Classical Liouville's theorem, due to Liouville and Cauchy, asserts that all bounded harmonic functions in $\mathbb{R}^d$ are constant. Bôcher and Picard relaxed the assumptions and only assumed semi-boundedness; in other words, they proved that all positive harmonic functions in $\mathbb{R}^d$ are constant. Yet another variant of Liouville's theorem asserts that polynomially bounded harmonic functions in $\mathbb{R}^d$ are necessarily polynomials.

I will discuss my recent work with Mateusz Kwaśnicki, where we study similar problems for functions harmonic with respect to an arbitrary Lé'vy process $X_t$ in $\mathbb{R}^d$. That is, we investigate the class of (bounded, positive or polynomially bounded) functions $f$ such that the generator of $X_t$ applied to $f$ is equal to zero in an appropriate sense; or, in probabilistic terms, such that $f (X_t)$ is a local martingale.

Our work extends previous results due to Fall (on isotropic stable processes), Fall and Weth (on processes similar to stable processes), Alibaud, del Teso, Endal and Jakobsen (on bounded functions), and Berger and Schilling (on positive functions).

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