QUASI-SURE EXISTENCE OF BROWNIAN ROUGH PATHS AND A CONSTRUCTION OF BROWNIAN PANTS

2006 ◽  
Vol 09 (04) ◽  
pp. 513-528 ◽  
Author(s):  
YUZURU INAHAMA
Keyword(s):  
Loop Space ◽  
Compact Manifold ◽  
Continuous Process ◽  
Rough Paths ◽  
Finite Dimensional ◽  
Rough Path

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.

scholarly journals SHORT TIME FULL ASYMPTOTIC EXPANSION OF HYPOELLIPTIC HEAT KERNEL AT THE CUT LOCUS

2017 ◽  
Vol 5 ◽  
Author(s):  
YUZURU INAHAMA ◽  
SETSUO TANIGUCHI
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Euclidean Space ◽  
Heat Kernel ◽  
Compact Manifold ◽  
Cut Locus ◽  
Finite Set ◽  
Rough Path ◽  
Hypoelliptic Heat Kernel ◽  
Short Time

In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on a Euclidean space and a compact manifold. We study the ‘cut locus’ case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assumptions we obtain an asymptotic expansion of the heat kernel up to any order. Our approach is probabilistic and the heat kernel is regarded as the density of the law of a hypoelliptic diffusion process, which is realized as a unique solution of the corresponding stochastic differential equation. Our main tools are S. Watanabe’s distributional Malliavin calculus and T. Lyons’ rough path theory.

scholarly journals A min-max characterization of Zoll Riemannian metrics

2021 ◽  
pp. 1-25
Author(s):  
MARCO MAZZUCCHELLI ◽  
STEFAN SUHR
Keyword(s):  
Loop Space ◽  
Closed Geodesic ◽  
Riemannian Metrics ◽  
Closed Manifold ◽  
Finite Dimensional ◽  
Simply Connected

Abstract We characterise the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coincide, every point lies on a closed geodesic.

scholarly journals Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths

2021 ◽  
Author(s):  
Kistosil Fahim ◽  
Erika Hausenblas ◽  
Debopriya Mukherjee
Keyword(s):  
Maximal Regularity ◽  
Correction Term ◽  
Regularity Property ◽  
One Dimension ◽  
Rough Paths ◽  
Point Of Interest ◽  
Convergence Results ◽  
Regular Approximation ◽  
Rough Path ◽  
Rough Path Theory

AbstractWe adapt Lyon’s rough path theory to study Landau–Lifshitz–Gilbert equations (LLGEs) driven by geometric rough paths in one dimension, with non-zero exchange energy only. We convert the LLGEs to a fully nonlinear time-dependent partial differential equation without rough paths term by a suitable transformation. Our point of interest is the regular approximation of the geometric rough path. We investigate the limit equation, the form of the correction term, and its convergence rate in controlled rough path spaces. The key ingredients for constructing the solution and its corresponding convergence results are the Doss–Sussmann transformation, maximal regularity property, and the geometric rough path theory.

Integration of controlled rough paths via fractional calculus

2017 ◽  
Vol 29 (5) ◽  
pp. 1163-1175 ◽  
Author(s):  
Yu Ito
Keyword(s):  
Fractional Calculus ◽  
Path Analysis ◽  
Path Integration ◽  
Rough Paths ◽  
Rough Path

AbstractWe develop a fractional calculus approach to rough path analysis, introduced by Y. Hu and D. Nualart [6], and show that our integration can be generalized so that it is consistent with the rough path integration introduced by M. Gubinelli [5].

scholarly journals PATH INTEGRALS ON A COMPACT MANIFOLD WITH NON-NEGATIVE CURVATURE

2007 ◽  
Vol 19 (09) ◽  
pp. 967-1044 ◽  
Author(s):  
ADRIAN P. C. LIM
Keyword(s):  
Compact Manifold ◽  
Compact Riemannian Manifold ◽  
Path Integrals ◽  
Energy Functional ◽  
Normalization Constant ◽  
Finite Dimensional ◽  
Riemannian Volume ◽  
Volume Measure ◽  
Constant E ◽  
Geodesic Paths

A typical path integral on a manifold, M is an informal expression of the form [Formula: see text] where H(M) is a Hilbert manifold of paths with energy E(σ) < ∞, f is a real-valued function on H(M), [Formula: see text] is a "Lebesgue measure" and Z is a normalization constant. For a compact Riemannian manifold M, we wish to interpret [Formula: see text] as a Riemannian "volume form" over H(M), equipped with its natural G1 metric. Given an equally spaced partition, [Formula: see text] of [0, τ], let [Formula: see text] be the finite dimensional Riemannian submanifold of H(M) consisting of piecewise geodesic paths adapted to [Formula: see text]. Under certain curvature restrictions on M, it is shown that [Formula: see text] where [Formula: see text] is a "normalization" constant, E : H(M) → [0,∞) is the energy functional, [Formula: see text] is the Riemannian volume measure on [Formula: see text], ν is Wiener measure on continuous paths in M, and ρ is a certain density determined by the curvature tensor of M.

scholarly journals NEW EXAMPLES OF FINITELY PRESENTED GROUPS WITH STRONG FIXED POINT PROPERTIES

2009 ◽  
Vol 01 (01) ◽  
pp. 1-12 ◽  
Author(s):  
INDIRA CHATTERJI ◽  
MARTIN KASSABOV
Keyword(s):  
Fixed Point ◽  
Compact Manifold ◽  
Finitely Presented Groups ◽  
Finite Presentation ◽  
Finite Dimensional ◽  
Fixed Point Properties ◽  
Finitely Presented ◽  
Contractible Manifold

We give an explicit finite presentation of a group normally generated by SL∞(ℤ). As a consequence, such a group cannot act on e.g. a finite dimensional contractible manifold or on a compact manifold.

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