arXiv subjects
Publications and source records attributed to Kohki Iba.
We study the Kac killing penalization for a one-dimensional recurrent L\'{e}vy process. We prove that the limit process possesses two equivalent characterizations by a Fredholm equation and by a Sturm--Liouville equation.
We investigate the long-time limit of an elephant random walk conditioned to avoid zero and show that the resulting process admits a Skorokhod-type embedding into a three-dimensional Bessel process. Using this embedding, we further establish several limit theorems for the conditioned elephant random walk, including a scaling limit to a deterministically rescaled and time-changed three-dimensional Bessel process, and a law of the iterated logarithm.
Tanaka's formula is a classical identity for Brownian motion, and Tsukada (2018) extended it to L\'{e}vy processes not necessarily symmetric. From a potential-theoretic point of view, this formula shows that the invariant function for the process killed upon hitting a singleton can be decomposed into the sum of a martingale part and a local time. In this paper, we generalize this singleton setting and derive a Tanaka-type formula for a compact set $B$. To this end, we introduce the equilibrium measure, defined as the rescaled limit of the $q$-capacity measures, and show that the invariant function for the process killed upon hitting $B$ can be represented as the integral, with respect to the equilibrium measure, of the invariant functions associated with processes killed upon hitting singletons, up to an additive constant called the Robin constant. Moreover, when $B$ is an interval, we obtain explicit representations of the equilibrium measure, the Robin constant, and the martingale part for recurrent stable processes as well as for recurrent spectrally negative L\'{e}vy processes. Finally, we discuss how an analogous Tanaka-type formula can also be established for transient L\'{e}vy processes.
For a one-dimensional L\'{e}vy process, we derive an explicit formula for the probability of first hitting a specified point among a fixed finite set. Moreover, using this formula, we obtain an explicit expression for each entry of the $Q$-matrix of the trace process on the finite set. These formulas involve solely the renormalized zero resolvent.
We study the penalization problem with various clocks where the weight is given as the exponential functional of multi-point local times for one-dimensional Lévy processes. The limit processes may vary according to the choice of random clock, and are singular to the original process.
For several classes of bounded sets $A$, the limit of a one-dimensional Lévy process conditioned to avoid $A$ up to a parametrized random time which tends to infinity. For $A$ we take the set of finite points with several clocks and a bounded $F_σ$-set with exponential clock. We also take an integer lattice with exponential clock.
We discuss conditionings to avoid two points and one-point local time penalizations with conditioning to avoid another point, for which we adopt various clocks. We also give corrections to some of the previous results of Takeda--Yano for one-point local time penalizations.
Long-time limit of one-dimensional L\'{e}vy processes weighted and normalized with respect to the exponential functional of two-point local times are studied. The limit processes may vary according to the choice of random clocks.