A Singular Parabolic Anderson Model

Carl E Mueller (University of Rochester)
Roger Tribe (University of Warwick)

Abstract


We consider the heat equation with a singular random potential term. The potential is Gaussian with mean 0 and covariance given by a small constant times the inverse square of the distance. Solutions exist as singular measures, under suitable assumptions on the initial conditions and for sufficiently small noise. We investigate various properties of the solutions using such tools as scaling, self-duality and moment formulae. This model lies on the boundary between nonexistence and smooth solutions. It gives a new model, other than the superprocess, which has measure-valued solutions.

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Pages: 98-144

Publication Date: February 25, 2004

DOI: 10.1214/EJP.v9-189

References

  1. Albeverio, S. and Rockner, M. (1989), Dirichlet forms, quantum fields and stochastic quantization, in Stochastic analysis, path integration and dynamics (Warwick, 1987), Pitman Res. Notes Math. Ser., 200, pages 1--21, Harlow, Longman Sci. Tech. MR 90j:81097
  2. Bentley, P.W. (1999), Regularity and inverse SDE representations of some stochastic PDEs, PhD Thesis at the University of Warwick.
  3. Carmona, R.A. and Molchanov, S.A. (1994). Parabolic Anderson problem and intermittency, AMS Memoir 518, Amer. Math. Soc. MR 94h:35080
  4. Cox, J.T., Fleischmann, K., and Greven, A. (1996), Comparison of interacting diffusions and an application to their ergodic theory, Prob. Th. Rel. Fields 105, 513-528. MR 97h:60073
  5. Cox, J.T., Klenke A., and Perkins E.A. (2000), Convergence to equilibrium and linear systems duality, in Stochastic models (Ottawa, ON, 1998), pages 41-66 CMS Conf. Proc., 26, Amer. Math. Soc., Providence, RI.
  6. Dawson, D.A. (1993), Measure-valued Markov processes. in Ecole d'et'e de probabilit'es de Saint-Flour, XXI-1991, Springer Lecture Notes in Mathematics} 1180, 1-260. MR 94m:60101
  7. Dawson, D.A. and Salehi, H. (1980), Spatially homogeneous random evolutions, Journal of Multivariate Analysis 10, 141-180. MR 82c:60102
  8. Da Prato, G. and Zabczyk, J. (1992), Stochastic Equations in Infinite Dimensions, Vol. 44 of Encyclopedia of mathematics and its applications, Cambridge University Press. MR 95g:60073
  9. Ethier, S. and Kurtz, T. (1986), Markov Processes, Characterization and Convergence, Wiley. MR 88a:60130
  10. Falconer, K.J. (1985), The Geometry of Fractal Sets, Vol. 85 of Tracts in mathematics, Cambridge University Press. MR 88d:28001
  11. Gartner J., Konig, W., and Molchanov S.A. (2000). Almost sure asymptotics for the continuous parabolic Anderson model, Prob. Th. Rel. Fields, 118, 547-573. MR 2002i:60121
  12. Holden, Helge; Oksendal, Bernt; Uboe, Jan, and Zhang, Tusheng, Stochastic partial differential equations, a modeling, white noise functional approach., Probability and its Applications, Birkhauser Boston Inc., Boston, MA. MR 98f:60124
  13. Ito, K. (1984), Foundations of stochastic differential equations in infinite dimensional spaces, Vol. 47 of CBMS-NSF Regional Conference Series in Applied Mathematics. MR 87a:60068
  14. Kunita, H. (1990), Stochastic flows and stochastic differential equations, Vol. 24 of Cambridge studies in advanced mathematics, Cambridge University Press. MR 91m:60107
  15. Liggett, T.M. (1985), Interacting particle systems, Springer-Verlag. MR 86e:60089
  16. Nualart David; and Rozovskii, Boris (1997) Weighted stochastic Sobolev spaces and bilinear SPDEs driven by space-time white noise. J. Funct. Anal., 149(1), 1997. MR 98m:60100
  17. Nualart, D. and Zakai M. (1989), Generalized Brownian functionals and the solution to a stochastic partial differential equation, J. Funct. Anal., 84, 279-296. MR 90m:60076
  18. Revuz, D. and Yor M. (1991), Continuous Martingales and Brownian Motion, Springer-Verlag. MR 2000h:60050
  19. Rogers, L.C.G. and Williams, D. (2000), Diffusions, Markov processes and martingales Vol. 2, Ito calculus, 2nd edition, Cambridge University Press. MR 2001g:60189
  20. Walsh, J.B. (1986), An introduction to stochastic partial differential equations, Ecole d'et'e de probabilit'es de Saint-Flour, XIV-1984, Springer Lecture Notes in Mathematics 1180, 265-439. MR 88a:60114
  21. Yor, M. (1980), Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson, Prob. Th. Rel. Fields, 53(1), 71-95. MR 82a:60120


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