Intermittence and Space-Time Fractional Stochastic Partial Differential Equations

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• 作者：Jebessa B. Mijena ; Erkan Nane
• 关键词：Caputo fractional derivative ; Time fractional SPDE ; Intermittency ; Intermittency fronts
• 刊名：Potential Analysis
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：44
• 期：2
• 页码：295-312
• 全文大小：314 KB
• 参考文献：1.Arponen, H., Horvai, P.: Dynamo effect in the Kraichnan magnetohydrodynamic turbulence. J. Stat. Phys. 129(2), 205–239 (2007)CrossRef MathSciNet MATH
  2.Baeumer, B., Meerschaert, M.M.: Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4, 481–500 (2001)MathSciNet MATH
  3.Baxendale, P.H., Rozovskiĭ, B.L.: Kinematic dynamo and intermittence in a turbulent flow. Geophys. Astrophys. Fluid Dyn. 73(1–4), 33–60 (1993). Magnetohydrodynamic stability and dynamos (Chicago, IL, 1992)CrossRef MathSciNet
  4.Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)MATH
  5.Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, Part II. Geophys. J. R. Astr. Soc. 13, 529–539 (1967)CrossRef
  6.Carmona, R.A., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Mem. Am. Math. Soc. 108(518), viii+125 (1994)MathSciNet
  7.Chen, L., Dalang, R.C.: The nonlinear stochastic heat equation with rough initial data: a summary of some new results, arXiv:1210.​1690v1.​pdf (2012)
  8.Chen, Z.-Q., Kim, K.-H., Kim, P.: Fractional time stochastic partial differential equations. Stoch. Process. Appl. 125, 1470–1499 (2015)CrossRef MathSciNet MATH
  9.Conus, D., Khoshnevisan, D.: On the existence and position of the farthest peaks of a family of stochastic heat and wave equations. Probab. Theory Relat. Fields 152(3–4), 681–701 (2012)CrossRef MathSciNet MATH
  10.Dalang, R.C., Quer-Sardanyons, L.: Stochastic integrals for spde’s: a comparison. Expo. Math. 29(1), 67–109 (2011)CrossRef MathSciNet MATH
  11.Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)CrossRef
  12.Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. Wiley, New York-London-Sydney (1971)MATH
  13.Foondun, M., Khoshnevisan, D.: Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14(21), 548–568 (2009)MathSciNet MATH
  14.Galloway, D.: Fast Dynamos, Advances in Nonlinear Dynamos. Fluid Mech. Astrophys. Geophys., Vol. 9, pp 37–59. Taylor & Francis (2003)
  15.Georgiou, N., Joseph, M., Khoshnevisan, D., Mahboubi, P., Shiu, S.-Y: Semi-discrete semi-linear parabolic SPDEs (2013)
  16.Haubold, H.J., Mathai, A.M., Saxena, R.K.: Review article: Mittag-Leffler functions and their applications. J. Appl. Math. 2011(298628), 51 (2011)MathSciNet
  17.Hu, G., Hu, Y.: Fractional diffusion in Gaussian noisy environment. Mathematics 3, 131–152 (2015)CrossRef
  18.Khoshnevisan, D.: Analysis of stochastic partial differential equations. CBMS Regional Conference Series in Mathematics, 119. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence (2014)
  19.Kochubei, A.N.: The Cauchy problem for evolution equations of fractional order. Differ. Equ. 25, 967–974 (1989)MathSciNet
  20.Mathai, A.M., Haubold, H.J.: Special Functions for Applied Scientists. Springer, Berlin (2007)
  21.Meerschaert, M.M., Nane, E., Vellaisamy, P.: Fractional Cauchy problems on bounded domains. Ann. Probab. 37, 979–1007 (2009)CrossRef MathSciNet MATH
  22.Meerschaert, M.M., Nane, E., Xiao, Y.: Fractal dimensions for continuous time random walk limits. Stat. Probab. Lett. 83, 1083–1093 (2013)CrossRef MathSciNet MATH
  23.Meerschaert, M.M., Scheffler, H.P.: Limit theorems for continuous time random walks with infinite mean waiting times. J. Appl. Probab. 41(3), 623–638 (2004)CrossRef MathSciNet MATH
  24.Meerschaert, M.M., Straka, P.: Inverse stable subordinators. Math. Model. Nat. Phenom. 8(2), 1–16 (2013)CrossRef MathSciNet MATH
  25.Mijena, J., Nane, E.: Space time fractional stochastic partial differential equations. To Appear. Stoch. Process Appl. (2015) doi:10.​1016/​j.​spa.​2015.​04.​008
  26.Nane, E.: Fractional Cauchy problems on bounded domains: survey of recent results. In: Baleanu, D. et al. (eds.) Fractional Dynamics and Control, pp 185–198. Springer, New York (2012)CrossRef
  27.Nigmatullin, R.R.: The realization of the generalized transfer in a medium with fractal geometry. Phys. Status Solidi B 133, 425–430 (1986)CrossRef
  28.Orsingher, E., Beghin, L.: Fractional diffusion equations and processes with randomly varying time. Ann. Probab. 37, 206–249 (2009)CrossRef MathSciNet MATH
  29.Simon, T.: Comparing Fréchet and positive stable laws. Electron. J. Probab. 19(16), 1–25 (2014)MathSciNet
  30.Walsh, J.B.: An Introduction to Stochastic Partial Differential Equations, École d’été de Probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, pp. 265–439. Springer, Berlin (1986)
  31.Wyss, W.: The fractional diffusion equations. J. Math. Phys. 27, 2782–2785 (1986)CrossRef MathSciNet MATH
  
