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arXiv · 2606.25336

Invariant Measures of L\'evy-driven Stochastic Differential Equations

V. Knopova·Y. Mokanu·R. L. Schilling
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Abstract

We study the structure and regularity of (infinitesimally) invariant measures of the solutions to stochastic differential equations $dX_t = b(X_t)\,dt + dZ_t$, where $(Z_t)_{t\geq 0}$ is a L\'evy process. We show, in particular, that the invariant measure has to satisfy a Volterra-type convolution equation; since we can obtain the kernels explicitly, we are able to apply regularity methods from harmonic analysis. As an application, we get a very short proof -- in any dimension -- of a classic result due to Sato and Yamazato on the form of the invariant measure of a generalized Ornstein--Uhlenbeck process.

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V. Knopova, Y. Mokanu, R. L. Schilling. 2026-06-24. Invariant Measures of L\'evy-driven Stochastic Differential Equations. https://arxiv.org/abs/2606.25336

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