Limit theorems for infinite-dimensional Piecewise Deterministic Markov Processes
Sprache des Vortragstitels:
10th German Probability and Statistics Days
Sprache des Tagungstitel:
Infinite-dimensional dynamical systems play an important role in modelling spatio-temporal behaviour of a large number of applications in, e.g., physics, biology and neuroscience, which in reality are intrinsically stochastic. The deterministic equations are understood to capture the averaged dynamics of a large number of individual stochastic events. Often arises the situation that the deterministic evolution is perturbed by randomly occurring individual events which persistently change the dynamics. For example, one may think of switching systems or multi-scale models where fast dynamics are modelled in a deterministically and slow dynamics are modelled by a jump process. Prominent examples for the latter are slow and fast reactions in chemical reaction systems with spatially non-homogeneous concentrations or recently developed stochastic hybrid models of excitable media. These models can be regularly cast into the general framework of Hilbert space valued Piecewise Deterministic Markov Processes (PDMPs).
We present limit theorems which, on the one hand, establish a precise mathematical connection between PDMPs in Hilbert spaces and deterministic evolution equations in the sense of a weak law of large numbers. On the other hand, we have derived a central limit theorem that characterises the internal fluctuations in the limit to be of a diffusive nature. These limit theorems provide the basis for a general Langevin approximation to PDMPs, i.e., certain stochastic partial differential equations that are expected to be similar in their dynamics to PDMPs. Finally, we exemplify the application of these results to a spatially extended hybrid stochastic model of a neuronal membrane. Here, the PDMP formulation provides a rigorous framework for a stochastic spatial model in which discrete random events are globally coupled via continuous space-dependent variables solving a partial differential equations.