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A practical global existence and uniqueness result for stochastic differential equations on Riemannian manifolds of bounded geometry

Matthias Rakotomalala

TL;DR

The paper addresses global existence and uniqueness of stochastic differential equations on Riemannian manifolds with bounded geometry under time-inhomogeneous tensor coefficients and stochastic drift. It develops a moving-frame (horizontal lift) framework on the orthonormal frame bundle $OM$, combined with regular localization and Itô switching, to construct a global, non-explosive solution and to obtain integrability and flow estimates. The main contributions are a global existence-uniqueness result for the lifted SDE on $OM$ under precise geometric and regularity conditions, and stochastic flow estimates that control the distance between flow points in time, preserving initial integrability. This work provides a practical foundation for stochastic control problems and modeling on manifolds beyond compact settings, with explicit geometric-analytic control via bounded geometry.

Abstract

In this paper, we establish a result for existence and uniqueness of stochastic differential equations on Riemannian manifolds, for regular inhomogeneous tensor coefficients with stochastic drift, under geometrical hypothesis on the manifold, so-called manifolds of bounded geometry. Furthermore, we provide stochastic flow estimates for the solutions.

A practical global existence and uniqueness result for stochastic differential equations on Riemannian manifolds of bounded geometry

TL;DR

The paper addresses global existence and uniqueness of stochastic differential equations on Riemannian manifolds with bounded geometry under time-inhomogeneous tensor coefficients and stochastic drift. It develops a moving-frame (horizontal lift) framework on the orthonormal frame bundle , combined with regular localization and Itô switching, to construct a global, non-explosive solution and to obtain integrability and flow estimates. The main contributions are a global existence-uniqueness result for the lifted SDE on under precise geometric and regularity conditions, and stochastic flow estimates that control the distance between flow points in time, preserving initial integrability. This work provides a practical foundation for stochastic control problems and modeling on manifolds beyond compact settings, with explicit geometric-analytic control via bounded geometry.

Abstract

In this paper, we establish a result for existence and uniqueness of stochastic differential equations on Riemannian manifolds, for regular inhomogeneous tensor coefficients with stochastic drift, under geometrical hypothesis on the manifold, so-called manifolds of bounded geometry. Furthermore, we provide stochastic flow estimates for the solutions.
Paper Structure (7 sections, 6 theorems, 72 equations)

This paper contains 7 sections, 6 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Riemannian Geometry
  3. Orthonormal frame bundle and horizontal lift
  4. Moving frame strong solution
  5. Main results
  6. Existence and uniqueness result
  7. Stochastic flow estimate and integrability

Key Result

Proposition 1.5

Suppose that $(M,g)$ is of bounded geometry, then there exists a constant $C_M>0$, such that for any $x\in M$ and for any normal charts around $x$ the following estimates holds in coordinates,

Theorems & Definitions (21)

  • Definition 1.1: Smooth Manifold
  • Definition 1.2: Riemann Curvature Tensor
  • Definition 1.3
  • Definition 1.4: Manifold of bounded Geometry
  • Proposition 1.5
  • Definition 1.6
  • Definition 1.7: Horizontal Lift
  • Definition 1.8: Regular Localization
  • Definition 1.9
  • Remark 1.10
  • ...and 11 more