Maximum principles for nonlocal parabolic Waldenfels operators

As a class of L\'evy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under L\'evy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for `parabolic' equations with nonlocal Waldenfels operators. Applications in stochastic differential equations with $\alpha$-stable L\'evy processes are presented to illustrate the maximum principles.


Introduction
The usual maximum principle concerns with second-order differential operators of elliptic or parabolic type. It is a basic property of solutions to boundary value problems for the associated elliptic or parabolic partial differential equations (PDEs) in a bounded domain. See [22,24]) for a general study of maximum principles. Classically, the maximum principle states that the maximum of the solution of a second-order elliptic or parabolic equation in a domain is to be found on the boundary of that domain. In particular, the strong maximum principle says that if the solution achieves its maximum in the interior of the domain, the solution must be a constant, while the weak maximum principle indicates that the maximum is to be found on the boundary but may re-occur in the interior as well. Let us also mention [19] where both weak and strong maximum principle for symmetric Markov generators are discussed via (local) Dirichlet forms. Moreover, a maximum principle for nonlocal operators generated by nonnegative kernels defined on topological groups acting continuously on a Hausdorff space was considered by Coville [7]. The strong maximum principle for semicontinuous viscosity solution of fully nonlinear second-order parabolic integro-differential equations was studied in [5].
A fairly large class of Markov processes on R d are governed analytically by their infinitesimal generators, called Lévy type generators or pseudodifferential operators associated with negative definite symbols (cf. e.g. [11]), either via martingale problem (cf. e.g. [15,16,28,29,14]) or via Dirichlet form (cf. e.g. [17,18,9,11]). From [11,6], these operators are usually integro-differential operators or nonlocal operators, consisting of a combination of second-order elliptic differential operators and integral operators of Lévy type. The nonlocal operator here corresponds to the jump component of a Markov process; in fact, it is an integral with respect to a jump measure.
The well-known Hille-Yosida theorem and the semigroup approach, which can be found in e.g. [12], provide an intrinsic link between Markov processes and partial differential equations, in particular second-order elliptic differential operators, as in the pioneering work of Feller in early 1950s. The monograph [30] (also references therein) explores the functional analytic approach to constructing Markov processes in a prescribed region of R d , via the elliptic boundary value problems for the associated Lévy-type generators.
Due to the nature of pseudo-differential operators (involving integral operators), the Lévy-type generators are nonlocal operators. This kind of integro-differential operators was initiated by Waldenfels [32] in 1960s. It was elucidated in [30] that a Markov process associated with such an operator as infinitesimal operator could be interpreted with a physical picture: A Markovian particle moves both by jumps and continuously in a certain region of the state space R d .
The present paper is devoted to the weak and strong maximum principles for the following nonlocal parabolic Waldenfels operator − ∂ ∂t + L: where the kernel {ν(t, x, ·) | (x, t) ∈ R d × [0, ∞)} behaves as the jump measure for the associated Markov process. The operator L is called an elliptic Waldenfels operator. Note that Waldenfels operators L and − ∂ ∂t + L appear in the generator and in the Fokker-Planck equation, respectively, for a stochastic differential equation with Lévy motions [26,3,8,27]. We would like to point out that Waldenfels operators also appear in nonlocal conservation laws [31]. Certain properties for diffusion generators perturbed by the nonlocal Laplacian operator have also been studied recently [1,2].
We will prove the new weak and strong maximum principles for the nonlocal parabolic operator − ∂ ∂t + L, and they do not require any "nondegeneracy" conditions. In order to cover the general case with either bounded or unbounded support of the jump measure ν, we will introduce two open sets D and E (with D ⊂ E), where D is the set where the maximum is achieved, and the stochastic process ("Markovian particle") cannot jump from D to the complement of E.
As a preparation for proving these maximum principles, we will prove the maximum principles for nonlocal elliptic Waldenfels operator L. These maximum principles are important for the construction of Markov processes. In [30,Appendix C], weak and strong maximum principles for such elliptic Waldenfels operators were proven, but under stringent conditions, that is, the jump measure has to have bounded support. The results in [5] includes a strong maximum principle for viscosity solutions of certain nonlinear nonlocal partial differential equations under a "nondegeneracy" condition.
The rest of this paper is organised as follows. In Section 2, we will present our results on maximum principles for elliptic Waldenfels operators. As a corollary, we also obtain the Hopf's Lemma about the sign of the gradient on the boundary. Section 3 is devoted to prove the maximum principles for parabolic Waldenfels operators. Some consequences and examples are presented in Section 4. Finally in Section 5, we present the proofs of some technical lemmas for the sake of completeness.

