%0 Journal Article %1 oelschlager1989derivation %A Oelschläger, Karl %D 1989 %J Probab. Theory Related Fields %K diffusion_approximation fisher-kpp interacting_particle_systems measure_valued_process reaction-diffusion travelling_wave %N 4 %P 565--586 %R 10.1007/BF00341284 %T On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes %U http://dx.doi.org/10.1007/BF00341284 %V 82 %X We study for any $NınN$N∈N a population of about $N$N individuals in $R^d$Rd, which is divided into $K$K subpopulations. The individuals may move around in $R^d$Rd, may die, or give birth to new individuals, and may change their subpopulation. We rescale the interaction between the individuals in a suitable (moderate) way, as the population size tends to infinity. Essentially, this means that for any fixed particle the drift coefficients, the birth, death and transition rates depend on the configuration of the remaining particles in a neighbourhood, which is macroscopically small, i.e., its volume tends to 0 as $N\toınfty$N→∞, and microscopically large, i.e., it contains an arbitrarily large number of individuals as $N\toınfty$N→∞. It is shown that for large $N$N the empirical processes of the different subpopulations converge to the solution of a system of reaction diffusion equations. For that we consider regularized versions of these empirical processes and study their asymptotic properties as $L_2(R^d)$L2(Rd)-valued stochastic processes.

@article{oelschlager1989derivation, abstract = {We study for any $N\in\bold N$N∈N a population of about $N$N individuals in $\bold R^d$Rd, which is divided into $K$K subpopulations. The individuals may move around in $\bold R^d$Rd, may die, or give birth to new individuals, and may change their subpopulation. We rescale the interaction between the individuals in a suitable (moderate) way, as the population size tends to infinity. Essentially, this means that for any fixed particle the drift coefficients, the birth, death and transition rates depend on the configuration of the remaining particles in a neighbourhood, which is macroscopically small, i.e., its volume tends to 0 as $N\to\infty$N→∞, and microscopically large, i.e., it contains an arbitrarily large number of individuals as $N\to\infty$N→∞. It is shown that for large $N$N the empirical processes of the different subpopulations converge to the solution of a system of reaction diffusion equations. For that we consider regularized versions of these empirical processes and study their asymptotic properties as $\bold L_2(\bold R^d)$L2(Rd)-valued stochastic processes.}, added-at = {2011-04-18T22:26:41.000+0200}, author = {Oelschl{\"a}ger, Karl}, biburl = {https://www.bibsonomy.org/bibtex/2d94f14453df388eb65f109187b3a0f8e/peter.ralph}, coden = {PTRFEU}, doi = {10.1007/BF00341284}, fjournal = {Probability Theory and Related Fields}, interhash = {74235d9641d8d1aa2cd36974859c1557}, intrahash = {d94f14453df388eb65f109187b3a0f8e}, issn = {0178-8051}, journal = {Probab. Theory Related Fields}, keywords = {diffusion_approximation fisher-kpp interacting_particle_systems measure_valued_process reaction-diffusion travelling_wave}, mrclass = {60K35 (35K57 60J60)}, mrnumber = {1002901 (91c:60144)}, mrreviewer = {Ralf Manthey}, number = 4, pages = {565--586}, timestamp = {2013-09-12T22:23:01.000+0200}, title = {On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes}, url = {http://dx.doi.org/10.1007/BF00341284}, volume = 82, year = 1989 }