Some conditional limiting theorems for symmetric Markov processes with tightness property

Let X be an µ -symmetric irreducible Markov process on I with strong Feller property. In addition, we assume that X possesses a tightness property. In this paper, we prove some conditional limiting theorems for the process X . The emphasis is on conditional ergodic theorem. These results are also discussed in the framework of one-dimensional diffusions.


Introduction
Let X = (Ω, X t , P x , T ) be an µ-symmetric irreducible Markov process on a locally compact separable metric space I, where µ is a positive Radon measure with full support and T is the lifetime. We denote by B(I) the Borel σ-field. Adjoining an extra point ∂ to the measurable set (I, B(I)). Set I ∂ = I ∪ {∂} and B(I ∂ ) = B(I) ∪ {A ∪ {∂} : A ∈ B(I)}. As usual, we denote by P x the law of the process starting from x and by P π the law of the process starting from a distribution π. The corresponding expectations are respectively denoted by E x and E π . We assume that the process X is a right Markov process on I with a finite lifetime T := inf{t > 0 : X t = ∂}, i.e., for all x ∈ I, For such a process, one of the fundamental problems is to study its long-term asymptotic behavior conditional on {T > t}. A closely related topics is quasi-ergodic distribution (see, e.g., [1]), i.e., a probability distribution m on I satisfying that for any x ∈ I and A ∈ B(I), where 1 A denotes the indicator function of the set A.
When T = ∞, under some suitable conditions, it is well-known that m coincides with ρ 0 . When T < ∞, it's a little surprising that they have this difference. This difference is worth further study. This paper is a continuation of studying quasi-ergodic distributions for symmetric Markov processes. Our aim of this work is to study the existence of a quasi-ergodic distribution and prove some mixing properties for the symmetric Markov processes.
Recently, the study of quasi-ergodic distributions has received more and more attention. When the absorbing boundary is fixed, the study on quasi-ergodic distribution in a very general framework can be found in [1,4,5,6,7,10,19]. In these works, some works need to assume that the process is λ-positive (see, e.g., [1,6,7]). Most of these works assume that the reference measure is a finite measure. If the reference measure is an infinite measure, it is more difficult to prove that the λ-invariant measure is a finite measure, which is the basis for the existence of quasi-stationary distributions and quasi-ergodic distributions. In the present paper, we will consider this case. To be exact, our main results are true whether the reference measure is finite or infinite. When the absorbing boundary is moving, the quasi-stationarity and quasi-ergodicity of discrete-time Markov chains were also studied by Oçafrain [14].
In this work, under suitable assumptions, we prove some mixing properties and provide a conditional ergodic theorem for the symmetric Markov processes. These results can give some interpretation of 'quasi-stationarity' of the quasi-ergodic distribution m.
These results also exhibit a phase transition due to the limiting distribution changes from 0 < p < 1 to p = 1. Finally, we are committed to studying these results in the framework of one-dimensional diffusions absorbed at 0.
The content of this paper is organized as follows. In Section 2, we present some preliminaries that will be needed in the sequel. Our main results and their proofs are presented in Section 3. In Section 4, we study the case of one-dimensional diffusions taking values in [0, ∞), where 0 is an absorbing regular boundary and +∞ is an entrance boundary.

Preliminaries
Before we state the main results of this paper, let us present some preliminaries. We define the semigroup and the resolvent by where B b (I) denotes the space of bounded Borel functions on I. Denote by r(t, x, y) the transition density function of the process X. We assume that the process X is symmetric with respect to the reference measure µ, i.e., for f, g ∈ B b (I), P t f, g µ = f, P t g µ , where the inner product f, g µ := I f (u)g(u)µ(du).
(H2) (Strong Feller property) For each t, P t (B b (I)) ⊂ C b (I), where C b (I) is the space of bounded continuous functions on I.
(H3) (Tightness) For any ε > 0, there exists a compact set K such that where K c denotes the complement of the compact set K.
The hypothesis (H) implies that for any a > 0, there exists a compact set K such that where τ K c denotes the first exit time from K c (see [17]). Let (E, D(E)) be the Dirichlet form on L 2 (I, µ) generated by X: Let λ be the bottom of spectrum of the infinitesimal operator of the process X, defined where · 2 denotes the L 2 -norm. Studying the existence of quasi-stationary distributions and quasi-ergodic distributions, it needs the condition that λ > 0. Under hypothesis (H), from [15, Corollary 3.8], we know that λ > 0 and for 0 < a < λ, sup Under the conditions that hypothesis (H) holds and φ 0 ∈ L 1 (I, µ) ∩ L ∞ (I, µ), Miura proved that for all A ∈ B(I), He has also proved that ν is the unique quasi-stationary distribution of the process X (see [ From the above results, we can see that, under hypothesis (H), φ 0 ∈ L 1 (I, µ)∩L ∞ (I, µ). Only under hypothesis (H), Takeda also showed that ν is the unique quasi-stationary distribution of the process X (see [16,Theorem 3.1]). Inspired by [13] and [16], in this paper, we study quasi-ergodic distributions for the process X, which is substantially different from the above limit distribution.