Maximum principles for elliptic Waldenfels operators
In this section, we consider the weak and strong maximum principles for the elliptic Waldenfels operator L (decomposed into local and nonlocal components) where A and K are defined as Note that the coefficients are taken to be independent of time t. Note that the operator K is actually the nonlocal Laplacian operator −(−∆) α 2 , when the jump measure ν is the α-stable type; see [8,Ch. 7].
The elliptic Waldenfels operator L plays an important role [30] in the theory of Markov processes constructed in a given domain of R d . In that context, the second-order differential operator describes the diffusion part of the associated Markov process and the integral operator of Lévy type corresponds to the jump behavior of the Markov process. Finally, there is an assumption in that context which indicates that a Markovian particle cannot move by jumps from any interior point of certain domain to the outside of closure of the domain. For further remarks and discussions, we refer e.g. to Bony, Courrège and Priouret [4] and Taira [30].
To cover more general situations, we introduce two open sets D and E in R d , with D ⊂ E and E not necessarily bounded. As usual, we denote the boundary of D by ∂D, its closure by D := D ∪ ∂D and its complement by We make following assumptions: 1. Continuity condition: a jk , b j , c ∈ C(E) (j, k = 1, ..., d).
Uniform ellipticity condition: there exists a constant γ > 0 such that for all x ∈ D, ξ ∈ R d .

Lévy measures: The kernel
and moreover, for Remark 2.1. The support condition (4) means in probability sense that a Markovian particle cannot move by jumps from a point x ∈ D to the outside of E. The motivation for this condition is that the maximizer point will propagate between connected components of the set in which the subsolution achieves maximum. The details will be discussed again in Remark 2.9 below. When the set E is the whole space R d , E − x is still the whole space, and then there are actually no extra restrictions on the support of each measure ν(x, ·). In the case that E = D, the support condition is supp ν(x, ·) ⊂ D −x, and this is related to the assumption in [30] that a Markovian particle cannot move by jumps from a point x ∈ D to the outside of D.
For convenience, the notation a = (a jk ) j,k=1,...,d means a is a matrix with (j, k)-th entry a jk , and b = (b 1 , ..., b d ) T is regarded as a row vector. We also recall the gradient operator (for space variable) ∇ x = ∂ ∂x 1 , ..., ∂ ∂x d T and the ∂x j ∂x k j,k=1,...,d , where ⊗ means the tensor product. The variables or subscripts will be omitted when there is no ambiguity. Then we can rewrite the operator L as where "tr" denote the trace of a matrix. Both x T y and x · y, for two vectors x, y ∈ R d , denote the scalar product. Moreover, we denote the positive and negative part of function u by u + := u ∨ 0 and u − := −(u ∧ 0) = (−u) ∨ 0, respectively. Then u = u + − u − and |u| = u + + u − .
In this section, L is the elliptic Waldenfels operator as defined in (1).

