Infinitely-many-species Lotka-volterra Equations Arising from Systems of Coalescing Masses

We consider nonlinear, probability measure{valued dynamical systems that generalise those classical Lotka{Volterra equations in which, to use ecological terminology, the total size of a nite number of populations of interacting species is conserved. In our generalisation there is a \di erent species" at each point of an arbitrary measurable space. Such in nitely{many{species analogues of the classical Lotka{Volterra equations appear as hydrodynamic{type limits of stochastic systems of randomly coalescing masses related to those that have been used to model physical and chemical processes of agglomeration, coagulation and condensation. One natural instance of our generalisation has closed form solutions, including a family of solutions that exhibit soliton{like behaviour. The large time asymptotics of other classes of examples can be completely described using analogues of Lyapunov function techniques. Moreover, there are conserved quantities in the form of relative entropies that generalise those found by Volterra in the classical case. Finally, each solution has a series expansion as a time-varying, geometric mixture of a xed sequence of probability measures. The existence of this expansion is related to the fact that the system is in martingale problem duality with a function{valued Markov process. Running head: LOTKA-VOLTERRA EQUATIONS. Mathematics Subject Classi cation: Primary 35F20; secondary 34F05, 34G20, 60H10. Research supported in part by Presidential Young Investigator Award