Uniqueness theorem for Fourier transformable measures on LCA groups

We show that if points of supports of two discrete ”not very thick” Fourier transformable measures on locally compact abelian (LCA) groups tend to one another at infinity and the same is true for the masses at these points, then these measures coincide. The result is valid for discrete almost periodic measures on LCA groups too. Also, we show that the result is false for some discrete ”thick” measures. To do this, we construct a discrete almost periodic measure on the real axis, whose masses at the points of support tend to zero as these points approach infinity.

On the Fourier Transformability of Strongly Almost Periodic Measures

In this paper we characterize the Fourier transformability of strongly almost periodic measures in terms of an integrability condition for their Fourier–Bohr series. We also provide a necessary and sufficient condition for a strongly almost periodic measure to be the Fourier transform of a measure. We discuss the Fourier transformability of a measure on $\mathbb{R}^{d}$ in terms of its Fourier transform as a tempered distribution. We conclude by looking at a large class of such measures coming from the cut and project formalism.

Periodic Measures of Mean-Field Stochastic Predator-Prey System

This paper discusses the dynamics of the mean-field stochastic predator-prey system. We prove the existence and pathwise uniqueness of the solution for stochastic predator-prey systems in the mean-field limit. Then we show that the solution of the mean-field equation is a periodic measure. Finally, we study the fluctuations of the periodic in distribution processes when the white noise converges to zero.

Infinite partitions and Rokhlin towers

We find a countable partition P on a Lebesgue space, labeled {1,2,3,…}, for any non-periodic measure-preserving transformation T such that P generates T and, for the T,P process, if you see an n on time −1 then you only have to look at times −n,1−n,…−1 to know the positive integer i to put at time 0 . We alter that proof to extend every non-periodic T to a uniform martingale (i.e. continuous g function) on an infinite alphabet. If T has positive entropy and the weak Pinsker property, this extension can be made to be an isomorphism. We pose remaining questions on uniform martingales. In the process of proving the uniform martingale result we make a complete analysis of Rokhlin towers which is of interest in and of itself. We also give an example that looks something like an independent identically distributed process on ℤ2 when you read from right to left but where each column determines the next if you read left to right.