Weak maximum principle for elliptic case
We now prove the weak maximum principle. 1. If c ≡ 0 in D, then sup Here the supremum may be infinity.
Proof. Assertion 1. We first consider the case with the strict inequality Suppose that on the contrary sup E u > sup E\D u. Then there exists a point x 0 ∈ D with u(x 0 ) = sup E u, and Thus at the maximizer point x 0 , we have where the last inequality means that the symmetric matrix ∇ 2 u(x 0 ) is nonpositive definite. In particular, Since the matrix a = (a jk ) is symmetric and positive definite at x 0 , there exists an orthogonal matrix P such that where "diag" means the diagonal matrix with diagonal entries λ j > 0, j = 1, ..., d, which are eigenvalues of a(x 0 ). Then by changing variables In light of (8), we find that at point x 0 , Thus, combining (7), (9) and the assumption c ≡ 0, together with the fact that u attains a maximum at x 0 , we obtain that at x 0 , Therefore, we get a contradiction in light of (6) and (10), which leads to sup E u = sup E\D u.
For the general case that Lu ≥ 0, we introduce a function where β > 0 will be selected below and is a positive parameter. Note that a 11 ≥ γ > 0, by substituting z = e 1 = (1, 0, ..., 0) into condition (2). Then by Taylor expansion and the moment condition (3) of kernel ν, we have provided β > 0 is large enough, where θ is a constant with 0 < θ < 1. Then by the previous conclusion, sup E u = sup E\D u . Let → 0 to find sup E u = sup E\D u by the continuity. This proves Assertion 1.
Assertion 2. If u ≤ 0 everywhere in D, the second assertion is trivially true. Hence we set D + := {x ∈ D | u(x) > 0} = ∅. Then The new operator L − c has no zeroth-order term and consequently Assertion 1 implies that This completes the proof.
Using the latter as an example, namely, where D is open and bounded but not necessarily connected, then the following conclusions holds: Corollary 2.6. Let D be an open and bounded set but not necessarily connected, and E be an open set satisfying 3. If c ≤ 0 and Lu = 0 both hold in D, then In all the three expressions, the supremum and infimum may be infinity.
Then it follows that This completes the proof.
Going one step further, we suppose E is bounded and then apply Corollary 2.6 to u − v, yielding the following corollary which is often used in applications.
Proof. The two results immediately follow by using the last two assertions of Corollary 2.6 for u − v.

Strong maximum principle for elliptic case
This section is devoted to the strong maximum principle for the elliptic Waldenfels operator L.
If one of the following conditions holds: 1. c ≡ 0 in D and u achieves a (finite) maximum over E at an interior point in D; 2. c ≤ 0 in D and u achieves a (finite) nonnegative maximum over E at an interior point in D;

