The discrete infinitesimal generator and the long-term behavior of dynamical systems

We are interested in the long-term behavior of deterministic and non-deterministic continuous-time dynamical systems. On the one hand, the statistical long-term behavior is well characterized by the stationary distribution of the system and by almost-invariant sets, i.e. sets from which the rate of outward mass transport is small. The computation of such objects supports the analysis of systems arising for example in oceanography, astrodynamics and molecular dynamics. On the other hand, engineering applications often require knowledge about the domain of attraction of some asymptotically stable fixed point.

It turns out that all this information can be computed from eigenpairs of the transfer operator associated with the underlying system. Transfer operator methods are often more reliable and efficient than techniques based on direct simulation. Sophisticated approaches have been developed for the discretization of transfer operators, however they still all require the simulation of (short) trajectories. In order to design computationally more efficient methods, we propose to discretize the infinitesimal generator of the transfer operator semigroup.

In my talk I will show the development of a discretization which has a physical meaning, and uses no trajectory simulation at all. Further, I will discuss its relation to the original system, and other properties as well.