Space-Time Velocity Correlation Function for Random Walks

Space-time velocity correlation function for random walks.

Space-time correlation functions constitute a useful instrument from the research toolkit of continuous-media and many-body physics. Here we adopt this concept for single-particle random walks and demonstrate that the corresponding space-time velocity autocorrelation functions reveal correlations which extend in time much longer than estimated with the commonly employed temporal correlation fun...

Correlation matrices of continuous-time random walks

We present eigenvalue spectra of correlation matrices of synthetic uncorrelated continuous-time random walks. Processes of this kind are used e.g. to model high-frequency financial time series, but are also rigorously connected to the time and space fractional diffusion equation. A new method for the fast production of Mittag-Leffler random numbers already used by us to solve the fractional dif...

Random walks in space time mixing environments

We prove that random walks in random environments, that are exponentially mixing in space and time, are almost surely diffusive, in the sense that their scaling limit is given by the Wiener measure. 1 The results Random walks in random environments are walks where the transition probabilities are themselves random variables (see [22, 23] for recent reviews of the literature). The environments c...

Time of random walks

An almost sure invariance principle for random walks in a space-time random environment

We consider a discrete time random walk in a space-time i.i.d. random environment. We use a martingale approach to show that the walk is diffusive in almost every fixed environment. We improve on existing results by proving an invariance principle and considering environments with an L2 averaged drift. We also state an a.s. invariance principle for random walks in general random environments wh...