u achieves a zero maximum over E at an interior point in D,
then u is constant on D.
Before proving this theorem, let us first give some comments on it.
Remark 2.9. The propagation of maximizer point by translation of measure support mentioned in [5,7] is similar in our case. That is, if the assumptions in Theorem 2.8 hold, then u is a constant on the set ∞ n=0 Λ n , where Λ n 's are defined by induction, This result depends on the support of every measure ν(x, ·), it can be easily proved by induction and continuity. It is noteworthy that in this scheme, the set D may not be connected, since jumps from one connected component to another might occur when measure supports overlap two or more connected components.
In conclusion, it is the integro-differential term, or jump diffusion term that leads to the propagation of maximizer point between those connected components. Therefore, we need to restrict that the Markovian point can move by jumps only inside the set E, i.e., the support condition (4), to obtain the propagation of maximizer (over E) point. Remark 2.10. As shown in Remark 2.1, our results on the weak and strong maximum principles formulated in Theorem 2.2 and 2.8, respectively, cover the situations when the support of jump measure is either bounded or unbounded, especially for E = D or E = R d in the setting. While Taira [30] only considered the situation for E = D. Furthermore, our assumptions are less restrictive than Taira's: In our work, the connectedness is not needed for the weak maximum principle while the boundedness is not necessary for the strong maximum principle. Moreover, the continuity of mapping x → ν(x, ·) is necessary only in the strong case but not for the weak maximum principle.
Like the weak case, by applying directly Theorem 2.8 to −u, one can conclude the strong maximum principle for the converse case Lu ≤ 0.
If one of the following conditions holds: 1. c ≡ 0 in D and u achieves a (finite) minimum over E at an interior point in D; 2. c ≤ 0 in D and u achieves a (finite) nonnegative minimum over E at an interior point in D; 3. u achieves a zero minimum over E at an interior point in D, then u is constant on D.
Now we start to prove Theorem 2.8.
Since u achieves its maximum at x 0 ∈ D, we have ∇u(x 0 ) = 0. We will create a contradiction by proving that where n is the unit outer normal vector of B at x 0 . Then by this contradiction, u must be constant within D, and the result follows by continuity. Now the rest of the proof is devoted to (12). We divide it into three steps. Step Then Φ is an embedding whose rank at x 0 equals to d, equivalently, if we denote by JΦ the Jacobian matrix of Φ, i.e., JΦ := ∇ x Φ, then JΦ is non-degenerate. As a result, the tangent mapping Φ * induced by Φ at point x 0 is an isomorphism. Now we consider the function u restricted in U . We defineû(y) := u(Φ −1 (y)), y ∈ Φ(U ). Thenû attains its maximum at Hence at the maximizer point y 0 , We also denote the image tangent vector of ∂ ∂n under tangent mapping Φ * by ∂ ∂n We compute at y 0 (or x 0 ) where Φ * is denoted as the cotangent mapping induced by Φ at point x 0 , ·, · is the dual product between the tangent space and cotangent space at y 0 (or x 0 ). Now recall that Φ * is an isomorphism. The tangent vector ∂ ∂n is independent of { ∂ ∂y j | j = 1, ..., d − 1} and consequently by (14), Combining (13) and (15) together with the fact thatû attains its maximum at y 0 , we have Combining (13), (15) and (16), we have at x 0 , whereâ := (JΦ)a(JΦ) T . Since a(x) is symmetric and positive definite and the matrix JΦ is non-degenerate, we see the matrixâ(x 0 ) is also symmetric and positive definite. Hence, as explained in the proof of Theorem 2.2 and by (16), we have Define Recall that u attains its maximum over E at x 0 . Now we have From (17), (18) and (20), we obtain By recalling the assumption on u, we have Lu(x) ≥ 0 for each x ∈ D, and thus Step 2. We set for β > 0 as selected below. Then Consider next the open set D 0 := B(x 1 , R) ∩ B(x 0 , r) (see Figure 1) with some r ∈ (0, R) which will be chosen later. When β is large enough, we have for x ∈ D 0 , where C 1 , C 2 are two positive constants. Moreover, by recalling (21), we have For the term I, it is clear that E 0 ∩ B = {x 0 } and consequently For the term II, using the Taylor expansion, and for x 0 + z ∈ E 0 and β large enough, with some θ ∈ (0, 1) and a constant C 4 > 0. Hence, Thus, combining the results of (23), (24), (25) and (26), we find that provided β > 0 is fixed large enough. Since Lv(x) is continuous in x ∈ D in light of the continuity of ν(x, ·), we have for x ∈ D 0 , provided r is small enough.
Step 3. Define for a constant > 0. We can choose so small that For the first two cases, c ≡ 0 in D, or c ≤ 0 in D also u(x 0 ) ≥ 0, from (27) and the fact that Lu ≥ 0 in D, we see that In view of the weak maximum principle of elliptic Waldenfels operator, Theorem 2.2, we know that u ≤ 0 in E. Note that u (x 0 ) = 0. Thus we have, as required.
For the third case that u(x 0 ) = 0, obviously u ≤ 0 in D. We find Notice that the zeroth-order coefficient of operator L − c + is c − c + , which is nonpositive in D. Hence we apply the result of the second case by replacing L and c respectively with L − c + and c − c + to get the same result for this case.
We have thus completed the proof.
Some comments will be helpful for understanding the long proof of Theorem 2.8.
Remark 2.12. In Theorem 2.8, we restrict the set D to be connected to ensure ∂D < ∩ D = ∅. More generally, if D is not connected, one may merely replace D with the connected component of D which contains the maximizer point, and we thus conclude that u is constant in this connected component.
Recalling Remark 2.13, we could see that the diffusion term gives rise to the propagation of maximizer point in the corresponding connected component. This is why we need the set D to be connected.
Remark 2.13. We can see from (23), (25) and (26) that, it is the secondorder differential term tr[a T (∇ 2 )], namely, the diffusion term that plays a leading role in Step 2 in the proof of Theorem 2.8. Remark 2.14. Theorem 2.8 still holds if the matrix a(x) = (a jk (x)) j,k=1,...,d is only positive semidefinite and the unit outer normal vector n is not in the nullspace of a(x 0 ).
In fact, recall thatâ = (JΦ)a(JΦ) T is also semidefinite as the Jacobian matrix JΦ is invertible. Due to the reason mentioned in Remark 2.4, we confirm that (18) still holds. Moreover, noting that there exists a positive constant γ such that n T a(x 0 )n ≥ γ > 0 with n not in the nullspace of a(x 0 ), and consequently ( with a positive constant γ 1 . Hence (22) holds with γ 1 in placing of γ and (23) also holds for some other constants C 1 , C 2 .
By a similar way to prove (12), we can easily obtain the following version of Hopf's boundary point lemma, which is a generalization of [30, Lemma C.3].

