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Small noise problems are quite important for all types of stochastic differential equations. In this paper, we focus on rough differential equations driven by scaled fractional Brownian rough path with Hurst parameter H∈(1/4,1/2]$H\in (1/4, 1/2]$. We prove a moderate deviation principle for this equation as the scale parameter tends to zero. [ABSTRACT FROM AUTHOR]

In this paper, we prove a support theorem of Stroock–Varadhan type for pinned diffusion processes. To this end, we use two powerful results from stochastic analysis. One is quasi-sure analysis for Brownian rough path. The other is Aida–Kusuoka–Stroock's positivity theorem for the densities of weighted laws of non-degenerate Wiener functionals. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the averaging principle for a fast-slow system of rough differential equations driven by mixed fractional Brownian rough path. The fast component is driven by Brownian motion, while the slow component is driven by fractional Brownian motion with Hurst index H (1 / 3 < H ≤ 1 / 2). Combining the fractional calculus approach to rough path theory and Khasminskii's classical time discretization method, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the L 1 -sense. The averaging principle for a fast-slow system in the framework of rough path theory seems new. [ABSTRACT FROM AUTHOR]

In this paper we prove that the derivative process of a rough differential equation driven by a Brownian rough path has finite Lr-moment for any r ≥ 1. This kind of problem is easy in the usual SDE theory, thanks to Burkholder-Davis-Gundy's inequality. In the context of rough path theory, however, it does not seem so obvious. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we establish asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths under the condition that the phase function has finitely many non-degenerate minima. Our main tool is the Banach space-valued rough path theory of T. Lyons. We use a large deviation principle and the stochastic Taylor expansion with respect to the topology of the space of geometric rough paths. This is a continuation of a series of papers by Inahama [Y. Inahama, Laplace''s method for the laws of heat processes on loop spaces, J. Funct. Anal. 232 (2006) 148–194] and by Inahama and Kawabi [Y. Inahama, H. Kawabi, Large deviations for heat kernel measures on loop spaces via rough paths, J. London Math. Soc. 73 (3) (2006) 797–816], [Y. Inahama, H. Kawabi, On asymptotics of certain Banach space-valued Itô functionals of Brownian rough paths, in: Proceedings of the Abel Symposium 2005, Stochastic Analysis and Applications, A Symposium in Honor of Kiyosi Itô, Springer, Berlin, in press. Available at: http://www.abelprisen.no/no/abelprisen/deltagere_2005.html]. [Copyright &y& Elsevier]

In this paper we will prove the quasi-sure existence of the Brownian rough path for finite-dimensional cases. As an application we will give a construction of Brownian pants, that is a certain continuous process on the continuous loop space over a compact manifold. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we will consider Laplace''s method for a class of heat processes on loop spaces. We will obtain the first term of the asymptotics under assumptions that the function under consideration attains its minimum at a unique point and that the Hessian at the point is non-degenerate. This kind of process was first introduced by P. Malliavin in 1990 for the loop group case and then gradually generalized by various authors. Our tool is the rough path theory of T. Lyons. This technique was pioneered by S. Aida for finite-dimensional processes in his unpublished paper. [Copyright &y& Elsevier]

In this paper we consider the Schrödinger operator HV=- 1/2△H+V on the hyperbolic plane H={z=(x,y) | x∈R,y>0}, where △H is the hyperbolic Laplacian and V is a scalar potential on H. It is proven that, under an appropriate condition on V at ‘infinity’, the number of eigenvalues of HV less than λ is asymptotically equal to the canonical volume of the quasi-classically allowed region {(x,y;ξ,η)∈T*H | y2(ξ2+η2)/2+V(x,y)<λ} as λ→∞. Our proof is based on the probabilistic methods and the standard Tauberian argument as in the proof of Theorem 10.5 in Simon (Functional Integration and Quantum Physics, Academic Press, New York, 1979). [Copyright &y& Elsevier]

We study the large eigenvalue asymptotics for the Schro¨dinger operator HV=−1/2Δ+V on the real and the complex hyperbolic n-spaces. Here Δ is the Laplace–Beltrami operator and V is a scalar potential. We assume that V is real-valued, continuous, semi-bounded from below and diverges at infinity in an appropriate sense. Then it is proven that the number of eigenvalues of HV less than λ behaves semi-classically as λ↗∞. This is a natural generalization of the result obtained by Inahama and Shirai [Eigenvalue asymptotics for the Schro¨dinger operators on the hyperbolic plane, J. Funct. Anal., submitted for publication]. [Copyright &y& Elsevier]

In this paper we prove the logarithmic Sobolev inequality on pinned loop groups over compact Lie groups equipped with [formula]-metric (s>1/2). We also compute the Ricci tensor with respect to [formula]-metric and investigate its asymptotic behavior as s decreases to 1/2. [ABSTRACT FROM AUTHOR]

We investigate the essential spectrum of the Pauli operators (and the Dirac and the Schrödinger operators) with magnetic fields on the Poincaré upper-half plane. The magnetic fields under consideration are asymptotically constant (which may be equal to zero), or diverge at infinity. Moreover, the Aharonov–Casher type result is also considered. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In this paper we consider heat kernel measure on loop groups associated to the H1/2-metric. Unlike Hs-case (s>1/2), there is a difficulty that H1/2 is not contained in the space of continuous loops. So we take limits. There are two limiting methods. One is to use delta functions and to let s go down to 1/2. The other is to fix s at 1/2 and to approximate the delta functions. For the second approach, a generalization of heat kernel measures is needed. Then, the first approach can be obtained as a special case of the second one. The limit in the sense of finite dimensional distribution is the fictitious infinite dimensional Haar measure. [Copyright &y& Elsevier]

We study the essential spectrum of the magnetic Schrödinger operators on the Poincaré upper-half plane and establish a hyperbolic analog of Iwatsuka's result [J. Math. Kyoto Univ. 23(3), 475-480 (1983)] on the stability of the essential spectrum under perturbations from constant magnetic fields. [ABSTRACT FROM AUTHOR]

Stochastic differential equations (SDEs) on compact foliated spaces were introduced a few years ago. As a corollary, a leafwise Brownian motion on a compact foliated space was obtained as a solution to an SDE. In this paper we construct stochastic flows associated with the SDEs by using rough path theory, which is something like a "deterministic version" of Itô's SDE theory. [ABSTRACT FROM AUTHOR]