Stability for a random evolution equation with Gau

J. Math. Anal. Appl. 272 (2002) 458–472 www.academicpress.com Stability for a random evolution equation with Gaussian perturbation ✩ Fubao Xi Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China Received 25 July 2000 Submitted by M. Iannelli Abstract In this paper we consider a random evolution equation which is perturbed by Gaussian type noise, and show how to construct its coupled solutions. On the basis of the coupling results, we discuss the asymptotic flatness and the stability in total variation norm for the solutions of the equation. In addition, we also prove the Feller continuity, and the existence and uniqueness of invariant measures of the solutions.  2002 Elsevier Science (USA). All rights reserved. Keywords: Coupling; Wasserstein metric; Stability; Asymptotic flatness; Random evolution equation 1. Introduction Let {(X(t),Z(t))} be a strong Markov process with the phase space Rd ×N , where N := {1,2, . . . , n0}. The first component {X(t)} satisfies the following random evolution equation with Gaussian perturbation: dX(t)= b(X(t),Z(t)) dt + σ (X(t),Z(t)) dB(t). (1.1) Let (Ω,F ,P ) be a probability space, {Ft } an increasing family of sub-σ - algebras of F , and {B(t)} an Ft -adapted Rd -valued Brownian motion and also ✩ Partially supported by the National Natural Science Foundation of China, Grant 19901001. E-mail address: xifubao@xinhuanet.com. 0022-247X/02/$ – see front matter  2002 Elsevier Science (USA). All rights reserved. PII: S0022-247X(02)0 01 63 -4 F. Xi / J. Math. Anal. Appl. 272 (2002) 458–472 459 a martingale with respect to {Ft }. The second component {Z(t)}, which is inde- pendent of {B(t)} and Ft -adapted, is a right-continuous Markov chain with finite state space N and Q-matrix (qkl) (see [4]). Moreover, suppose that qkl > 0 for all k �= 1. For x ∈Rd and σ = (σij ) ∈ Sd (the space of d × d matrices), define |x| = (∑ |xi |2 )1/2 , |σ | = (∑ |σij |2 )1/2 . For the existence and uniqueness of the solution to (1.1) we make the following assumption. Assumption 1.1. Both b(x, k) and σ(x, k) satisfy the Lipschitz condition and the linear growth condition as follows:∣∣b(x, k)− b(y, k)∣∣+ ∣∣σ(x, k)− σ(y, k)∣∣�H |x − y|, x, y ∈Rd, k ∈N, (1.2)∣∣b(x, k)∣∣+ ∣∣σ(x, k)∣∣�H (1+ |x|), (x, k) ∈Rd ×N, (1.3) for some constant H > 0. It is known (cf. [11]) that under Assumption 1.1, the stochastic differential equation (SDE) (1.1) has a unique continuous solution X(t). In the sequel we will always assume that Assumption 1.1 holds. Hence (1.1) and {Z(t)} together determine a unique strong Markov process {(X(t),Z(t))}. Stability for the random evolution equation with Gaussian perturbation has been well studied by many authors (see [3,7,9] and references therein). Basak et al. [3] studied the stability in distribution for the model (1.1) with linear drifts, that is, b(x, k)= A(k)x for every (x, k) ∈ Rd ×N , where A(k) ∈ Sd for each k ∈N . They discussed the asymptotic flatness for the flow X(t) and proved that the transition probability P(t, (x, k), ·) of {(X(t),Z(t))} converges weakly to some probability measure π(·) as t →∞, for every (x, k) ∈ Rd ×N . Previously, Basak and Bhattacharya [2] discussed the similar problems