Proposition 2.15 (Hopf's boundary point lemma for elliptic Waldenfels operators). Let D be an open set (not necessarily bounded or connected)
with boundary ∂D being C 2 . Assume that u ∈ C 2 (D), Lu ≥ 0 in D, and supp ν(x, ·) ⊂ D−x for each x ∈ D, and furthermore the mapping x → ν(x, ·) is continuous in D. Suppose that u achieves its (finite) maximum over D at point x 0 ∈ ∂D such that u(x 0 ) > u(x) for all x ∈ D, and that one of the following conditions holds: Then the outer normal derivative is positive: ∂u ∂n (x 0 ) > 0. In fact, if we let E = D and replace B by D in Step 1 in the proof of Theorem 2.8, also replace D < by D in Step 3, then the three-step argument also works in the context of Proposition 2.15 and the result follows.

Maximum principles for parabolic Waldenfels operators
We assume that D, E are two open sets in R d and D ⊂ E, where E is not necessarily bounded. Set D T := D × (0, T ] and E T := E × (0, T ] for arbitrarily fixed T > 0.
Uniform ellipticity condition: There exists a constant γ > 0 such that 3. Lévy measures: The kernel {ν(t, x, ·) | (x, t) ∈ R d × [0, T ]} is a family of Lévy measures, namely, each ν(t, x, ·) is a Borel measure on and moreover, for fixed Here we further assume that for each (x, t) ∈ D T , the measure ν(t, x, ·) is supported in The Markov process associated with such a generator L can be determined as a solution to the martingale problem induced by L (see, e.g., [29]). However, it is not clear if the Markov process determined by the martingale problem is linked to a stochastic differential equation with certain boundary conditions. Now we consider the parabolic Waldenfels operator with L being defined in (28), and we are concerned with the maximum principles for such a parabolic operator.

