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Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients

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Abstract

In this paper we develop a new weak convergence and compact embedding method to study the existence and uniqueness of the \(L_{\rho}^{2p}({\mathbb{R}^{d}};{\mathbb{R}^{1}})\times L_{\rho}^{2}({\mathbb{R}^{d}};{\mathbb{R}^{d}})\) valued solution of backward stochastic differential equations with p-growth coefficients. Then we establish the probabilistic representation of the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.

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References

  1. Bally, V., Matoussi, A.: Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theor. Probab. 14, 125–164 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barles, G., Lesigne, E.: SDE, BSDE and PDE. In: Backward Stochastic Differential Equations. Pitman Res. Notes Math., vol. 364, pp. 47–80. Longman, Harlow (1997)

    Google Scholar 

  3. Briand, Ph., Hu, Y.: BSDE with quadratic growth and unbounded terminal value. Probab. Theory Relat. 136, 604–618 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dalang, R.C., Mueller, C., Tribe, R.: A Feynman–Kac-type formula for the deterministic and stochastic wave equations and other P.D.E.’s. Trans. Am. Math. Soc. 360, 4681–4703 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Darling, R.W.R.: Constructing Gamma-martingales with prescribed limits, using backward SDE. Ann. Probab. 23, 1234–1261 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time, I and II. Commun. Pure Appl. Math. 28, 1–47 (1975) 279–301

    Article  MathSciNet  MATH  Google Scholar 

  7. Elworthy, K.D., Truman, A., Zhao, H.Z., Gaines, J.G.: Approximate travelling waves for generalized KPP equations and classical mechanics. Proc. R. Soc. Lond., Sect. A 446, 529–554 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948)

    Article  MathSciNet  Google Scholar 

  9. Freidlin, M.I.: Functional Integration and Partial Differential Equations. Annals of Mathematics Studies, vol. 109. Princeton University Press, Princeton (1985)

    MATH  Google Scholar 

  10. Elworthy, K.D.: Geometric aspects of diffusions on manifolds. In: Ecole d’Eté de Probabilités de Saint Flour, XVII. Lecture Notes in Math., vol. 1362, pp. 276–425. Springer, Berlin (1987)

    Google Scholar 

  11. El-Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance 7, 1–72 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kac, M.: On distributions of certain Wiener functionals. Trans. Am. Math. Soc. 65, 1–13 (1949)

    Article  MATH  Google Scholar 

  13. Kobylanski, M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28, 558–602 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kunita, H.: Stochastic flow acting on Schwartz distributions. J. Theor. Probab. 7, 247–278 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lepeltier, J.P., San Martin, J.: Backward stochastic differential equations with continuous coefficient. Stat. Probab. Lett. 32, 425–430 (1997)

    Article  MATH  Google Scholar 

  16. Malliavin, P., Stroock, D.W.: Short time behaviour of the heat kernel and its logarithmic derivatives. J. Differ. Geom. 44, 550–570 (1996)

    MathSciNet  MATH  Google Scholar 

  17. Matoussi, A., Xu, M.: Sobolev solution for semi-linear PDE with obstacle under monotonicity condition. Electron. J. Probab. 13, 1035–1067 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55–61 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pardoux, E.: BSDE’s weak convergence and homogenization of semilinear PDE’s. In: Clarke, F., Stern, R. (eds.) Nonlin. Analy., Diff. Equa. and Control, pp. 503–549. Kluwer Academic, Dordrecht (1999)

    Google Scholar 

  20. Pardoux, E., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Rozuvskii, B.L., Sowers, R.B. (eds.) Stochastic Partial Differential Equations. Lect. Notes Control Inf. Sci., vol. 176, 200–217. Springer, Berlin (1992)

    Google Scholar 

  21. Robinson, J.C.: Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  22. Simon, B.: Functional Integration and Quantum Physics, 2nd edn. AMS, Chelsea (2005)

    MATH  Google Scholar 

  23. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1988)

    Book  MATH  Google Scholar 

  24. Wentzell, A.D., Freidlin, M.I.: On small random perturbations of dynamical system. Russ. Math. Surv. 25, 1–55 (1970)

    Google Scholar 

  25. Zhang, Q., Zhao, H.Z.: Stationary solutions of SPDEs and infinite horizon BDSDEs. J. Funct. Anal. 252, 171–219 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, Q., Zhao, H.Z.: Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients. J. Differ. Equ. 248, 953–991 (2010)

    Article  MATH  Google Scholar 

  27. Zhang, Q., Zhao, H.Z.: SPDEs with Polynomial Growth Coefficients: Weak Solutions via BDSDEs and Stationary Solutions. Preprint

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Qi Zhang

  2. Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK

    Huaizhong Zhao

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  1. Qi Zhang
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  2. Huaizhong Zhao
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Correspondence to Huaizhong Zhao.

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Zhang, Q., Zhao, H. Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients. J Theor Probab 25, 396–423 (2012). https://doi.org/10.1007/s10959-011-0350-y

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  • DOI: https://doi.org/10.1007/s10959-011-0350-y

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