Main results
We will present our main results and their proofs in this section. Note that, our main results are true whether the reference measure is finite or infinite. The following theorem is one of our main results. Under the assumptions that the reference measure is a finite measure and the semigroup {P t } t≥0 is ultracontractive, that is, for any t > 0, there exists a constant c t > 0 such that such a form of theorem has been given in [19,Theorem 3.2]. Compared with [19], we don't need harsh constraints and our approach only uses elementary probability tools and can be easily applied to many other settings. Theorem 3.1. Assume that hypothesis (H) holds. Then, for any bounded and measurable functions f, g on I and 0 < p < q < 1, we have and ν is as in (2.1).
Proof. (i) Our results depend on the existence of the ground state φ 0 . Thanks to Proposition 2.2, we know that the ground state φ 0 of the Dirichlet form (E, D(E)) exists. Thus, we have I φ 2 0 (y)µ(dy) = 1. Then, m is a probability measure on I. Besides, from Proposition 2.3 and Proposition 2.4, we know that the ground state φ 0 ∈ L 1 (I, µ) ∩ L ∞ (I, µ).
For all r ≥ u, by the definition of h u (x), we get Therefore, the function f h u φ 0 is bounded and measurable. Thus, by (3.1), we obtain From the definition of h u (x), we can see that Then, letting u → ∞, by the monotone convergence theorem, we obtain Conversely, since f and g are bounded, we can repeat the argument, replacing f (X pt )g(X t ) by   We can extend (3.4) to arbitrary bounded f and g by subtraction. So (i) holds.
(ii) For each u, when (1 − q)t ≥ u and (q − p)t ≥ u, by the Markov property, we obtain For all r ≥ u, by the definition of h u (x) and h u (x), we get Therefore, the function f h u φ 0 is bounded and measurable. Moreover, as seen from the proof of (i), we have h u (x) ↑ φ 0 , 1 µ I g(y)m(dy), as u → ∞.
Thus, the proof of (ii) then follows from the same arguments as in the proof of (i). (iii) By (i) or (ii) and the dominated convergence theorem, we get Hence, there exists a quasi-ergodic distribution for the process X. As an interesting application of Theorem 3.1, we obtain the following conditional functional weak laws of large numbers. Under an admittedly more restricted setting, such a form of result has appeared in [19,Theorem 3.6]. However, compared with [19], we don't need harsh constraints. Proposition 3.3. Assume that hypothesis (H) holds. Then, for any bounded and measurable function f on I and any ε > 0, we have Proof. First note that for any bounded and measurable function f on I, we have This completes the proof of the proposition.
Next, we remark that if the semigroup {P t } t≥0 is intrinsically ultracontractive, then for any probability measure π on I, Theorem 3.1 is still holds. The semigroup {P t } t≥0 is said to be intrinsically ultracontractive, if for any t > 0, there exist two constants α t , c t > 0 such that α t φ 0 (x)φ 0 (y) ≤ r(t, x, y) ≤ c t φ 0 (x)φ 0 (y) for x, y ∈ I. Before we give the result, let's first prove the following result.
Proof. The main method of the proof is similar to that in [2, Theorem 5.1]. The approach is to use the following condition, which is actually equivalent to the exponential convergence (3.6) (see [3, Theorem 2.1])
As seen from the proof of Theorem 3.1, the proof only uses the fact that the equality (3.1) holds and φ 0 ∈ L 1 (I, µ) ∩ L ∞ (I, µ). If the semigroup {P t } t≥0 is intrinsically ultracontractive, we know from Proposition 3.4 or [16,Lemma 3.7] that for any initial distribution π on I, lim t→∞ I e λt P t g(x)π(dx) φ 0 , 1 π = I g(y)φ 0 (y)µ(dy). (3.8) That is to say, for any initial distribution π on I, the equality (3.1) holds. Thus, due to the reason mentioned above, by using a similar argument as in the proof of Theorem 3.1, we have the following result. We point out that Theorem 3.5 complements results of [19] for the reference measure being a finite measure and the semigroup {P t } t≥0 being ultracontractive.
Theorem 3.5. Assume that hypothesis (H) holds. If the semigroup {P t } t≥0 is intrinsically ultracontractive, then for any bounded and measurable functions f, g on I, any probability measure π on I and 0 < p < q < 1, we have where m and ν are as in Theorem 3.1.

One-dimensional diffusions
This section is devoted to study the case of one-dimensional diffusions taking values in [0, ∞), where 0 is an absorbing regular boundary and +∞ is an entrance boundary. Let Y = (Y t , t ≥ 0) be a one-dimensional drifted Brownian motion in [0, ∞) such that 0 is an absorbing boundary. More formally, Y is defined as the solution of the stochastic differential equation where (B t , t ≥ 0) is a standard one-dimensional Brownian motion and q(x) ∈ C 1 [0, ∞). Define Q(y) := y 0 2q(x)dx. In this section, we will use the following hypothesis.
According to [18], we know that if hypothesis (B) holds, then 0 is an absorbing regular boundary and +∞ is an entrance boundary. And, for all x ∈ (0, ∞), we have  Notice that µ is the speed measure of the process Y . From [18], we know that if hypothesis (B) holds, then µ(0, ∞) < ∞.
It is well known (see, e.g., [8]) that the one-dimensional diffusion process Y is symmetric with respect to µ and satisfies (H1) and (H2). When studying quasi-stationary distributions and quasi-ergodic distributions of one-dimensional diffusion processes, it is often necessary to classify the boundary: regular boundary, exit boundary, entrance boundary and natural boundary. Following [11,Chapter 5], we know that: We remark that if hypothesis (B) holds, then for all x ∈ (0, ∞), the equality (3.1) holds (see [18]). Hence, if hypothesis (B) holds, then for the process Y , "µ-a.e. x" in Theorem 3.1 can be strengthened to "all x".
This means that for any initial distribution π on (0, ∞), the equality (3.1) holds and then ν(dy) = φ 0 (y)µ(dy) is the unique quasi-stationary distribution of the process Y . Due to the same reason mentioned in Section 3, by using a similar argument as in the proof of Theorem 3.1, we have the following result. where m(dy) = φ 2 0 (y)µ(dy) and ν is as in (4.3).