FIRST EIGENVALUE OF ONE-DIMENSIONAL DIFFUSION PROCESSES

We consider the ﬁrst Dirichlet eigenvalue of diffusion operators on the half line. A criterion for the equivalence of the ﬁrst Dirichlet eigenvalue with respect to the maximum domain and that to the minimum domain is presented. We also describle the relationships between the ﬁrst Dirichlet eigenvalue of transient diffusion operators and the standard Muckenhoupt’s conditions for the dual weighted Hardy inequality. Pinsky’s result [ 17 ] and Chen’s variational formulas [ 8 ] are reviewed, and both provide the original motivation for this research.


Introduction and Main Results
In this paper, we deal with explicit bounds of the first Dirichlet eigenvalue for diffusion operators on the half line + := [0, ∞). The work is a continuation or a supplement of [17,5,6], and is also inspired by analogous research for birth-death processes in [7,19]. Let a(x) be positive everywhere on (0, ∞). For any measurable function b(x) on (0, ∞), define C(x) = on + with the Dirichlet boundary condition at x = 0. Then, L is a non-negative, self-adjoint operator on ( + , 2 (µ)), and it corresponds to a non-negative, Markovian symmetric and closable bilinear form (see [10]) defined for f , g ∈ C ∞ 0 ( + ), the space of smooth functions with compact support on + . As usual, denote by · and (·, ·) the norm and the inner product on 2 (µ), respectively. Let λ 0 be the first eigenvalue of Dirichlet diffusion operator −L. The classical Rayleigh-Ritz variational formula (c.f. 232 DOI: 10.1214/ECP.v14-1464 First Eigenvalue of One-dimensional Diffusion Processes 233 see [18,16]) gives us That is, Theorem 1.1 shows that the bounds for λ 0 take two possible forms depending on whether ∞ 0 e −C(x) d x is finite or infinite. For the clarity of exposition, we denote δ given in (1.2) and (1.3) by δ 1 and δ 2 , respectively. As mentioned in [17; Remark 3], it is δ 1 (not δ 2 ) that coincides with the standard Muckenhoupt's constant for the weighted Hardy inequality (H1): where u and v are non-negative weighted functions on + . Let C 1 be the optimal constant in (1.4). Then, [15] gives us Set u(x) = a(x) −1 e C(x) and v(x) = e C(x) . It follows that δ 1 = B 1 . Recently the Hardy inequality (1.4) has been extensively applied to studying the first non-trivial eigenvalue of diffusion operators and related functional inequalities (c.f. see [2,8,12,14,21]). For example, assume that the diffusion operator L is ergodic, i.e.
∞ 0 e −C(x) d x = ∞ and µ( + ) < ∞. The other first eigenvalue λ 0 , slightly different from λ 0 in (1.1), is given by where C 1 ( + ) denotes the space of continuously differential functions. Then, it has been proven in [8; The same estimations hold for λ 0 and λ 0 when the operator L is ergodic. However, since the class of admissible functions in (1.5) is larger than that in (1.1), it only follows that λ 0 ≥ λ 0 . In view of these facts, it is natural to question that whether λ 0 = λ 0 in this case. The answer is positive, and in fact we have a stronger assertion.   [11,13]). Thus, in this case the diffusion operator L determines the process uniquely ( [1]). Another viewpoint of (1.6) comes from the theory of Dirichlet forms. Let X min t be the minimal process generated by (D, C ∞ 0 ( + )), and X max  However, we will see that these bounds are closely connected with the dual Hardy inequality (H2): The difference between (1.4) and (1.7) only lies on small change (i.e. the range of integral inside) in the left hand side of these two equalities, but the assertions are significantly distinct. Actually, let C 2 be the optimal constant in (1.7). Then, by [15], The constant B 2 is completely different from B 1 associated with the Hardy inequality (1.4). Just like δ 1 in Theorem 1.1, we define δ T = B 2 by letting u(x) = a(x) −1 e C(x) and v(x) = e C(x) , i.e.
Then the following conclusion holds for λ 0 given by (1.1).

Theorem 1.3. For transient diffusion operators L on the half line
The absence of the condition f (0) = 0 in the definition (1.8) indicates that λ 0,T is in fact the first Neumann eigenvalue of transient diffusion operator L on + ; the proof of this equivalence being deferred to Section 2.2. As an alternative probabilistic viewpoint (c.f. see [7; Proposition 1.1]), λ 0,T is the rate of the exponential decay of transient diffusion process, and it is the largest ǫ such that P t f ≤ f e −ǫ t for all f ∈ C ∞ 0 ( + ). Theorem 1.3 establishes the relationships among the dual Hardy inequality (1.7), the first Dirichlet eigenvalue λ 0 and the first eigenvalue λ 0,T of transient diffusion operators. As a byproduct, Theorem 1.3 implies that λ 0 > 0 iff λ 0,T > 0, though it is true on general grounds that λ 0 ≥ λ 0,T .
The proofs of our theorems are presented in the next section. Here, Chen's variational formulas (c.f. [4,5,6]) for the first eigenvalue of ergodic diffusion processes are reviewed. We also use them to improve Theorem 1.1. Although we restrict ourselves on the half line in this paper, the corresponding results for the whole line, higher-dimensional situations and Riemannian manifolds follow from similar ideas in [17,5,6].

