3) (2015), pp. 360–397.
   Google Scholar
• L. Chen, Y. Hu, and D. Nualart, Nonlinear stochastic time-fractional slow and fast diffusion equations, Stoch. Process. Appl. preprint (2015).
   Web of Science ®Google Scholar
• Z.Q. Chen, K.H. Kim, and P. Kim, Fractional time stochastic partial differential equations, Stoch. Process. Appl. 125(4) (2015), pp. 1470–1499.
   Web of Science ®Google Scholar
• R.C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e’.s, Electron. J. Probab. 4(6) (1999), 29 pp.
   Google Scholar
• K. Diethelm, The Analysis of Fractional Differential Equations, An Application-oriented Exposition using Differential Operators of Caputo Type, Lecture Notes in Mathematics Vol. 2004, Springer-Verlag, Berlin, 2010.
   Google Scholar
• S. Eidelman and A. Kochubei, Cauchy problem for fractional diffusion equations, J. Differ. Equ. 199(2) (2004), pp. 211–255.
   Web of Science ®Google Scholar
• M. Foondun and D. Khoshnevisan, Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab. 14 (2009), pp. 548–568.
   Web of Science ®Google Scholar
• M. Foondun and E. Nane, Asymptotic properties of some space-time fractional stochastic equations, preprint (2015). Available at arXiv:1505.04615.
   Google Scholar
• Y. Hu, Heat equations with fractional white noise potentials, Appl. Math. Optim. 43(3) (2001), pp. 221–243.
   Web of Science ®Google Scholar
• Y. Hu, Chaos expansion of heat equations with white noise potentials, Potential Anal. 16(1) (2002), pp. 45–66.
   Web of Science ®Google Scholar
• G. Hu and Y. Hu, Fractional diffusion in Gaussian noisy environment, Mathematics 3 (2015), pp. 131–152. doi:10.3390/math3020131.
   Web of Science ®Google Scholar
• Y. Hu, J. Huang, K. Le, D. Nualart, and S. Tindel, Stochastic heat equation with rough dependence in space, preprint (2015). Available at arXiv:1505.04924.
   Google Scholar
• Y. Hu, J. Huang, D. Nualart, and S. Tindel, Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency, Electron. J. Probab. 20(55) (2015), pp. 1–50.
   Google Scholar
• Y. Hu and J. Yan, Wick calculus for nonlinear Gaussian functionals, Acta Math. Appl. Sin. Engl. Ser. 25 (2009), pp. 399–414.
   Web of Science ®Google Scholar
• D. Khoshnevisan and Y. Xiao, Harmonic analysis of additive Lévy processes, Probab. Theory Related Fields 145(3–4) (2009), pp. 459–515.
   Web of Science ®Google Scholar
• A.A. Kilbas and M. Saigo, H-transforms: Theory and Applications, Analytical Methods and Special Functions Vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2004.
   Google Scholar
• A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies Vol. 204, Elsevier Science B.V., Amsterdam, 2006.
   Google Scholar
• A. Kochubeĭ, Diffusion of fractional order, Differ. Equ. 26(4) (1990), pp. 485–492.
   Web of Science ®Google Scholar
• F. Mainardi, Y. Luchko, and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal. 4(2) (2001), pp. 153–192.
   Google Scholar
• J. Mijena and E. Nane, Intermittence and time fractional stochastic partial differential equations, preprint (2014). Available at arXiv:1409.7468.
   Google Scholar
• J. Mijena and E. Nane, Space-time fractional stochastic partial differential equations, to appear in Stoch. Process. Appl. (2015).
   Web of Science ®Google Scholar
• D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Probability and its Applications, Springer-Verlag, Berlin.
   Google Scholar
• F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark (eds.), NIST Handbook of Mathematical Functions, US Department of Commerce National Institute of Standards and Technology, Washington, DC, 2010.
   Google Scholar
• I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Vol. 198, Academic Press Inc., San Diego, CA, 1999.
   Google Scholar
• A.V. Pskhu, The fundamental solution of a diffusion-wave equation of fractional order, Izv. Math. 73(2) (2009), pp. 351–392.
   Web of Science ®Google Scholar
• W. Schneider, Completely monotone generalized Mittag--Leffler functions, Exp. Math. 14(1) (1996), pp. 3–16.
   Google Scholar
• W. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30(1) (1989), pp. 134–144.
   Web of Science ®Google Scholar
• J. Song, On a class of stochastic partial differential equations, preprint (2015). Available at arXiv:1503.06525v2.
   Google Scholar
• E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series No. 30, Princeton University Press, Princeton, NJ, 1970.
   Google Scholar
• E. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series No. 32, Princeton University Press, Princeton, NJ, 1971.
   Google Scholar
• V.V. Uchaikin and V.M. Zolotarev, Chance and Stability, Stable Distributions and Their Applications, Modern Probability and Statistics, VSP, Utrecht, 1999.
   Google Scholar
• D.V. Widder, The Laplace Transform, Princeton Mathematical Series Vol. 6, Princeton University Press, Princeton, NJ, 1941.
   Google Scholar