A General Analytical Result for Non-linear SPDE's and Applications

Laurent Denis (Université du Maine, France)
L. Stoica (University of Bucharest, Romania)

Abstract


Using analytical methods, we prove existence uniqueness and estimates for s.p.d.e. of the type $$ du_t+Au_tdt+f ( t,u_t ) dt+R g(t, u_t ) dt=h(t,x,u_t) dB_t, $$ where $A$ is a linear non-negative self-adjoint (unbounded) operator, $f$ is a nonlinear function which depends on $u$ and its derivatives controlled by $\sqrt{A}u$, $Rg$ corresponds to a nonlinearity involving $u$ and its derivatives of the same order as $Au$ but of smaller magnitude, and the right term contains a noise involving a $d$-dimensional Brownian motion multiplied by a non-linear function. We give a neat condition concerning the magnitude of these nonlinear perturbations. We also mention a few examples and, in the case of a diffusion generator, we give a double stochastic interpretation.

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Pages: 674-709

Publication Date: October 11, 2004

DOI: 10.1214/EJP.v9-223

References

  • Bally, V.; Matoussi, A. Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theoret. Probab. 14 (2001), no. 1, 125--164. MR1822898
  • Bally, V.; Pardoux, E.; Stoica, L. Backward stochastic differential equations associated to a symmetric Markov process. Potential Anal. 22 (2005), no. 1, 17--60. MR2127730
  • Bouleau N., Hirsch F., textit''Dirichlet forms and analysis on Wiener space'', Kluwer, 1993.
  • Cardon-Weber, Caroline. Cahn-Hilliard stochastic equation: existence of the solution and of its density. Bernoulli 7 (2001), no. 5, 777--816. MR1867082
  • Chojnowska-Michalik, Anna. Stochastic differential equations in Hilbert spaces. Probability theory (Papers, VIIth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1976), pp. 53--74, Banach Center Publ., 5, PWN, Warsaw, 1979. MR0561468
  • Da Prato G., ''textitStochastic evolution equations by semigroups methods'', Centre de Recerca Matematica, Quaderns, num. 11/gener 1998.
  • Da Prato, Giuseppe; Zabczyk, Jerzy. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. xviii+454 pp. ISBN: 0-521-38529-6 MR1207136
  • Davies, Edward Brian. One-parameter semigroups. London Mathematical Society Monographs, 15. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. viii+230 pp. ISBN: 0-12-206280-9 MR0591851
  • Fukushima, Masatoshi; Ōshima, Yōichi; Takeda, Masayoshi. Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. x+392 pp. ISBN: 3-11-011626-X MR1303354
  • Gyöngy, István; Rovira, Carles. On $L^ p$-solutions of semilinear stochastic partial differential equations. Stochastic Process. Appl. 90 (2000), no. 1, 83--108. MR1787126
  • Krylov, N. V. An analytic approach to SPDEs. Stochastic partial differential equations: six perspectives, 185--242, Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI, 1999. MR1661766
  • Lions, J.-L. Équations différentielles opérationnelles et problèmes aux limites. (French) Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg 1961 ix+292 pp. MR0153974
  • Mikulevicius R., Rozovskii B.L. textit''A note on Krylov's Lp-theory for systems of SPDEs'' Electronic Journal of Probability, % 6, (2002) paper no.12, pp. 1-35.
  • Pardoux, E. Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3, no. 2, 127--167. (1979), MR0553909
  • Pardoux, Étienne; Peng, Shi Ge. Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Related Fields 98 (1994), no. 2, 209--227. MR1258986
  • Rozovskiĭ, B. L. Stochastic evolution systems. Linear theory and applications to nonlinear filtering. Translated from the Russian by A. Yarkho. Mathematics and its Applications (Soviet Series), 35. Kluwer Academic Publishers Group, Dordrecht, 1990. xviii+315 pp. ISBN: 0-7923-0037-8 MR1135324
  • Saussereau B. ''textitSur un Classe d'Equations aux Dériv% ées Partielles'', thèse de Doctorat de l'Université du Maine, 2001.
  • Stoica, I. L. A probabilistic interpretation of the divergence and BSDE's. Stochastic Process. Appl. 103 (2003), no. 1, 31--55. MR1947959
  • Walsh, John B. An introduction to stochastic partial differential equations. École d'été de probabilités de Saint-Flour, XIV—1984, 265--439, Lecture Notes in Math., 1180, Springer, Berlin, 1986. MR0876085


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