Time Scales and Exponential Trend to Equilibrium: Gaussian Model Problems

  • Lara NeureitherEmail author
  • Carsten Hartmann
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 282)


We review results on the exponential convergence of multidimensional Ornstein-Uhlenbeck processes and discuss notions of characteristic time scales by means of concrete model systems. We focus, on the one hand, on exit time distributions and provide explicit expressions for the exponential rate of the distribution in the small-noise limit. On the other hand, we consider relaxation time scales of the process to its equilibrium measure in terms of relative entropy and discuss the connection with exit probabilities. Along these lines, we study examples which illustrate specific properties of the relaxation and discuss the possibility of deriving a simulation-based, empirical definition of slow and fast degrees of freedom which builds upon a partitioning of the relative entropy functional in connection with the observed relaxation behaviour.


Multidimensional Ornstein-Uhlenbeck process Exponential convergence Relative entropy Large deviations Small noise asymptotics 



This research has been partially funded by Deutsche Forschungsgemeinschaft (DFG) through the grant CRC 1114 “Scaling Cascades in Complex Systems”, Projects A05 “Probing scales in equilibrated systems by optimal nonequilibrium forcing” and B05 “Origin of the scaling cascades in protein dynamics”.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für MathematikBrandenburgische Technische Universität Cottbus-SenftenbergCottbusGermany

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