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Hörmander properties of discrete time Markov processes

Clément Rey

Abstract

We present an abstract framework for establishing smoothing properties within a specific class of inhomogeneous discrete-time Markov processes. These properties, in turn, serve as a basis for demonstrating the existence of density functions for our processes or more precisely for regularized versions of them. They can also be exploited to show the total variation convergence towards the solution of a Stochastic Differential Equation as the time step between two observations of the discrete time Markov processes tends to zero. The distinctive feature of our methodology lies in the exploration of smoothing properties under some local weak Hörmander type conditions satisfied by the discrete-time Markov processes. Our Hörmander properties are demonstrated to align with the standard local weak Hörmander properties satisfied by the coefficients of the Stochastic Differential Equations which are the total variation limits of our discrete time Markov processes.

Hörmander properties of discrete time Markov processes

Abstract

We present an abstract framework for establishing smoothing properties within a specific class of inhomogeneous discrete-time Markov processes. These properties, in turn, serve as a basis for demonstrating the existence of density functions for our processes or more precisely for regularized versions of them. They can also be exploited to show the total variation convergence towards the solution of a Stochastic Differential Equation as the time step between two observations of the discrete time Markov processes tends to zero. The distinctive feature of our methodology lies in the exploration of smoothing properties under some local weak Hörmander type conditions satisfied by the discrete-time Markov processes. Our Hörmander properties are demonstrated to align with the standard local weak Hörmander properties satisfied by the coefficients of the Stochastic Differential Equations which are the total variation limits of our discrete time Markov processes.
Paper Structure (25 sections, 20 theorems, 376 equations)

This paper contains 25 sections, 20 theorems, 376 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Organization of the paper
  4. Notations.
  5. Main results
  6. A Class of Markov Semigroups
  7. An alternative regularization property
  8. An invariance principle
  9. A Malliavin-inspired approach to prove smoothing properties
  10. A generic discrete time Malliavin calculus
  11. Regularization properties for approximations of the semigroup
  12. Localization
  13. The regularization property for a modified measure
  14. Malliavin tools and estimates
  15. The integration by parts formula
  16. ...and 10 more sections

Key Result

Theorem 2.1

Let $T \in \pi^{\delta,\ast}$, let $L \in \mathbb{N}$ and let $f \in \mathcal{C}_{pol}^{\infty}(\mathbb{R}^{d} ; \mathbb{R})$ satisfying: there exists $\mathfrak{D}_{f} \geqslant 1$ and $\mathfrak{p}_{f} \in \mathbb{N}$ such that for every $x \in \mathbb{R}^{d}$, Then we have the following properties:

Theorems & Definitions (46)

  • Remark 2.1
  • Theorem 2.1
  • proof
  • Corollary 2.1
  • Theorem 2.2
  • Remark 2.2
  • Example 2.1
  • proof : Proof of Theorem \ref{['th:invariance_main_result']}
  • Remark 3.1
  • Theorem 3.1
  • ...and 36 more