arXiv subjects

Seiichiro Kusuoka

Publications and source records attributed to Seiichiro Kusuoka.

24 records · Page 2Linked to original sources

arXiv2015

Recurrence of the Brownian motion in multidimensional semi-selfsimilar environments and Gaussian environments

Asymptotic behavior of the one-dimensional Brownian motion in general random environments has been investigated by many researchers. However, many of the methods used in the argument are available only for the one-dimensional case. In this paper the multidimensional case of the problem is considered, and we obtain some sufficient conditions for recurrence of the multi-dimensional Brownian motion in random environments. By using the sufficient conditions we show that the recurrence of the Brownian motion in Gaussian environments under some conditions on the correlation functions.

math.PR↗

arXiv2014

Hölder continuity and bounds for fundamental solutions to non-divergence form parabolic equations

We consider the non-degenerate second-order parabolic partial differential equations of non-divergence form with bounded measurable coefficients (not necessary continuous). Under some assumptions it is known that the fundamental solution to the equations exists uniquely, has the Gaussian bounds and is locally Hölder continuous. In the present paper, we concern the Gaussian bounds and the lower bound of the index of the Hölder continuity with respect to the initial point. We use the pinned diffusion processes for the probabilistic representation of the fundamental solutions and apply the coupling method to obtain the regularity of them. Under some assumptions weaker than the Hölder continuity of the coefficients, we obtain the Gaussian bounds and the $(1-\varepsilon)$-Hölder continuity of the fundamental solution in the initial point.

math.PR↗

arXiv2012

Diffusion processes in thin tubes and their limits on graphs

The present paper is concerned with diffusion processes running on tubular domains with conditions on nonreaching the boundary, respectively, reflecting at the boundary, and corresponding processes in the limit where the thin tubular domains are shrinking to graphs. The methods we use are probabilistic ones. For shrinking, we use big potentials, respectively, reflection on the boundary of tubes. We show that there exists a unique limit process, and we characterize the limit process by a second-order differential generator acting on functions defined on the limit graph, with Kirchhoff boundary conditions at the vertices.

math.PR↗

arXiv2012

On smoothing properties of transition semigroups associated to a class of SDEs with jumps

We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) driven by additive pure-jump Lévy noise. In particular, we assume that the Lévy process driving the SDE is the sum of a subordinated Wiener process and of an independent arbitrary Lévy process, that the drift coefficient is continuous (but not necessarily Lipschitz continuous) and grows not faster than a polynomial, and that the SDE admits a Feller weak solution. By a combination of probabilistic and analytic methods, we provide sufficient conditions for the Markovian semigroup associated to the SDE to be strong Feller and to map $L_p$ to continuous bounded functions. A key intermediate step is the study of regularizing properties of the transition semigroup associated to the subordinated Wiener process in terms of negative moments of the subordinator.

math.PR↗

arXiv2012

Stochastic ranking process with space-time dependent intensities

We consider the stochastic ranking process with space-time dependent jump rates for the particles. The process is a simplified model of the time evolution of the rankings such as sales ranks at online bookstores. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution, and also the tagged particle processes converge almost surely, in the infinite particle limit. The limit distribution is characterized by a system of inviscid Burgers-like integral-partial differential equations with evaporation terms, and the limit process of a tagged particle is a motion along a characteristic curve of the differential equations except at its Poisson times of jumps to the origin.

math.PR↗

arXiv2012

Stein method for invariant measures of diffusions via Malliavin calculus

Given a random variable $F$ regular enough in the sense of the Malliavin calculus, we are able to measure the distance between its law and almost any continuous probability law on the real line. The bounds are given in terms of the Malliavin derivative of $F$. Our approach is based on the theory of Itô diffusions and the stochastic calculus of variations. Several examples are considered in order to illustrate our general results.

math.PR↗