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Diffusion limits for a nonlinear density dependent space-time population model. (English) Zbl 0868.60078

The author considers a reaction-diffusion model and constructs a population density process by using approximately \(Nl\) particles performing rate \(N^2\) random walks between \(N\) cells distributed on the unit interval. Within a cell each particle gives birth at rate \(b_1+ \gamma Nl/2\) and dies at rate \(d_2n_k/l+ d_1+\gamma Nl/2\), where \(n_k\) \((1\leq k\leq N)\) is the number of particles in the occupied cell and \(b_1\), \(d_1\), \(d_2\), \(\gamma\) are fixed nonnegative parameters. Particles also immigrate into the system according to a rate \(b_0 Nl\) Poisson process and are then uniformly distributed among the \(N\) cells. For the case \(\gamma=0\), it is well-known that with suitable scaling, deterministic limits satisfy the PDE \[ {\partial\psi\over\partial t}= \Delta\psi(t)- d_2\psi^2(t)+ c\psi(t)+ b_0,\tag{1} \] where \(c=b_1-d_1-\varepsilon_0 d_2l^{-1}\) with \(\varepsilon_0=0\) or 1. Let \(W\) denote a cylindrical Brownian motion on \(L_2([0,1])\). For the case of particle interaction where \(d_2>0\) and \(\gamma>0\), the limit by passage to \(l\to\infty\) as \(N\to\infty\) leads to the SPDE \[ d\psi(t)= \{\Delta\psi- d_2\psi^2+ (b_1-d_1)\psi+ b_0\}dt+ \sqrt{\gamma\psi(t)}dW(t),\tag{2} \] and the other limit with \(l\) being held constant as \(N\to\infty\) leads to the SPDE \[ d\psi(t)= \{\Delta\psi- d_2\psi^2+ (b_1- d_1-d_2l^{-1})\psi+b_0\}dt+ \sqrt{\gamma\psi}dW(t).\tag{3} \] Both of them are nonlinear perturbations of the equation (1) satisfied by the density process of super Brownian motion. Convergence in distribution holds in \(D([0,T];L_2([0,1]))\) with the Skorokhod topology if \(l\to\infty\) as \(N\to\infty\), and convergence holds in \(D([0,T];H_{-\alpha})\) for any \(\alpha>0\) if \(l\) is constant as \(N\to\infty\). Holding \(l\) constant requires that the cells be averaged together before obtaining the diffusion process limit. If \(l\to\infty\) as \(N\to\infty\), then a diffusion limit occurs in each cell. In this case, one obtains the same limit as first letting \(l\to\infty\), giving a system of \(N\) coupled diffusion processes, and then letting \(N\to\infty\) to obtain a limit satisfying an SPDE. The extra linear term \(d_2l^{-1}\psi\) in (3) can be seen from the case of the deterministic limit (1) when \(\gamma=0\) and \(l\) is constant. In this case, the random walk jumps occur so much faster than reaction jumps that one may expect that cell numbers at a fixed time \(t>0\) will be approximately distributed as independent Poisson random variables. The proof is technically based on the result done by the author [ibid. 22, No. 4, 2040-2070 (1994; Zbl 0843.60057)]. An analogue of the SPDE (2) has been obtained by C. Mueller and R. Tribe [Probab. Theory Relat. Fields 102, No. 4, 519-545 (1995; Zbl 0827.60050)].
Reviewer: I.Dôku (Urawa)

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60B10 Convergence of probability measures
60J60 Diffusion processes
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[1] ARNOLD, L. and THEODOSOPULU, M. 1980. Deterministic limit of the stochastic model of chemical reactions with diffusion. Adv. in Appl. Probab. 12 367 379. Z. JSTOR: · Zbl 0429.60025
[2] BLOUNT, D. 1991. Comparison of stochastic and deterministic models of a linear chemical reaction with diffusion. Ann. Probab. 19 1440 1462. Z. · Zbl 0741.92022
[3] BLOUNT, D. 1992. Law of large numbers in the supremum norm for a chemical reaction with diffusion. Ann. Appl. Probab. 2 131 141. Z. · Zbl 0747.60033
[4] BLOUNT, D. 1993. Limit theorems for a sequence of reaction diffusion sy stems. Stochastic Process. Appl. 45 193 207. Z. · Zbl 0773.60052
[5] BLOUNT, D. 1994. Density dependent limits for a nonlinear reaction diffusion model. Ann. Probab. 22 2040 2070. Z. · Zbl 0843.60057
[6] BOLDRIGHINI, C., DE MASI, A. and PELLEGRINOTTI, A. 1992. Nonequilibrium fluctuations in particle sy stems modeling reaction diffusion equations. Stochastic Process. Appl. 42 1 30. Z. · Zbl 0758.60107
[7] BURKHOLDER, D. 1973. Distribution function inequalities for martingales. Ann. Probab. 1 19 42. Z. · Zbl 0301.60035
[8] DAWSON, D. 1993. Measure valued Markov processes. Ecole d’Ete de Probabilites de Saint-Flour \' \' XX. Lecture Notes in Math. 1541 1 260. Springer, Berlin. Z. · Zbl 0799.60080
[9] ETHIER, S. N. and KURTZ, T. G. 1986. Markov Processes, Characterization and Convergence. Wiley, New York.Z. · Zbl 0592.60049
[10] EVANS, S. and PERKINS, E. 1994. Measure-valued diffusions with singular interactions. Canad. J. Math. 46 120 168. Z. · Zbl 0806.60039
[11] KONNO, N. and SHIGA, T. 1988. Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 201 225. Z. · Zbl 0631.60058
[12] KOTELENEZ, P. 1986. Law of large numbers and center limit theorem for linear chemical reactions with diffusion. Ann. Probab. 14 173 193. Z. · Zbl 0661.60053
[13] KOTELENEZ, P. 1987. Fluctuations near homogeneous states of chemical reactions with diffusion. Adv. in Appl. Probab. 19 352 370. Z. JSTOR: · Zbl 0625.60068
[14] KOTELENEZ, P. 1988. High density limit theorems for nonlinear chemical reactions with diffusion. Probab. Theory Related Fields 78 11 37. Z. · Zbl 0628.60108
[15] KOTELENEZ, P. 1992a. Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations. Stochastics Stochastics Rep. 41 177 199. Z. · Zbl 0766.60078
[16] KOTELENEZ, P. 1992b. Comparison methods for a class of function valued stochastic partial differential equations. Probab. Theory Related Fields 93 1 19. Z. · Zbl 0767.60053
[17] MUELLER, C. and TRIBE, T. 1995. Stochastic p.d.e.’s arising from the long range contact and long range voter processes. Probab. Theory Related. Fields 102 519 545. Z. · Zbl 0827.60050
[18] TRIBE, R. 1994. The long time behavior of a stochastic P.D.E. Preprint. Z.
[19] WALSH, J. 1986. An introduction to stochastic partial differential equations. Lecture Notes in Math. 1180 265 439. Springer, Berlin. · Zbl 0608.60060
[20] TEMPE, ARIZONA 85287-1804 E-mail: blount@math.la.asu.edu
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