Stochastic interacting particle systems and nonlinear kinetic equations
- Eibeck, Andreas
- Wagner, Wolfgang
2010 Mathematics Subject Classification
- 60K40 65C35
- Stochastic particle systems, regularity of jump processes, kinetic equations, existence of solutions, coagulation, fragmentation, source and efflux, dissipative collisions
We present the stochastic approach to nonlinear kinetic equations (without gradient terms) in a unifying general framework, which covers many interactions important in applications, like coagulation, fragmentation, inelastic collisions, as well as source and efflux terms. We provide conditions for the existence of corresponding stochastic particle systems in the sense of regularity (non-explosion) of a jump process with unbounded intensity. Using an appropriate space of measure-valued functions, we prove relative compactness of the sequence of processes and characterize the weak limits in terms of solutions to the nonlinear equation. As a particular application, we derive existence theorems for Smoluchowski's coagulation equation with fragmentation, efflux and source terms, and for the Boltzmann equation with dissipative collisions.
- Ann. Appl. Probab., 13 (2003), pp. 845-889