A Liouville theorem for L\'evy generators

Under mild assumptions, we establish a Liouville theorem for the"Laplace"equation $Au=0$ associated with the infinitesimal generator $A$ of a L\'evy process: If $u$ is a weak solution to $Au=0$ which is at most of (suitable) polynomial growth, then $u$ is a polynomial. As a by-product, we obtain new regularity estimates for semigroups associated with L\'evy processes.


Introduction
The classical Liouville theorem states that any bounded solution u ∶ R d → R to the Laplace equation ∆u = 0 is constant. There is an extension for unbounded functions: If ∆u = 0 and u is at most of polynomial growth, say, u(x) ≤ C(1 + x k ) for some constants C > 0 and k ∈ N 0 , then u is a polynomial of degree at most k. In this paper, we extend this result to a wide class of integro-differential operators. More precisely, we establish a Liouville theorem for equations Au = 0 where A is of the form for some b ∈ R d , a positive semi-definite matrix Q ∈ R d×d and a measure ν on (R d {0}, B(R d {0})) satisfying ∫ y≠0 min{1, y 2 } ν(dy) < ∞. Equivalently, A can be written as a pseudo-differential operator, wheref (ξ) = (2π) −d ∫ R d f (x)e −ix⋅ξ dx denotes the Fourier transform of f and the symbol ψ is a continuous negative definite function with Lévy-Khintchine representation Since A is the infinitesimal generator of a Lévy process, see below, we also call A a Lévy generator. The family of Lévy generators includes many interesting and important operators, e.g. the Laplacian ∆, the fractional Laplacian −(−∆) α 2 , α ∈ (0, 2), and the free relativistic Hamiltonian m − √ −∆ + m 2 , m > 0. If A is a local operator, i.e. ν = 0, then the Liouville theorem is classical, and so the focus is on the non-local case ν ≠ 0. For Lévy generators with a sufficiently smooth symbol, there is a Liouville theorem by Fall & Weth [5]; the required regularity of ψ increases with the dimension d ∈ N. Ros-Oton & Serra [18] established a general Liouville theorem for symmetric stable operators, where α ∈ (0, 2) and µ is a non-negative finite measure on the unit sphere S d−1 satisfying an ellipticity condition. The recent papers [1,11] give necessary and sufficient conditions for the Liouville property, i.e. conditions under which the implication holds. Choquet & Deny [4] characterized the bounded solutions u to convolution equations of the form u = u * µ; these equations play a central role in the study of the "Laplace" equation Au = 0, see Lemma 2.3. Since the Liouville theorem is an assertion on the smoothness of harmonic functions, there is a close connection between the Liouville theorem and Schauder estimates; see [13,18] and the references therein for recent results. We would like to mention that there are also Liouville theorems in the half-space, see e.g. [3,18], and Liouville theorems for certain Lévy-type operators, see e.g. [2,16,17,22]. In this paper, we use a probabilistic approach, inspired by [18], to prove a Liouville theorem for a wide class of Lévy generators. Before stating the result, let us briefly recall some material from probability theory. It is well known, cf. [19,8,9], that there is a one-to-one correspondence between continuous negative definite functions and Lévy processes, i.e. stochastic processes with càdlàg (right-continuous with finite left-hand limits) sample paths and stationary and independent increments. Given a continuous negative definite function ψ ∶ R d → C, there exists a Lévy process which means that A = −ψ(D) is the infinitesimal generator of (X t ) t≥0 . The Lévy process (X t ) t≥0 is uniquely determined by ψ, the so-called characteristic exponent of (X t ) t≥0 , and by the associated Lévy triplet (b, Q, ν). The following theorem is our main result.
, then u is a polynomial of degree at most ⌊γ⌋. In particular, A has the Liouville property (3).

1.2.
Remark. (i) Weak solutions to Au = 0 are only determined up to a Lebesgue null set, cf. Section 2. When we write "u is a polynomial", this means that u has a representative which is a polynomial, i.e. there is a polynomialũ such that u =ũ Lebesgue almost everywhere. (ii) If (X t ) t≥0 is a Brownian motion, then (C1) is trivial and (C2) holds for all β > 0; consequently, we recover the classical Liouville theorem for the Laplacian. (iv) Condition (C2) is equivalent to assuming that E( X t β ) = ∫ R d x β p t (x) dx is finite for some (all) t > 0, cf. [19]. Consequently, (C2) implies, in particular, that P t u(x) = Eu(x + X t ) is well defined for any measurable function u satisfying the growth condition u(x) ≤ M (1 + x β ).
(v) The conditions (C1) and (C2) are quite mild assumptions, which hold for a large class of pseudo-differential operators. The recent paper [11] does, however, indicate that our conditions are not sharp; it is shown that A = −ψ(D) has the Liouville property ( Secondly, we use that the convolution operator P t has smoothing properties, i.e. P t u has a higher regularity than u. If u is a solution to Au = 0, and hence to P t u = u, then these regularizing properties of P t allow us to establish suitable Hölder estimates for u which lead, by iteration, to the conclusion that u is smooth; thus a polynomial. The remaining article is structured as follows. In Section 2 we introduce the notion of weak solutions and study the connection between the "Laplace" equation Au = 0 and the convolution equation P t u = u. In Section 3 we establish regularity estimates for the semigroup (P t ) t≥0 , which are of independent interest. The Liouville theorem is proved in Section 4.