Improvement of Theorem 1.1 and Proof of Theorem 1.2
We begin with the proof of Theorem 1.2.
To prove our conclusion, it suffices to verify that Adopting the standard Feller's notations (c.f. see [11,13]) for one-dimensional diffusion operator, µ(d x) and s(x) := x 0 e −C(u) du are called speed measure and scale function, respectively. Moreover, L can be expressed as and the boundary behavior of the corresponding process is also described by speed measure and scale function. Particularly, ∞ is said to be not regular if and only if On the other hand, assume that then, the integration by parts formula yields and so it also holds that The required assertion follows by the above facts.
According to Theorem 1.2, in recurrent settings we can use Chen's variational formulas to improve the estimations for λ 0 in Theorem 1.1. For instance, define four classes of functions: Then, [4,5,6] give us the following two powerful variational formulas for λ 0 given by (1.1).

Moreover, both inequalities in (2.3) become equalities if a and b are continuous.
Next, we turn to the transient situation. To handle this case, [17] employs the h-transformed When written out, we get Letting Thus, the diffusion operator L h with diffusion coefficient a and the new drift b * is recurrent. Note that the spectrum is variant under h-transforms. So, λ 0 = λ h 0 , where λ h 0 is the first Dirichlet eigenvalue of L h on + with an absorbing boundary at x = 0. That is, the transformed operator L h reduces transient cases into recurrent ones. It immediately follows that (4δ Furthermore, which is just δ 2 defined in (1.3).
Again we use Chen's variational formulas to refine the estimations for λ 0 in Theorem 1.1. Theorem 2.1 along with the remark above yields that the following statement for λ 0 given by (1.1). For the completeness, we will prove that the formula (2.5) implies the second assertion in Theorem 1.1. By similar arguments, Theorem 2.1 also improves the first assertion in Theorem 1.1. Firstly, the proof of [5; Theorem 1.1] yields

Theorem 2.2. For transient diffusion operator L,
(2.6) In fact, take f (x) = Thus, The required assertion (2.6) follows. Secondly, we claim that For fixed du. Then f ∈ F I . For any y < x, f ′ ( y) = e −C * ( y) , and Since x is arbitrary, which gives us (2.7). Therefore, according to (2.6) and (2.7), the required conclusion follows. We end this subsection with an illustration of the improvements offered by Theorem 2.1 over the bounds provided in Theorem 1.

Proof of Theorem 1.3. (1) Define
Then, it holds that x 0 e C( y) d y. Thanks to the facts that h ′ ≤ 0 and h(∞) = 0, for x > 0, That is, This yields the first inequality in (2.9) upon taking the supremum with respect to x > 0. For any Thus, The second required inequality in (2.9) also follows. Now, if δ 2 = ∞, then δ * 2 = ∞. Assume that δ * 2 = ∞. If by using the decreasing property of h. Hence, the qualities δ 2 and δ * 2 are equivalent. The first required conclusion follows by this assertion and (2.9). Letting n → ∞ and then taking infimum with respect to m > 0, the second required assertion follows. The proof is complete. Remark 2.3. As shown by part (2) in the proof above, the admissible function f ∈ C ∞ 0 ( + ) for the definition of λ 0,T only satisfies that f (∞) = 0 not f (0) = 0. This is the crucial distinction between λ 0,T and λ 0 given by (1.5). Just due to this difference, one only links λ 0,T with the dual Hardy inequality (1.7), but connects λ 0 with the other Hardy inequality (1.4).
To conclude this section, we give a stronger conclusion (i.e. the variational formula) for λ 0,T than that in Theorem 1.3. Similar to Theorem 2.1, we need other four classes of functions: Theorem 2.4. The following Chen-type variational formulas are satisfied for λ 0,T : It is easy to check that lim l→∞ λ l 0,T = λ 0,T . (2.11) Note that λ l 0,T is just the first eigenvalue of −L with Neumann boundary at x = 0 and Dirichlet boundary at x = l. So λ l 0,T can be written as Then, where ← − I (resp. ← − II ) is the operator I(resp. II) in (2.3) by replacing a(x) and b(x) with a(l − x) and b(l − x), respectively. Changing variables yields the following Chen-type variational formulas for λ l 0,T : Now, the required assertion follows by combining (2.11) with (2.12) and letting l → ∞.
By using Theorem 2.4, one can also present the second proof of (1.9). We only give a sketch here.