in the diffusion context, that is, the particular case of (1.1) with Z(t) removed. In the present paper, we consider Eq. (1.1) with nonlinear drifts, and show how to construct its coupled solutions. On the basis of the coupling results, we discuss the asymptotic flatness for both the flow X(t) in the pth mean (p � 1) and the transition probability P(t, (x, k), ·) of the Markov process {(X(t),Z(t))} in the Wasserstein metric, and also discuss the stability in total variation norm for {(X(t),Z(t))} (more precisely, the convergence in the total variation norm of the transition probability P(t, (x, k), ·) to the unique invariant measure π(·) of {(X(t),Z(t))} as t →∞, for every (x, k) ∈Rd ×N). Since the convergence in total variation norm leads to the weak convergence, so the stability in total variation norm implies the stability in distribution. In addition, we also prove the Feller continuity of {(X(t),Z(t))} and the existence and uniqueness of invariant measure for {(X(t),Z(t))}. To 460 F. Xi / J. Math. Anal. Appl. 272 (2002) 458–472 get these results, we mainly use the coupling methods which were well studied by [4,5] and references therein. The Wasserstein metric and the total variation norm are both defined by coupling (see Sections 2 and 3), and the total variation norm actually is also a kind of Wasserstein metric (cf. [4,5]). From [4] we know that the convergence in the Wasserstein metric is usually stronger than the weak convergence, so one can use the Wasserstein metric to study the asymptotic problems relating to probability metrics. For example, Barrio et al. [1] applied the Wasserstein metric to the studying of some central limit theorems. In this paper, we employ the Wasserstein metric to investigate the stability problems for Eq. (1.1). 2. Coupling and flatness In this section we construct the coupling of {(X(t),Z(t))} and itself. Using the coupling results, we also discuss the asymptotic flatness for the flow X(t) and the transition probability P(t, (x, k), ·) of {(X(t),Z(t))}. Now we first construct the basic coupling of the discrete component {Z(t)} and itself as follows. Let {(Z(t),Z′(t))} be the Markov process with phase space N × N and coupling operator (see [4]) Ωf (k, l)= ∑ m (qkm − qlm)+ ( f (m, l)− f (k, l)) + ∑ m (qlm − qkm)+ ( f (k,m)− f (k, l)) + ∑ m qkm ∧ qlm ( f (m,m)− f (k, l)), where f is a bounded function on N ×N . Set S = inf{t � 0: Z(t)=Z′(t)}. Then {Z(t)} and {Z′(t)} will move together from S onward. From [4] we know that one can construct many kinds of couplings of {Z(t)} and itself. However, in the sequel we shall always let {(Z(t),Z′(t))} be just the basic coupling constructed above. Noting that {Z(t)} is a finite state Markov chain with Q-matrix (qkl) satisfying qkl > 0 for all k �= l, and using [4, Corollary 5.21], it is not difficult to prove the following lemma. Lemma 2.1. For any k, l ∈N , we have P (k,l)(S <∞)= 1, where P (k,l) denotes the distribution of {(Z(t),Z′(t))} starting from (k, l). F. Xi / J. Math. Anal. Appl. 272 (2002) 458–472 461 For (x, k) ∈ Rd ×N , set a(x, k)= σ(x, k)σ (x, k)∗, (2.1) where σ(x, k)∗ denotes the transpose of σ(x, k). Now we make the following assumption. Assumption 2.2. For each k ∈ N , assume that