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AIMS Mathematics

2017, Volume 2, Issue 2: 348-364. doi: 10.3934/Math.2017.2.348
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Research article

Large Deviations for Stochastic Fractional Integrodifferential Equations

  • Department of Mathematics, Bharathiar University, Coimbatore 641046, India
  • Received: 20 January 2017 Accepted: 25 May 2017 Published: 12 June 2017

  • MSC : 34A08, 45J05, 60F10, 60H10

  • In this work we establish a Freidlin-Wentzell type large deviation principle for stochastic fractional integrodifferential equations by using the weak convergence approach. The compactness argument is proved on the solution space of corresponding skeleton equation and the weak convergence is done for Borel measurable functions whose existence is asserted from Yamada-Watanabe theorem. Examples are included which illustrate the theory and also depict the link between large deviations and optimal controllability.

    Citation: Murugan Suvinthra, Krishnan Balachandran, Rajendran Mabel Lizzy. Large Deviations for Stochastic Fractional Integrodifferential Equations[J]. AIMS Mathematics, 2017, 2(2): 348-364. doi: 10.3934/Math.2017.2.348

    Related Papers:

  • In this work we establish a Freidlin-Wentzell type large deviation principle for stochastic fractional integrodifferential equations by using the weak convergence approach. The compactness argument is proved on the solution space of corresponding skeleton equation and the weak convergence is done for Borel measurable functions whose existence is asserted from Yamada-Watanabe theorem. Examples are included which illustrate the theory and also depict the link between large deviations and optimal controllability.


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    [1] K. Balachandran, S. Divya, M. Rivero and J.J. Trujillo, Controllability of nonlinear implicit neutral fractional Volterra integrodifferential systems, Journal of Vibration and Control, 22 (2016), 2165-2172.
    [2] K. Balachandran, V. Govindaraj, L. Rodrguez-Germa and J.J. Trujillo, Controllability results for nonlinear fractional-order dynamical systems, Journal of Optimization Theory and Applications, 156 (2013), 33-44.
    [3] K. Balachandran, M. Matar and J. J. Trujillo, Note on controllability of linear fractional dynamical systems, Journal of Control and Decision, 3 (2016), 267-279.
    [4] L. Bo and Y. Jiang, Large deviation for the nonlocal Kuramoto-Sivashinsky SPDE, Nonlinear Analysis: Theory, Methods and Applications, 82 (2013), 100-114.
    [5] M. Boue and P. Dupuis, A variational representation for certain functionals of Brownian motion, Annals of Probability, 26(1998), 1641-1659.
    [6] A. Budhiraja P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Annals of Probability, 36 (2008), 1390-1420.
    [7] A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probability of Mathematics and Statistics, 20 (2000), 39-61.
    [8] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, New York: Springer, 2007.
    [9] A. Di Crescenzo and A. Meoli, On a fractional alternating Poisson process, AIMS Mathematics, 1(2016), 212-224.
    [10] N. Dunford and J. Schwartz, Linear Operators, Part Ⅰ, New York: Wiley-Interscience, 1958.
    [11] P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, New York: Wiley-Interscience, 1997.
    [12] W.H. Fleming, A stochastic control approach to some large deviations problems, Recent Mathematical Methods in Dynamic Programming, Springer Lecture Notes in Math., 1119 (1985), 52-66.
    [13] M.I. Freidlin and A.D. Wentzell, On small random perturbations of dynamical systems, Russian Mathematical Surveys, 25 (1970), 1-55.
    [14] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, New York: Springer, 1984.
    [15] R. Joice Nirmala and K. Balachandran, Controllability of nonlinear fractional delay integrodifferential system, Discontinuity, Nonlinearity, and Complexity, 5 (2016), 59-73.
    [16] M. Kamrani, Numerical solution of stochastic fractional differential equations, Numerical Algorithms, 68 (2015), 81-93.
    [17] E. Kreyszig, Introductory Functional Analysis with Applications, New York: JohnWiley and Sons Inc, 1978.
    [18] W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise, Applied Mathematics and Optimization, 61 (2010), 27-56.
    [19] A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier, 2006.
    [20] R. Mabel Lizzy, K. Balachandran and M. Suvinthra, Controllability of nonlinear stochastic fractional systems with distributed delays in control, Journal of Control and Decision, 4 (2017), 153-167.
    [21] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: John-Wiley, 1993.
    [22] C. Mo and J. Luo, Large deviations for stochastic differential delay equations, Nonlinear Analysis: Theory, Methods and Applications, 80 (2013), 202-210.
    [23] S.A. Mohammed and T.S. Zhang, Large deviations for stochastic systems with memory, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 881-893.
    [24] J. C. Pedjeu and G. S. Ladde, Stochastic fractional differential equations: Modelling, method and analysis, Chaos, Solitons and Fractals, 45 (2012), 279-293.
    [25] I. Podlubny, Fractional Differential Equations, London: Academic Press, 1999.
    [26] J. Ren and X. Zhang, Schilder theorem for the Brownian motion on the diffeomorphism group of the circle, Journal of Functional Analysis, 224 (2005), 107-133.
    [27] J. Ren and X. Zhang, Freidlin-Wentzell large deviations for homeomorphism flows of non-Lipschitz SDE, Bulletin of Science, 129 (2005), 643-655.
    [28] S.S. Sritharan and P. Sundar, Large deviations for two dimensional Navier-Stokes equations with multiplicative noise, Stochastic Processes and their Applications, 116 (2006), 1636-1659.
    [29] M. Suvinthra and K. Balachandran, Large deviations for nonlinear Itôtype stochastic integrodifferential equations, Journal of Applied Nonlinear Dynamics, 6 (2017), 1-15.
    [30] M. Suvinthra, K. Balachandran and J.K. Kim, Large deviations for stochastic differential equations with deviating arguments, Nonlinear Functional Analysis and Applications, 20 (2015), 659-674.
    [31] S.R.S. Varadhan, Asymptotic probabilities and differential equations, Communications on Pure and Applied Mathematics, 19 (1966), 261-286.
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