Weak maximum principle for parabolic case
We are in the position to present both weak and strong maximum principles for parabolic Waldenfels operator − ∂ ∂t + L. First we prove the weak one.
Here the supremum may be infinity.
Proof. Assertion 1. We prove this by contradiction. Suppose that the strict inequality holds, i.e., but there exists a point (x 0 , t 0 ) ∈ D T such that On one hand, as explained in the proof of Theorem 2.2, we note that Lu ≤ 0 at point (x 0 , t 0 ). On the other hand, if 0 < t 0 < T , then (x 0 , t 0 ) ∈ (D T ) • and consequently Thus we always have − ∂u ∂t + Lu ≤ 0 at point (x 0 , t 0 ), a contradiction to (31). In the general case that − ∂u ∂t + Lu ≥ 0 holds in D T , define with a positive parameter . Then and hence sup E T u = sup E T \D T u . Now Assertion 1 follows by setting → 0. Assertion 2. If u is nonpositive throughout D, Assertion 2 is trivially true. Hence we may assume on the contrary that u achieves a positive maximum at a point (x 0 , t 0 ) ∈ D T over E T .
We first consider the case with strict inequality − ∂u ∂t + Lu > 0 in D T . Since u(x 0 , t 0 ) > 0 and c ≤ 0, we derive the contradiction to Assertion 1, More generally, if − ∂u ∂t + Lu ≥ 0 in D T , then set as before u (x, t) := u(x, t) − t with > 0, which leads to Moreover, if u achieves a positive maximum at a point (x 0 , t 0 ) ∈ D T over E T , then by the continuity, u also achieves a positive maximum at a point (x 0 , t 0 ) ∈ D T over E T , provided that is small enough. However, as in the previous proof, we obtain a contradiction. This completes the proof. Remark 3.4. From Theorem 2.2 and Remark 2.4, we have already known that, for u ∈ C 2,2 (D T ) ∩ C(E T ), the supremum (or respectively, positive supremum) is achieved on E T \ (D T ) • . The alert reader could notice that we may have appeared to be cheating here, as we should also verify that the kernel ν still satisfy the third assumption in the definition of elliptic Waldenfels operator (1) when regarding it as a kernel in R d+1 . In fact, the modified kernelν((x, t), dzds) := ν(t, x, dz)δ 0 (ds) does satisfy the moment condition (3), which is enough for us even thoughν is not supported inside R d+1 \ {0}. See the proof of Lemma 3.11 for details.
Moreover, if − ∂u ∂t + Lu > 0 in D T , we see that the maximum (or respectively, positive maximum) cannot be achieved on the upper boundary D ×{T } by the same argument as in the proof of Theorem 3.1. Hence, it is clear that Theorem 3.1 holds for u ∈ C 2,2 (D T ) ∩ C(E T ), which is a natural consequence of Theorem 2.2.
However, the result for u ∈ C 2,1 (D T ) ∩ C(E T ) cannot be obtained in this way. We only need the first-order differentiability in t, benefiting from the form of the operator − ∂ ∂t + L. This is evident in the different forms of u in (11) and (32). 1. If c ≡ 0 in D T , then max Here Γ T is the parabolic boundary of D T , i.e., Γ T : There are some consequences of the weak maximum principle for a parabolic Waldenfels operator. We only highlight the following results.
3. If c ≤ 0 and − ∂u ∂t + Lu = 0 both hold in D T , then Here the supremum and infimum may be infinity.
There is no sign condition on c.

(Comparison Principle) If
Proof. In the case that c ≤ 0 in D T , the two conclusions are trivially followed by applying Corollary 3.6 to u − v.
For general case without any assumption on the sign of c, we only need to prove that if − ∂u ∂t + Lu ≤ 0 in D T and u ≥ 0 on E T \ D T , then u ≥ 0 in E T . Define u β := ue −βt . Then u ≥ 0 is equivalent to u β ≥ 0. We calculate Hence − ∂u ∂t + Lu ≤ 0 is equivalent to − ∂u β ∂t + (L − β)u β ≤ 0. Choose a sufficiently large β, we can ensure c − β, the zeroth-order coefficient of operator L − β, to be nonpositive in D T . By the preceding statements, we know u β ≥ 0 in E T , equivalently, u ≥ 0 in E T .
Proof. We consider the function u β := ue −βt like in the previous corollary. By the same argument we know that − ∂u ∂t + Lu = 0 is equivalent to − ∂u β ∂t + (L − β)u β = 0. The zeroth-order coefficient of operator L − β, i.e., c − β, is nonpositive in D T . Therefore, Assertion 3 of Corollary 3.6 implies that Our result follows.

