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Smoothness of martingale observables and generalized Feynman-Kac formulas

Alex Karrila, Lauri Viitasaari

TL;DR

This work proves that, under Hörmander’s criterion, all martingale observables of an Itô diffusion are smooth and satisfy the associated parabolic PDEs, even without ellipticity. It then provides a generalized Feynman–Kac formula allowing degenerate diffusions and boundary stopping, along with a corresponding X-harmonic theory, under mild boundary regularity. A key technical device is a slowed-down random time-change that confines dynamics and yields smooth particle densities, enabling a rigorous combination of PDE hypoellipticity and stochastic calculus. The results are applied to Schramm–Loewner evolutions, showing smoothness of conformally covariant SLE observables and enabling Girsanov-based constructions of perturbed SLE models and multiple-SLE frameworks.

Abstract

We prove that, under the Hörmander criterion on an Itô process, all its martingale observables are smooth. As a consequence, we also obtain a generalized Feynman-Kac formula providing smooth solutions to certain PDE boundary-value problems, while allowing for degenerate diffusions as well as boundary stopping (under very mild boundary regularity assumptions). We also highlight an application to a question posed on Schramm-Loewner evolutions, by making certain Girsanov transform martingales accessible via Itô calculus.

Smoothness of martingale observables and generalized Feynman-Kac formulas

TL;DR

This work proves that, under Hörmander’s criterion, all martingale observables of an Itô diffusion are smooth and satisfy the associated parabolic PDEs, even without ellipticity. It then provides a generalized Feynman–Kac formula allowing degenerate diffusions and boundary stopping, along with a corresponding X-harmonic theory, under mild boundary regularity. A key technical device is a slowed-down random time-change that confines dynamics and yields smooth particle densities, enabling a rigorous combination of PDE hypoellipticity and stochastic calculus. The results are applied to Schramm–Loewner evolutions, showing smoothness of conformally covariant SLE observables and enabling Girsanov-based constructions of perturbed SLE models and multiple-SLE frameworks.

Abstract

We prove that, under the Hörmander criterion on an Itô process, all its martingale observables are smooth. As a consequence, we also obtain a generalized Feynman-Kac formula providing smooth solutions to certain PDE boundary-value problems, while allowing for degenerate diffusions as well as boundary stopping (under very mild boundary regularity assumptions). We also highlight an application to a question posed on Schramm-Loewner evolutions, by making certain Girsanov transform martingales accessible via Itô calculus.
Paper Structure (15 sections, 21 theorems, 105 equations)

This paper contains 15 sections, 21 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Smoothness of martingale observables
  4. Feynman--Kac formulas
  5. $X$-harmonic problems
  6. Application examples
  7. Applications to SLEs
  8. Some verifications of the assumptions of Theorem \ref{['thm:X-harmonic FK']}
  9. PDE connections for processes with finite life-time
  10. Proofs
  11. Preliminaries
  12. A slowed-down process
  13. Proof of Theorem \ref{['thm:mgale observables solve PDEs']}
  14. Proofs of Theorems \ref{['thm:Feynman--Kac']} and \ref{['thm:X-harmonic FK']}
  15. Proofs of the technical lemmas

Key Result

Theorem 2.2

Fix $T>0$ and let $f: \Lambda \times [0,T) \to \mathbb{R}$ be locally bounded and Borel measurable, and $g: \Lambda \to \mathbb{R}$ and $h: \Lambda \times(0,T) \to \mathbb{R}$ continuous. If $f$ is a $(g,h)$-martingale observable for the process $X$, and condition eq:Hormander criterion is satisfie i.e., for all smooth, compactly-supported $\varphi: \Lambda \times (0, T) \to \mathbb{R}$, we have

Theorems & Definitions (41)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Definition 3.4
  • Proposition 3.5
  • ...and 31 more