• 作者单位：Jebessa B. Mijena (1)
  Erkan Nane (2)
  
  1. Department of Mathematics, Georgia College & State University, Milledgeville, GA, 31061, USA
  2. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Potential Theory
  Probability Theory and Stochastic Processes
  Geometry
  Functional Analysis
  
• 出版者：Springer Netherlands
• ISSN：1572-929X

文摘

We consider time fractional stochastic heat type equation $$\partial^{\beta}_{t}u_{t}(x)=-\nu(-{\Delta})^{\alpha/2} u_{t}(x)+I^{1-\beta}_{t}[\sigma(u)\overset{\cdot}{W}(t,x)] $$in (d + 1) dimensions, where ν > 0, β ∈ (0, 1), α ∈ (0, 2], \(d<\min \{2,\beta ^{-1}\}\alpha \), \(\partial ^{\beta }_{t}\) is the Caputo fractional derivative, −(−Δ) α/2 is the generator of an isotropic stable process, \(\overset {\cdot }{W}(t,x)\) is space-time white noise, and \(\sigma :\mathbb {R}\to \mathbb {R}\) is Lipschitz continuous. The time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove: (i) absolute moments of the solutions of this equation grows exponentially; and (ii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. These results extend the results of Foondun and Khoshnevisan (Electron. J. Probab. 14(21), 548–568, 2009) and Conus and Khoshnevisan (Probab. Theory Relat. Fields 152(3–4), 681–701, 2012) on the parabolic stochastic heat equations. Keywords Caputo fractional derivative Time fractional SPDE Intermittency Intermittency fronts Mathematics Subject Classification (2010) 60H15 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (31) References1.Arponen, H., Horvai, P.: Dynamo effect in the Kraichnan magnetohydrodynamic turbulence. J. Stat. Phys. 129(2), 205–239 (2007)CrossRefMathSciNetMATH2.Baeumer, B., Meerschaert, M.M.: Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4, 481–500 (2001)MathSciNetMATH3.Baxendale, P.H., Rozovskiĭ, B.L.: Kinematic dynamo and intermittence in a turbulent flow. Geophys. Astrophys. Fluid Dyn. 73(1–4), 33–60 (1993). Magnetohydrodynamic stability and dynamos (Chicago, IL, 1992)CrossRefMathSciNet4.Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)MATH5.Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, Part II. Geophys. J. R. Astr. Soc. 13, 529–539 (1967)CrossRef6.Carmona, R.A., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Mem. Am. Math. Soc. 108(518), viii+125 (1994)MathSciNet7.Chen, L., Dalang, R.C.: The nonlinear stochastic heat equation with rough initial data: a summary of some new results, arXiv:1210.​1690v1.​pdf (2012)8.Chen, Z.-Q., Kim, K.-H., Kim, P.: Fractional time stochastic partial differential equations. Stoch. Process. Appl. 125, 1470–1499 (2015)CrossRefMathSciNetMATH9.Conus, D., Khoshnevisan, D.: On the existence and position of the farthest peaks of a family of stochastic heat and wave equations. Probab. Theory Relat. Fields 152(3–4), 681–701 (2012)CrossRefMathSciNetMATH10.Dalang, R.C., Quer-Sardanyons, L.: Stochastic integrals for spde’s: a comparison. Expo. Math. 29(1), 67–109 (2011)CrossRefMathSciNetMATH11.Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)CrossRef12.Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. Wiley, New York-London-Sydney (1971)MATH13.Foondun, M., Khoshnevisan, D.: Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14(21), 548–568 (2009)MathSciNetMATH14.Galloway, D.: Fast Dynamos, Advances in Nonlinear Dynamos. Fluid Mech. Astrophys. Geophys., Vol. 9, pp 37–59. Taylor & Francis (2003)15.Georgiou, N., Joseph, M., Khoshnevisan, D., Mahboubi, P., Shiu, S.