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 Electronic Journal of Probability > Vol. 8 (2003) > Paper 6 open journal systems 


On Convergence of Population Processes in Random Environments

Anja Sturm, Technische Universität Berlin


Abstract
We consider the stochastic heat equation with a multiplicative colored noise term on the real space for dimensions greater or equal to 1. First, we prove convergence of a branching particle system in a random environment to this stochastic heat equation with linear noise coefficients. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems, which are infinite dimensional stochastic differential equations with correlated noise terms, provided that uniqueness of the limit is known. In the course of the proof, we establish existence and uniqueness of solutions to the lattice systems, as well as a new existence result for solutions to the stochastic heat equation. The latter are shown to be jointly continuous in time and space under some mild additional assumptions.


Full text: PDF

Pages: 1-39

Published on: April 11, 2003


Bibliography
  1. Blount, D. (1996), Diffusion limits for a nonlinear density dependent space-time population model. Annals of Probability 24(2) ,639-659. Math. Review 98c:60138
  2. Brzezniak, Z. and Peszat, S. (1999), Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process. Studia Mathematica, 137 , 261-299. Math. Review 2000m:60077
  3. Burkholder, D.L. (1973), Distribution function inequalities for martingales. Annals of Probability, 1 , 19-42. Math. Review 51:1944
  4. Carmona, R.A. and Molchanov, S.A. (1994), Parabolic Anderson problem and intermittency. Mem. Americ. Math. Soc. 108 (518) viii+125 pp. Math. Review 94h:35080
  5. Dalang, R.C. (1999), Extending martingale measure stochastic integrals with applications to spatially homogeneous SPDEs. Electronic Journal of Probability, 4 1-29. Math. Review 2000b:60132
  6. Dawson, D. (1975), Stochastic evolution equations and related measure processes. J. Multivariate Analysis, 5 1-52. Math. Review 52:9375
  7. Dawson, D. (1990), Infinitely divisible random measures and superprocesses. Stochastic Analysis and Related Topics (Silivri 1990), Progr. Probab. 31 , Birkh"auser 1-129, Math. Review 52:9375
  8. Dawson, D. (1991), Measure-Valued Markov Processes. Lecture Notes in Mathematics 1541 , Springer. Math. Review 94m:60101
  9. Dawson, D., Etheridge, A.M., Fleischmann, K., Mytnik, L., Perkins, E. and Xiong, J. (2002), Mutually catalytic branching in the plane: Infinite measure states. Electronic Journal of Probability, 7(15) 1-61, Math. Review 1921744
  10. Dawson, D. and Perkins, E. (1999), Measure valued processes and renormalization of branching particle systems. Mathematical Surveys and Monographs 64 , 45-106. American Mathematical Society. Math. Review 2000a:60098
  11. Dawson, D. and Salehi, H. (1980), Spatially homogeneous random evolutions. Journal of Multivariate Analysis, 10 , 141-180. Math. Review 82c:60102
  12. Da Prato, G., Kwapien, S. and Zabczyk, J. (1987), Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics, 23 , 1-23. Math. Review 89b:60148
  13. Da Prato, G. and Zabczyk, J. (1992), Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its applications, Cambridge University Press. Math. Review 95g:60073
  14. Dunford, N. and Schwartz, J.T. (1958), Linear Operators I General Theory. Interscience Publishers. Math. Review 22:8302
  15. Ethier, S.N. and Kurtz, T.G. (1986), Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics, Wiley. Math. Review 88a:60130
  16. Etheridge, A.M. (2000), An Introduction to Superprocesses. University Lecture series 20 , American Mathematical Society. Math. Review 2001m:60111
  17. Feller, W. (1951), Diffusion processes in genetics. In Proc. Second Berkeley Symp. Math. Statist. Prob., 227-246 Berkeley. University of California Press. Math. Review 13,671c
  18. Funaki, T. (1983), Random motion of strings and related stochastic evolution equations. Nagoya Math. J., 89 , 129-193. Math. Review 85g:60063
  19. Gyöngy, I. (1998), Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise I. Potential Analysis, 9 , 1-25. Math. Review 99j:60091
  20. Gyöngy, I. (1998), Existence and uniqueness results for semilinear stochastic partial differential equations. Stochastic Processes and their Applications, 73 , 271-299. Math. Review 99b:60091
