The continuous-time random walk is defined as a Poissonization of discretetime random walk. We study the noncolliding system of continuous-time simple and symmetric random walks on Z. We show that the system is determinantal for any finite initial configuration without multiple point. The spatio-temporal correlation kernel is expressed by using the modified Bessel functions. We extend the system to the noncolliding process with an infinite number of particles, when the initial configuration has equidistant spacing of particles, and show a relaxation phenomenon to the equilibrium determinantal point process with the sine kernel.
In this paper a general theorem for constructing infinite particle systems of jump type with long range interactions is presented. It can be applied to the system that each particle undergoes an α-stable process and interaction between particles is given by the logarithmic potential appearing random matrix theory or potentials of Ruelle's class with polynomial decay. It is shown that the system can be constructed for any α ∈ (0, 2) if its equilibrium measure µ is translation invariant, and α is restricted by the growth order of the 1-correlation function of the measure µ in general case.
Infinite dimensional stochastic differential equations (ISDEs) describing systems with an infinte number of particles are considered. Each paticles undergo Lévy process, and interaction between particles is given by long range interaction poteintial, which is not necessary of Ruelle's class but also logarthmic. We discuss the existence and uniqueness of solutions of ISDEs.
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