-Y: Semi-discrete semi-linear parabolic SPDEs (2013)16.Haubold, H.J., Mathai, A.M., Saxena, R.K.: Review article: Mittag-Leffler functions and their applications. J. Appl. Math. 2011(298628), 51 (2011)MathSciNet17.Hu, G., Hu, Y.: Fractional diffusion in Gaussian noisy environment. Mathematics 3, 131–152 (2015)CrossRef18.Khoshnevisan, D.: Analysis of stochastic partial differential equations. CBMS Regional Conference Series in Mathematics, 119. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence (2014)19.Kochubei, A.N.: The Cauchy problem for evolution equations of fractional order. Differ. Equ. 25, 967–974 (1989)MathSciNet20.Mathai, A.M., Haubold, H.J.: Special Functions for Applied Scientists. Springer, Berlin (2007)21.Meerschaert, M.M., Nane, E., Vellaisamy, P.: Fractional Cauchy problems on bounded domains. Ann. Probab. 37, 979–1007 (2009)CrossRefMathSciNetMATH22.Meerschaert, M.M., Nane, E., Xiao, Y.: Fractal dimensions for continuous time random walk limits. Stat. Probab. Lett. 83, 1083–1093 (2013)CrossRefMathSciNetMATH23.Meerschaert, M.M., Scheffler, H.P.: Limit theorems for continuous time random walks with infinite mean waiting times. J. Appl. Probab. 41(3), 623–638 (2004)CrossRefMathSciNetMATH24.Meerschaert, M.M., Straka, P.: Inverse stable subordinators. Math. Model. Nat. Phenom. 8(2), 1–16 (2013)CrossRefMathSciNetMATH25.Mijena, J., Nane, E.: Space time fractional stochastic partial differential equations. To Appear. Stoch. Process Appl. (2015) doi:10.​1016/​j.​spa.​2015.​04.​008 26.Nane, E.: Fractional Cauchy problems on bounded domains: survey of recent results. In: Baleanu, D. et al. (eds.) Fractional Dynamics and Control, pp 185–198. Springer, New York (2012)CrossRef27.Nigmatullin, R.R.: The realization of the generalized transfer in a medium with fractal geometry. Phys. Status Solidi B 133, 425–430 (1986)CrossRef28.Orsingher, E., Beghin, L.: Fractional diffusion equations and processes with randomly varying time. Ann. Probab. 37, 206–249 (2009)CrossRefMathSciNetMATH29.Simon, T.: Comparing Fréchet and positive stable laws. Electron. J. Probab. 19(16), 1–25 (2014)MathSciNet30.Walsh, J.B.: An Introduction to Stochastic Partial Differential Equations, École d’été de Probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, pp. 265–439. Springer, Berlin (1986)31.Wyss, W.: The fractional diffusion equations. J. Math. Phys. 27, 2782–2785 (1986)CrossRefMathSciNetMATH About this Article Title Intermittence and Space-Time Fractional Stochastic Partial Differential Equations Journal Potential Analysis Volume 44, Issue 2 , pp 295-312 Cover Date2016-02 DOI 10.1007/s11118-015-9512-3 Print ISSN 0926-2601 Online ISSN 1572-929X Publisher Springer Netherlands Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Potential Theory Probability Theory and Stochastic Processes Geometry Functional Analysis Keywords Caputo fractional derivative Time fractional SPDE Intermittency Intermittency fronts 60H15 Authors Jebessa B. Mijena (1) Erkan Nane (2) Author Affiliations 1. Department of Mathematics, Georgia College & State University, Milledgeville, GA, 31061, USA 2. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA Continue reading... To view the rest of this content please follow the download PDF link above.