Definition. Let
A be a pseudo-differential operator with continuous negative definite symbol In (4) we implicitly assume that the integrals exist. For the integral on the right-hand side, the existence is evident from ϕ ∈ C ∞ c (U ) and f ∈ L 1 loc (U ). The other integral is harder to deal with because A * is a non-local operator, i.e. decay properties of ϕ (e.g. compactness of the support) do not carry over to A * ϕ. Our first result in this section shows that the decay of A * ϕ is closely linked to the existence of fractional moments ∫ y ≥1 y β ν(dy) of the Lévy measure ν, associated with ψ via (2); see [5, Lemma 2.1] for a related result.
More precisely, there exists for all R > 0 a constant Let us mention that ∫ y ≥1 y β ν(dy) < ∞ is actually equivalent to (5). Here, we need (and prove) only sufficiency for (5); for the converse implication see [6, Theorem 4.1]. Proposition 2.2 gives a sufficient condition such the integral on the left-hand side of (4) exists: Since the adjoint A * is a pseudo-differential operator with symbol ψ and triplet (−b, Q, ν(−⋅)), Proof of Proposition 2.2. Since the assertion is obvious for the local part of A, we may assume without loss of generality that b = 0 and for some constant C = C(R). Integrating with respect to x, we find by Tonelli's theorem that x ≥2R On the other hand, it is immediate from Taylor's formula that and this yields the required estimate for ∫ x <2R (1 + x β ) Aϕ(x) dx.
Next we establish a connection between the "Laplace" equation Au = 0 and the convolution equation P t u = u.

Lemma.
Let (X t ) t≥0 be a Lévy process with Lévy triplet (b, Q, ν), infinitesimal generator (A, D(A)) and semigroup (P t ) t≥0 . Assume that X t has for t > 0 a density p t ∈ C b (R d ) with respect to Lebesgue measure, and let β ≥ 0 be such that ∫ y ≥1 y β ν(dy) < ∞.
then there existsũ ∈ C(R d ) such that u =ū Lebesgue almost everywhere andũ = P tũ for all t > 0.
Note that the exceptional null set {ũ ≠ P tũ } does, in general, depend on t; for the application which we have in mind, that is, for the proof of Liouville's theorem, this is not a problem since we will use the result only for t = 1.
for some constant C 1 > 0 which does not depend on ε, x and u. As ∫ y ≥1 y β ν(dy) < ∞, the Lévy process has fractional moments of order β, i.e. E( X t β ) = ∫ y β p t (y) dy < ∞, see e.g. [19,Theorem 25.3] or [12,Theorem 4.1], and so P t u and P t u ε are well-defined. We have For fixed x ∈ R d , it follows from the dominated convergence theorem that the second term on the right-hand side is less than, say, > 0, for R large enough. Since u ε → u in L 1 loc (dx), the first term is less than for small ε > 0. Hence, P t u ε (x) → P t u(x) as ε → 0 for each x ∈ R d . Next we show that By the definition of P t u and u ε ,we have Because of the growth estimate in (7), we may apply Fubini's theorem: It remains to show that ∆ = 0. As ϕ ε ∈ C ∞ c (R d ), an application of Dynkin's formula gives Applying Lemma 2.2, using the growth condition on u and the fact that ∫ and therefore we may apply once more Fubini's theorem: we conclude that for each fixed ω ∈ Ω, s ∈ [0, t] and x ∈ R d , it follows from Au = 0 weakly that the inner integral on the right-hand side is zero, and so ∆ = 0. This finishes the proof of (8). As u ε → u in L loc 1 , there exists a subsequence converging Lebesgue almost everywhere. Letting ε → 0 in (8) along this subsequence, we get P t u = u Lebesgue almost everywhere. If we set u ∶= P 1 u, then u = P 1 u =ũ Lebesgue almost everywhere and u = u = P t u = P tũ a.e.
where the latter equality follows from the fact that P t does not see Lebesgue null sets since X t has a density with respect to Lebesgue measure. Finally, we note thatũ ∈ C(R d ). Indeed, given ε > 0 and r > 0, there is some R > r such that sup x∈B(0,r) y ≥R Hence, for all x, z ∈ B(0, r) i.e.ũ is continuous. Sinceũ and P tũ are continuous, it follows fromũ = P tũ Lebesgue almost everywhere thatũ(x) = P tũ (x) for all x ∈ R d .