c(x, y, k) is a d × d matrix such that a(x, y, k)= ( a(x, k) c(x, y, k) c(x, y, k)∗ a(y, k) ) is nonnegative definite for any (x, k) ∈ Rd ×N . For each family of {c(x, y, k): k ∈ N} satisfying Assumption 2.2, we can construct a coupling of {(X(t),Z(t))} and itself as follows. Recall that {(Z(t), Z′(t))} is the basic coupling constructed above. Set a ( t, x,Z(t), y,Z′(t) ) = ( a(x,Z(t)) c(t, x,Z(t), y,Z′(t)) c(t, x,Z(t), y,Z′(t))∗ a(y,Z′(t)) ) , where c(t, x,Z(t), y,Z′(t))= I[S,∞](t)c(x, y,Z(t)). Then let {(X(t), Y (t))} sat- isfy SDE in R2d d ( X(t) Y (t) ) = σ (t,X(t),Z(t), Y (t),Z′(t))dW(t)+( b(X(t),Z(t)) b(Y (t),Z′(t)) ) dt, (2.2) where σ ( t, x,Z(t), y,Z′(t) ) σ ( t, x,Z(t), y,Z′(t) )∗ = a(t, x,Z(t), y,Z′(t)), and {W(t)} is a Brownian motion independent of {(Z(t),Z′(t))} in R2d . Let P (x,k,y,l) denote the distribution of the coupling {(X(t),Z(t), Y (t),Z′(t))} starting from (x, k, y, l), and E(x,k,y,l) denote the corresponding expectation. Set T = inf{t � 0: X(t)= Y (t), Z(t)=Z′(t)}. Then T is the coupling time of {(X(t),Z(t))} and {(Y (t),Z′(t))}. Now we define a metric on Rd ×N as λ ( (x,m), (y,n) )= (ρ2(x, y)+ d(m,n))1/2, where ρ(x, y)= |x − y| = ( d∑ i=1 (xi − yi)2 )1/2 , d(m,n)= { 0, m= n, 1, m �= n. 462 F. Xi / J. Math. Anal. Appl. 272 (2002) 458–472 Let B(Rd × N) be the σ -algebra on Rd × N generated by λ(· , ·). We denote the transition probabilities of the strong Markov process {(X(t),Z(t))} by {P(t, (x, k),A): t � 0, (x, k) ∈Rd ×N , A ∈ B(Rd ×N)}. Definition 2.3. For p � 1, we define the Wasserstein metric between P(t, (x, k), ·) and P(t, (y, k), ·) as Wp ( P ( t, (x, k), ·),P (t, (y, k), ·)) = inf Q [∫ λp ( (x,m), (y,n) ) Q(dx,dm,dy, dn) ]1/p , where Q varies over all coupling probability measures with marginals P(t, (x, k), ·) and P(t, (y, k), ·). With the coupling constructed above and the Wasserstein metric in mind, we introduce the notion of asymptotic flatness for the flow X(t) and the transition probability P(t, (x, k), ·). Definition 2.4. The flow X(t) is asymptotically flat in the pth mean (p � 1) if there exists a coupling of {(X(t),Z(t))} and itself such that sup x,y∈K E(x,k,y,k)ρp(Xt , Yt )→ 0 as t →∞ (2.3) for every compact K ⊂Rd . Definition 2.5. The transition probability P(t, (x, k), ·) is asymptotically flat in Wp-metric (p � 1) if sup x,y,∈K Wp ( P ( t, (x, k), ·),P (t, (y, k), ·))→ 0 as t →∞ (2.4) for every compact K ⊂Rd . Assumption 2.6. For each k ∈ N , assume that Ck is a d × d symmetric positive definite matrix. For a family of {Ck: k ∈N} satisfying Assumption 2.6, set Dk(x)= √ x∗Ckx, Dk(x, y)= √ (x − y)∗Ck(x − y) for x, y ∈ Rd and k ∈ N . Then we can find positive constants √λmin and √λmax such that√ λmin |x|�Dk(x)� √ λmax |x|, (2.5)√ λmin ρ(x, y)�Dk(x, y)� √ λmax ρ(x, y) (2.6) F. Xi / J. Math. Anal. Appl. 272 (2002) 458–472 463 for x, y ∈ Rd and k ∈ N . Actually, for fixed k ∈ N , Dk(x, y) is the Mahalanobis distance between x and y . For a family of {c(x, y, k): k ∈ N} satisfying Assumption 2.2 and a family of {Ck: k ∈N} satisfying Assumption 2.6, set A(x,y, k)= a(x, k)+ a(y, k)− 2c(x, y, k), �A(x,y, k)= 〈Ck(x − y),A(x, y, k)Ck(x − y)〉/D2k (x, y), x �= y, Bˆ(x, y, k)= 〈Ck(x − y), b(x, k)− b(y, k)〉 for x, y ∈ Rd and k ∈N , where 〈· , ·〉 denotes the