Strong maximum principle for parabolic case
We now turn to the strong maximum principle for the parabolic Waldenfels operator − ∂ ∂t + L. Theorem 3.9 (Strong maximum principle for parabolic Waldenfels operators). Let D be an open and connected set but not necessarily bounded, and E be an open set satisfying D ⊂ E. Assume that u ∈ C 2,2 (D T ) ∩ C(E T ), − ∂u ∂t + Lu ≥ 0 in D T , and supp ν(t, x, ·) ⊂ E − x for each (x, t) ∈ D T . Moreover, assume that the mapping (x, t) → ν(t, x, ·) is continuous in D T . If one of the following conditions holds: 1. c ≡ 0 in D T and u achieves a (finite) maximum over E T at a point (x 0 , t 0 ) ∈ D T ; 2. c ≤ 0 in D T and u achieves a (finite) nonnegative maximum over E T at a point (x 0 , t 0 ) ∈ D T ; 3. u achieves a zero maximum over E T at a point ( A result of strong maximum principle for viscosity solutions of certain nonlinear nonlocal parabolic operators proved in [5] required a "nondegeneracy" condition, which is crucial in that context. But our strong maximum principle for linear nonlocal parabolic operator in Theorem 3.9 does not need this or any other conditions like this.
The converse case that − ∂u ∂t + Lu ≤ 0 in D T is immediate.
If one of the following conditions holds: 1. c ≡ 0 in D T and u achieves a (finite) minimum over E T at a point (x 0 , t 0 ) ∈ D T ; 2. c ≤ 0 in D T and u achieves a (finite) nonpositive minimum over E T at a point (x 0 , t 0 ) ∈ D T ; 3. u achieves a zero minimum over E T at a point (x 0 , t 0 ) ∈ D T , To prove the strong maximum principle, we will consider the horizontal propagation of maximizer point in space by the similar arguments in elliptic case, and further obtain the vertical propagation of maximizer point locally in time by the weak maximum principle in elliptic case.
Denote M := max E T u < ∞ for convenience. Under the assumptions in Theorem 3.9, that is, u ∈ C 2,2 (D T ) ∩ C(E T ), − ∂u ∂t + Lu ≥ 0 in D T , and u(x 0 , t 0 ) = M with point (x 0 , t 0 ) ∈ D T , supp ν(t, x, ·) ⊂ E − x for each (x, t) ∈ D T , and the mapping (x, t) → ν(t, x, ·) is continuous in D T . Furthermore, one of the following assumptions holds: The following lemma follows from Theorem 2.8 and Remark 2.14. Proof. If t 0 = T , the theorem is trivial. Hence we assume t 0 < T . Equivalently, (x 0 , t 0 ) is an interior point of D T .
Combining Remark 2.14 and the preceding arguments , we conclude that, as in Theorem 2.8, ∂u ∂n > 0 holds in the case that the unit outer normal vector n of (x 0 , t 0 ) over ∂B is not in the nullspace of a(x 0 , t 0 ), which is exactly the space N := {(x, t) | x = 0}. But this leads to a contradiction: since u attains a maximum at the interior point (x 0 , t 0 ), we have ∂u ∂n = 0. Therefore, (x 0 , t 0 ) must be a pole of ball B, whose unit outer normal vector is just in N . This completes the proof.
The next lemma shows that for every t ∈ (0, T ), we have either u(x, t) < M or u(x, t) = M for all x ∈ D. This means that the non-maximizer point (or the maximizer point) may propagation horizontally in space. The proof can be found in [24], we do not present it here and the main points of the proof can be found in Section 5.
Remark 3.13. In Lemma 3.12, we restrict the set D to be connected to make sure that the point (x 1 , t 0 ) can be chosen. More generally, if D is not connected, we may replace D in the previous proof with the connected component of D which contains the maximizer point. We thus conclude that for every fixed t ∈ (0, T ), either u(x, t) < M or u(x, t) = M holds in each connected component of D. Then as in Remark 2.12, we see that the diffusion term gives rise to the horizontal propagation of maximizer point in the corresponding connected component.
As in Remark 2.9, or from [5,7], we also see that the horizontal propagation of maximizer point by translation of measure support. Namely, if u(x 0 , t 0 ) = M with x 0 ∈ D and t 0 ∈ (0, T ), then u ≡ M on the set ∞ n=0 Λ n , whereΛ n 's are defined by induction, Thus in this scheme, the jump diffusion term leads to the horizontal propagation of maximizer point between those connected components, since jumps from one connected component to another might occur when measure supports overlap two or more connected components.
Furthermore, we present the final lemma we need. It means the maximizer point may propagate vertically in time in a local sense. The proof can also be found in Section 5 Lemma 3.14. Assume that u < M in D × (t 0 , t 1 ), with 0 ≤ t 0 < t 1 ≤ T . Then u < M in D × {t 1 }.
Finally we can prove Theorem 3.9.