Regularity estimates for semigroups associated with Lévy processes
Let (X t ) t≥0 be a Lévy process with transition density p t , t > 0, and semigroup If u ∶ R d → R is bounded and Borel measurable, then P t u is continuous, being convolution of a bounded function with an integrable function, cf. [20,Theorem 15.8]. In this section, we study the regularity of x ↦ P t u(x) for unbounded functions u. If u is unbounded, then we need some assumptions to make sense of the integral appearing in the definition of P t u. It is natural to assume that there exists a constant β > 0 such that the associated Lévy measure ν satisfies ∫ y ≥1 y β ν(dy) < ∞. This condition ensures that E( X t β ) < ∞ for all t ≥ 0, cf. Sato [19], and so P t u is well-defined for any function u satisfying u(x) ≤ M (1 + x β ), x ∈ R d , for some M > 0. Under the assumption that p t ∈ C 1 b (R d ), we will show that P t u is locally Hölder continuous for every function u satisfying u(x) ≤ M (1 + x γ ), x ∈ R d , for some γ < β. Before stating the result, let us give a word of caution. As and therefore one might suspect that P t u is differentiable (and not only locally Hölder continuous). In general, it is not possible to make this calculation rigorous, even if u is bounded. To start with, it is not clear that the integral ∫ R d u(y) ∇p t (y − x) dy is finite since the decay of p t does not necessarily carry over to its derivatives. However, there is an interesting -and wide -class of Lévy processes for which the above reasoning can be made rigorous, and we will work out the details in the second part of this section.
3.1. Lemma. Let (X t ) t≥0 be a Lévy process with Lévy triplet (b, Q, ν) and semigroup (P t ) t≥0 . Let β > 0 be such that ∫ y ≥1 y β ν(dy) < ∞, and assume that X t has for some t > 0 a density p t ∈ C 1 b (R d ) with respect to Lebesgue measure. If u is a measurable function satisfying u(x) ≤ M (1 + x γ ) for some M > 0 and γ ∈ [0, β), then where ∶= β−γ d+β ∈ (0, 1) and C = C(t, β) < ∞ is a constant which does not depend on u. In particular, x ↦ P t u(x) is Hölder continuous of order on any compact set K ⊆ R d and (B(0,r)) ≤ (C + 2)M r γ for all r ≥ 1.
Proof. Because of the growth assumption on u, it follows from E( X t β ) < ∞ that P t u is welldefined. Fix r, R ≥ 1 and x, h ∈ R d with h , x ≤ 1. By the definition of the semigroup, Applying the mean value theorem and using the growth condition on u, we find that For the second term, we use again the growth condition on u: Performing a change of variables and using the elementary estimate Note that the integral on the right-hand side is finite since E( X t β ) < ∞. Consequently, we have shown that there exists a constant C = C(β, t) > 0 such that Choosing R ∶= h −1 (d+β) gives (9). The remaining assertion is obvious from (9).
If (P t ) t≥0 is the semigroup associated with a subordinated Brownian motion (X t ) t≥0 , then the regularity estimate from Proposition 3.2 can be improved. We do not need this strengthened version for the proof of the Liouville theorem, but we present the proof since we believe that the result is of independent interest. Recall that a Lévy process (S t ) t≥0 is a subordinator if (S t ) t≥0 has non-decreasing sample paths.

3.2.
Proposition. Let (X t ) t≥0 be a Lévy process which is of the form X t = B St for a d-dimensional Brownian motion (B t ) t≥0 and a subordinator (S t ) t≥0 satisfying P(S t = 0) = 0 for all t > 0. Denote by (b, Q, ν) the Lévy triplet of (X t ) t≥0 , and let β > 0 be such that ∫ y ≥1 y β ν(dy) < ∞. If u ∶ R d → R is a measurable function satisfying u(x) ≤ M (1 + x γ ), x ∈ R d , for some M > 0 and γ ∈ [0, β], then x ↦ P t u(x) is smooth for all t > 0 and where C k = C k (t) is a finite constant, which does not depend on u and r.
Let us mention that P(S t = 0) = 0 is equivalent to assuming that (X t ) t≥0 has a density with respect to Lebesgue measure, cf. [14,Lemma 4.6].
St is a Lévy process with Lévy triplet, say, (b (k) , Q (k) , ν (k) ), cf. [21] or [19]. By definition, Consequently, and so the finiteness of the fractional moment ∫ y ≥1 y β ν (k) (dy) does not depend on the dimension k. By assumption, the moment is finite for k = d, and hence it is finite for all k ≥ 1. Thus, E( X (k) t β ) < ∞ for all k ≥ 1 and t ≥ 0. As P(S t = 0), the process (X (k)

