Stationary distributions for jump processes with memory (Q441237)

The authors analyse a two-dimensional Markov process whose first marginal is a symmetric stable Lévy process perturbed by a memory, and whose second marginal is the memory itself. This process is also a certain deterministic transformation of a two-dimensional Lévy process whose first marginal is the standard Poisson process, and whose second marginal is the above symmetric stable Lévy process. Wrapping the first marginal on the circle, they obtain a strong Markov process whose state space is the Cartesian product of the unit circle and the real axis. The authors show that the unique invariant probability of this process is the product of the uniform measure and a Gaussian measure more precisely given by NEWLINE\[CARRIAGE_RETURNNEWLINE\frac{1}{2\pi}e^{-\pi y^2} dxdy.CARRIAGE_RETURNNEWLINE\]