  21. Jacod, J. and Shiryaev, A.N. (1987), Limit Theorems for Stochastic Processes. Fundamental Principles of Mathematical Sciences, 288 , Springer. Math. Review 89k:60044
  22. Kotelenez, P. (1986), Law of large numbers and central limit theorems for linear chemical reactions with diffusion. Annals of Probability, 14(1) , 173-193. Math. Review 87b:60052
  23. Kotelenez, P. (1992), Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations. Stochastics and Stochastics Reports, 41(3) , 177-199. Math. Review 95c:60054
  24. Krylov, N.V. (1996), On Lp - theory of stochastic partial differential equations in the whole space. Siam J. Math. Anal. 27, 313-340. Math. Review 97b:60107
  25. Konno, N. and Shiga, T. (1988), Stochastic partial differential equations for some measure-valued diffusions. Probability Theory and Related Fields, 79 , 201-225. Math. Review 89i:60130
  26. Kallianpur, G. and Sundar, P. (2000), Hilbert-space-valued Super-Brownian motion and related evolution equations. Applied Mathematics and Optimization, 41 , 141-154. Math. Review 2001b:60079
  27. Manthey, R. and Mittmann, K. (1999), On the qualitative behaviour of the solution to a stochastic partial functional-differential equation arising in population dynamics. Stochastics and stochastics reports, 66 , 153-166. Math. Review 2000c:60093
  28. Mueller, C. and Perkins, E. (1992), The compact support property for solutions to the heat equation with noise. Probability Theory and Related Fields, 93(3) , 325-358. Math. Review 93k:60156
  29. Mytnik, L. (1996), Superprocesses in random environments. Annals of Probability, 24(4) , 1953-1978. Math. Review 97h:60046
  30. Noble, J.M. (1997), Evolution equation with gaussian potential. Nonlinear Analysis, 28(1) , 103-135. Math. Review 97j:60120
  31. Pardoux, E. (1993), Stochastic Partial Differential Equations, a review. Bulletin des Sciences Mathematiques, 2e serie, 117 , 29-47. Math. Review 94i:60071
  32. Perkins, E. (2002), Dawson-Watanabe Superprocesses and Measure-valued Diffusions. Lecture Notes in Mathematics 171 , Springer. Math. Review 1915445
  33. Peszat, S. and Zabczyk, J. (1997), Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Processes and their Applications, 72 , 187-204. Math. Review 99k:60166
  34. Peszat, S. and Zabczyk, J. (2000), Nonlinear stochastic wave and heat equations. Probability Theory and Related Fields, 116 , 421-443. Math. Review 2001f:60071
  35. Roelly-Coppoletta, S. (1986), A criterion of convergence of measure-valued processes: Application to measure branching processes. Stochastics, 17 , 43-65. Math. Review 88i:60132
  36. Reimers, M. (1989), One dimensional stochastic partial differential equations and the branching measure diffusion. Probability Theory and Related Fields, 81 , 319-340. Math. Review 90m:60077
  37. Shiga, T. (1994), Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canadian Journal of Mathematics, 46 , 415-437. Math. Review 95h:60099
  38. Shiga, T. and Shimizu, A. (1980), Infinite dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ., ,20(3) , 395-416. Math. Review 82i:60110
  39. Sanz-Solé, M. and Sarrà, M. (2002), Hoelder continuity for the stochastic heat equation with spatially correlated noise. Progress in Probability, Stochastic Analysis, Random Fields and Applications, Birkhaeuser Basel. Math. Review number not available.
  40. Sturm, A. (2002), On spatially structured population processes and relations to stochastic partical differential equations. PhD thesis, University of Oxford. Math. Review number not available.
  41. Tindel, S. and Viens, F. (1999), On space-time regularity for the stochastic heat equation on lie groups. Journal of Functional Analysis, 169(2) , 559-603. Math. Review 2001f:58076
  42. Viot, M. (1976), Solutions faibles d'équations aux derivées partielles stochastique non linéaires. PhD thesis, Universite Pierre et Marie Curie-Paris VI. Math. Review number not available.
  43. Walsh, J.B. (1986), An introduction to stochastic partial differential equations. Lecture Notes in Mathematics, 1180 , Springer. Math. Review 88a:60114
  44. Watanabe, S. (1968), A limit theorem of branching processes and continuous state branching. J. Math. Kyoto University, 8 , 141-167. Math. Review 38:5301
  45. Yamada, T. and Watanabe, S. (1971), On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ., 11 , 155-167. Math. Review 43:4150 Y
















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Electronic Journal of Probability. ISSN: 1083-6489