Proof of Liouville's theorem
In this section, we prove the Liouville theorem, cf. Theorem 1.1. First, we use a general result by Choquet & Deny [4] to show that the only bounded solutions to the convolution equation P t u = u are the trivial ones.
4.1. Proposition. Let (X t ) t≥0 be a Lévy process with characteristic exponent ψ and semigroup (P t ) t≥0 , and denote by Af = −ψ(D)f the associated Lévy generator. Assume that X t has a density p t ∈ C b (R d ) for some t > 0.
(i) If u is a bounded measurable function such that P t u = u a.e., then u is constant a.e.
Proof. (i) Without loss of generality, we may assume that P t u(x) = u(x) for all x ∈ R d ; otherwise replace u byũ ∶= P t u and note that P t u = P tũ as X t has a density with respect to Lebesgue measure. Since ∫ R d p t (y) dy = 1 and p t ≥ 0 is continuous, there exist x 0 ∈ R d and r > 0 such that p t (y) > 0 for all y ∈ B(x 0 , r). In particular, B(x 0 , r) is contained in the support of the distribution of X t . By [4, Theorem 1], this implies Hence, u is constant.
(ii) This is immediate from Lemma 2.3 and (i).
We are now ready to prove the Liouville theorem.
Proof of Theorem 1.1. By Lemma 2.3, we may assume without loss of generality that u is continuous and u(x) = P 1 u(x) for all x ∈ R d . Applying Lemma 3.1, we find that there exists a constant C > 0 such that for ∶= (β − γ) (d + β) > 0 and some constant C = C(β) > 0. This implies Indeed: If x ≤ 1, then this follows from (13) for r = 1, x ′ = x and h ′ = h; if x > 1 we choose r = x , h ′ = h r and x ′ = x r in (13). This means that for each fixed h ∈ R d , 0 Since the semigroup (P t ) t≥0 is invariant under translations, we have P 1 v = v, and therefore we can apply the above reasoning to v (instead of u) to obtain that Define iteratively ∆ h u(x) ∶= u(x + h) − u(x) and ∆ k h u(x) ∶= ∆ h (∆ k−1 h u)(x), k ≥ 2, then the previous inequality shows ∆ 2 h u(x) ≤ 4C 2 M 2 1 + x γ−2 h 2 , x ∈ R d , h ≤ 1. Iterating the procedure, we find that for the largest integer k ≥ 1 such that γ − k ≥ 0; the latter condition ensures that the constant γ in Lemma 3.1 is non-negative. Applying once more Lemma 3.1, we get If x, h ∈ R d are such that x ≥ 1 and h ≤ 1, then we obtain from this inequality for r = x , x ′ = x r and h ′ = h r that ∆ k h u(x + h) − ∆ k h u(x) ≤ (2CM ) k+1 x γ−(k+1) h (k+1) . As γ − (k + 1) < 0, this gives sup x >r ∆ k+1 h u(x) ≤ (2CM ) k+1 r γ−(k+1) h (k+1) r→∞ → 0.
Consequently, x ↦ w(x) ∶= ∆ k+1 h u(x) is for each fixed h ≤ 1 a continuous function which vanishes at infinity and which satisfies P 1 w = w. The Liouville property, cf. Proposition 4.1, yields w = 0, i.e. ∆ k+1 h u(x) = 0 for all x ∈ R d and h ≤ 1. We claim that this implies that u is a polynomial. Take ϕ ∈ C ∞ c (R d ) with ϕ ≥ 0 and ∫ R d ϕ(x) dx = 1, and set ϕ n (x) ∶= n d ϕ(nx). The convolution u n ∶= u * ϕ n satisfies ∆ k+1 h u n (x) = 0 for all x ∈ R d and h ≤ 1. Since u n is smooth, we have ∂ k+1 xj u n (x) = lim r↓0 ∆ k+1 rej u n (x) r k+1 = 0 for all x ∈ R d , j ∈ {1, . . . , d} and n ∈ N; here e j denotes the j-th vector in R d . Hence, ∂ α u n = 0 for all α ≥ N ∶= (k + 1)d, and so u n is a polynomial of degree at most N for each n ∈ N. Since u n converges pointwise to u, it follows that u is a polynomial of degree at most N . Recalling that u satisfies by assumption the growth condition u(x) ≤ M (1 + x γ ) for all x ∈ R d , we conclude that u is a polynomial of order at most ⌊γ⌋.