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Register a new userThe law of a point process of Brownian excursions in a domain is determined by the law of its trace
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We show the result that is stated in the title of the paper, which has consequences about decomposition of Brownian loop-soup clusters in two dimensions.
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We show that Brownian motion of a one-dimensional domain wall in a large but finite system yields a $omega^{-3/2}$ power spectrum. This is successfully applied to the totally asymmetric simple exclusion process (TASEP) with open boundaries. An excellent agreement between our theory and numerical results is obtained in a frequency range where the domain wall motion dominates and discreteness of the system is not effective.
For $alpha >0$, the $alpha$-Lipschitz minorant of a function $f : mathbb{R} rightarrow mathbb{R}$ is the greatest function $m : mathbb{R} rightarrow mathbb{R}$ such that $m leq f$ and $vert m(s) - m(t) vert leq alpha vert s-t vert$ for all $s,t in mathbb{R}$, should such a function exist. If $X=(X_t)_{t in mathbb{R}}$ is a real-valued Levy process that is not a pure linear drift with slope $pm alpha$, then the sample paths of $X$ have an $alpha$-Lipschitz minorant almost surely if and only if $mathbb{E}[vert X_1 vert]< infty$ and $vert mathbb{E}[X_1]vert < alpha$. Denoting the minorant by $M$, we consider the contact set $mathcal{Z}:={ t in mathbb{R} : M_t = X_t wedge X_{t-}}$, which, since it is regenerative and stationary, has the distribution of the closed range of some subordinator made stationary in a suitable sense. We provide a description of the excursions of the Levy process away from its contact set similar to the one presented in It^o excursion theory. We study the distribution of the excursion on the special interval straddling zero. We also give an explicit path decomposition of the other generic excursions in the case of Brownian motion with drift $beta$ with $vert beta vert < alpha$. Finally, we investigate the progressive enlargement of the Brownian filtration by the random time that is the first point of the contact set after zero.
An important property of Kingmans coalescent is that, starting from a state with an infinite number of blocks, over any positive time horizon, it transitions into an almost surely finite number of blocks. This is known as `coming down from infinity. Moreover, of the many different (exchangeable) stochastic coalescent models, Kingmans coalescent is the `fastest to come down from infinity. In this article we study what happens when we counteract this `fastest coalescent with the action of an extreme form of fragmentation. We augment Kingmans coalescent, where any two blocks merge at rate $c>0$, with a fragmentation mechanism where each block fragments at constant rate, $lambda>0$, into its constituent elements. We prove that there exists a phase transition at $lambda=c/2$, between regimes where the resulting `fast fragmentation-coalescence process is able to come down from infinity or not. In the case that $lambda<c/2$ we develop an excursion theory for the fast fragmentation-coalescence process out of which a number of interesting quantities can be computed explicitly.
The trace of a Markov process is the time changed process of the original process on the support of the Revuz measure used in the time change. In this paper, we will concentrate on the reflecting Brownian motions on certain closed strips. On one hand, we will formulate the concrete expression of the Dirichlet forms associated with the traces of such reflecting Brownian motions on the boundary. On the other hand, the limits of these traces as the distance between the upper and lower boundaries tends to $0$ or $infty$ will be further obtained.
In this paper we study the convergence in distribution and the local limit theorem for the partial sums of linear random fields with i.i.d. innovations that have infinite second moment and belong to the domain of attraction of a stable law with index $0<alphaleq2$ under the condition that the innovations are centered if $1<alphaleq2$ and are symmetric if $alpha=1$. We establish these two types of limit theorems as long as the linear random fields are well-defined, the coefficients are either absolutely summable or not absolutely summable.
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