Examples
We will give some examples in this section. These examples are all concerned with symmetric α-stable Lévy noise which are not covered in Taira's framework in [30], since the jump measure is of unbounded support.
where W t is a standard Wiener process, and L α t is a Lévy process with jump measure ν α (dz) = c α,d dz |z| d+α , for α ∈ (0, 2) and c α,d a positive constant depending on α and d, together with zero drift and zero diffusion. The generator for this system is the following elliptic Waldenfels operator Let D be a domain in R d . The mean exit time for X t , starting at x, exits firstly from D is denoted by τ (x). By Dynkin formula for such jump diffusion process [3,13], as shown in [8,21,25], we know that τ satisfies the following equation, By the strong maximum principle 2.8, or precisely Corollary 2.11 with the special case E = R d , we conclude that the mean exit time τ cannot take zero value inside D, unless it is constant (inside the domain D).
Example 4.2 (Escape probability). Similarly, let U be a subset of D c . The likelihood that X t , starting at x, exits firstly from D by landing in the target set U is called the escape probability from D to U , denoted by p(x). As shown in [23,20], the escape probability p satisfies the following equation, where W t is a standard Wiener process and L α t is a Lévy process with pure jump measure ν α (dz) = c α,d dz |z| d+α , for α ∈ (0, 2) and c α,d a positive constant depending on α and d. The Fokker-Planck equation for the probability density of the solution, as shown in [8,10], is Let D be a domain in R d . In this case, the coefficient of zeroth-order term is c = −∇ · b. We apply the strong maximum principle in Theorem 3.9 and Corollary 3.10 with E = R d . If ∇ · b ≡ 0, which means the deterministic vector field of stochastic system (34) is divergence-free, then the probability density p cannot attain its maximum (or minimum) over R d × [0, ∞) in D × [0, ∞), unless it is constant at all time before the maximizer (or minimizer) point. Moreover, if ∇ · b ≥ 0, then p cannot attain its maximum or zero minimum over R d × [0, ∞) in D × [0, ∞) (note that p only takes nonnegative values), unless it is constant at all time before this point as well.

Appendix: Proofs for lemmas
The proofs of some technical lemmas will be presented here for the sake of completeness.

Proof of Lemma 3.12
Assume on the contrary that D × {t 0 } contains some points at which u = M . Choose the point (x 1 , t 0 ) nearest to (x 0 , t 0 ) at the line {(x, t 0 ) | x ∈ D}, such that u(x 1 , t 0 ) = M , which is possible since D is connected and the set  Consider next the open ball B := B((x, t 0 ), ρ(x)) ⊂ R d+1 with center (x, t 0 ) and radius ρ(x). Then B ⊂ D T , u < M in B and ∂B contains points where u = M . Thus by Lemma 3.11 we conclude that either u(x, t 0 +ρ(x)) = M or u(x, t 0 − ρ(x)) = M . By the Pythagorean theorem, we assert ρ(x + e) 2 ≤ ρ(x) 2 + 2 , for sufficiently small | | > 0, where e is the unit vector along l. In the same way, ρ(x) 2 ≤ ρ(x + e) 2 + 2 . We note that which implies that This leads to for each x ∈ l 0 . Consequently, ρ(x) is a constant in l 0 , which is a contradiction since ρ(x) > 0 in l 0 and ρ(x) → 0 as x → x 1 . The proof is complete.
Using Taylor expansions and the fact that each ν(t, x, ·) is Lévy measure, we know that where C 2 is a positive constant independent of β. Hence, inB ∩ (E × (t 0 , t 1 )) provided β is sufficiently large. Figure 3: Sketch for Lemma 3.14.
This is a contradiction. Under Assumption 3, we may use the same argument as in the end of the proof of Theorem 2.8, that is, replacing L and c respectively with L − c + and c − c + and applying the acquired result for Assumption 2, the same result for this case follows.