[Paper Review] Exponential mixing under controllability conditions for SDEs driven by a degenerate Poisson noise

Vahagn Nersesyan, Renaud Raquépas|arXiv (Cornell University)|Mar 19, 2019

Stochastic processes and financial applications29 references2 citations

TL;DR

This paper establishes exponential mixing and uniqueness of the invariant measure for stochastic differential equations (SDEs) driven by degenerate compound Poisson noise under mild dissipativity and controllability conditions. By introducing a novel solid controllability condition weaker than the Hörmander condition, the authors prove exponential convergence in total variation for the semigroup, extending results to non-Gaussian, degenerate noise in infinite-dimensional and networked systems.

ABSTRACT

We prove existence and uniqueness of the invariant measure and exponential mixing in the total-variation norm for a class of stochastic differential equations driven by degenerate compound Poisson processes. In addition to mild assumptions on the distribution of the jumps for the driving process, the hypotheses for our main result are that the corresponding control system is dissipative, approximately controllable and solidly controllable. The solid controllability assumption is weaker than the well-known parabolic H\"ormander condition and is only required from a single point to which the system is approximately controllable. Our analysis applies to Galerkin projections of stochastically forced parabolic partial differential equations with asymptotically polynomial nonlinearities and to networks of quasi-harmonic oscillators connected to different Poissonian baths.

Motivation & Objective

• To establish exponential mixing and uniqueness of the invariant measure for SDEs driven by degenerate compound Poisson noise.
• To weaken the standard Hörmander condition by introducing a new solid controllability assumption that applies only from a single point.
• To extend results on ergodicity beyond Gaussian noise and full-rank diffusion, covering non-Markovian, jump-driven systems.
• To apply the framework to Galerkin approximations of stochastically forced parabolic PDEs and networks of quasi-harmonic oscillators.
• To provide a coupling-based proof strategy valid on non-compact state spaces using dissipativity and controllability.

Proposed method

• Use a coupling argument based on maximal couplings and transition time estimates to bound total variation distance.
• Introduce the concept of solid controllability: a compact set of controls ensures image contains a non-degenerate ball under perturbation.
• Apply exponential estimates on hitting times (Appendix A) to control mixing rates.
• Leverage a Lyapunov function candidate ∥x∥² under dissipativity (C1) to ensure pathwise stability.
• Use the Markov semigroup (P∗t) and its action on probability measures to prove convergence to the invariant measure.
• Apply measure-theoretic tools (Lemma C.2) to ensure image measures have positive densities under regular mappings.

Experimental results

Research questions

• RQ1 Can exponential mixing be established for SDEs with degenerate Poisson noise when the Hörmander condition fails globally?
• RQ2 What weaker controllability condition ensures exponential mixing in the absence of full-rank diffusion?
• RQ3 How does the combination of dissipativity and local controllability from a single point lead to global ergodicity?
• RQ4 Can the coupling method be adapted to non-Gaussian, jump-driven SDEs on non-compact spaces?
• RQ5 What are the implications of this framework for stochastic PDEs and oscillator networks with Poissonian heat baths?

Key findings

• Under conditions (C1)–(C3), the SDE admits a unique invariant measure µinv in the space of Borel probability measures on Rd.
• The semigroup (P∗t) converges exponentially fast to µinv in total variation norm: ∥P∗tµ − µinv∥var ≤ C e−ct (1 + ∫∥x∥µ(dx)) for all t ≥ 0.
• The solid controllability condition (C3) is strictly weaker than the parabolic Hörmander condition and applies only from a single point ˆx.
• The result holds even when rank(B) < d, i.e., for degenerate jump noise, under mild moment and density assumptions on the jump distribution.
• The framework applies to Galerkin approximations of stochastically forced parabolic PDEs with asymptotically polynomial nonlinearities.
• The method extends to networks of quasi-harmonic oscillators coupled to independent Poissonian heat baths, establishing exponential mixing in such systems.