inner product in Rd . Theorem 2.7. Assume that there exist a family of {c(x, y, k): k ∈ N} satisfying Assumption 2.2, a family of {Ck: k ∈N} satisfying Assumption 2.6, and constants p � 1 and γ > 0 such that tr ( A(x,y, k)Ck )− (2−p)�A(x,y, k)+ 2Bˆ(x, y, k) + 2 p D 2−p k (x, y) ∑ l �=k qkl ( D p l (x, y)−Dpk (x, y) ) �−γD2k (x, y) (2.7) for x, y ∈ Rd and k ∈N . Then E(x,k,y,k)ρp(Xt , Yt )� (λmin)−p/2Dpk (x, y) exp ( −p 2 γ t ) → 0 (2.8) as t →∞, and Wp ( P ( t, (x, k), ·),P (t, (y, k), ·)) � (λmin)−1/2Dk(x, y) exp ( −γ 2 t ) → 0 (2.9) as t →∞. Obviously, (2.8) and (2.9) imply that the flow X(t) is asymptotically flat in the pth mean and that the transition probability P(t, (x, k), ·) is asymp- totically flat in Wp-metric, respectively. Proof. Let L(k) denote the differential operator defined by a(x, y, k) and b(x, y, k)= (b(x, k), b(y, k))∗. Then L(k)D p k (x, y)= p 2 D p−2 k (x, y) [ tr ( A(x,y, k)Ck ) − (2− p)�A(x,y, k)+ 2Bˆ(x, y, k)]. Due to the marginality of the coupling operator Ω , we know that ΩD p k (x, y)= ∑ l �=k qkl ( D p l (x, y)−Dpk (x, y) ) 464 F. Xi / J. Math. Anal. Appl. 272 (2002) 458–472 (cf. [3, Section 5.2]). Therefore, (2.7) is equivalent to( L(k)+Ω)Dpk (x, y)�−p2 γDpk (x, y). Then, by the product rule, we have( L(k)+Ω)(Dpk (x, y) exp(p2 γ t )) = exp ( p 2 γ t )[( L(k)+Ω)Dpk (x, y)+ p2 γDpk (x, y) ] � 0 (2.10) for x, y ∈ Rd and k ∈N . Now set Sm = inf { t � 0: |Xt − Yt |>m } , TR = inf { t � 0: |Xt |2 + |Yt |2 >R } , V = Tr ∧ Sm. Using [11, Lemma 3 in Chapter II] and (2.10), we have E(x,k,y,k)D p Z(t∧V )(Xt∧V ,Yt∧V ) exp [ p 2 γ (t ∧ V ) ] −Dpk (x, y) =E(x,k,y,k) t∧V∫ 0 ( L ( Z(s) )+Ω)(DpZ(s)(Xs,Ys) exp(p2 γ s )) ds � 0, and so E(x,k,y,k)D p Z(t∧V )(Xt∧V ,Yt∧V ) exp [ p 2 γ (t ∧ V ) ] �Dpk (x, y). Letting R ↑∞, m ↑∞, we get E(x,k,y,k)D p Z(t)(Xt , Yt )�D p k (x, y) exp ( −p 2 γ t ) . Combining this with (2.6), we obtain (2.8). In view of Definition 2.3, we get (2.9) from (2.8). This completes the proof. ✷ Example 2.8. Let (1.1) be the following Ornstein–Uhlenbeck form equation with switching according to the movement of the Markov chain {Z(t)}: dX(t)=−βZ(t)X(t) dt + dB(t), (2.11) where each constant βk > 0 for k ∈ N . Let I denote the d × d identity matrix. Take c(x, y, k)= I and Ck = I for x, y ∈ Rd and k ∈ N . Then (2.7) holds with p = 2 and γ = 2 min{βk: k ∈ N}. Hence we obtain from Theorem 2.7 that the flow X(t) is asymptotically flat in the second mean and the transition probability P(t, (x, k), ·) of {(X(t),Z(t))} is asymptotically flat in W2-metric. F. Xi / J. Math. Anal. Appl. 272 (2002) 458–472 465 Before concluding this section, we present a result on the successful coupling which will be used for the studying of the stability in total variation norm in the next section. Recall the definition of A(x,y, k) and set A˜(x, y, k)= 〈x − y,A(x, y, k)(x− y)〉/|x − y|2, x �= y, B˜(x, y, k)= 〈x − y, b(x, k)− b(y, k)〉 for x, y ∈ Rd and k ∈N . Now, for a fixed family of {c(x, y, k): k ∈N} satisfying Assumption 2.2, choose γ (·) ∈ C(R+) such that γ (r)� sup {[ trA(x,y, k)− A˜(x, y, k)+ 2B˜(x, y, k)]/A˜(x, y, k): ρ(x, y)= r, k ∈N} and α(·) ∈C(R+) such that 0� α(r)� inf { A˜(x, y, k): ρ(x, y)= r, k ∈N}. Define C(r)= exp [ r∫ 1 γ (u) u du ] , f (r)= r∫ 1 C(s)−1 ds, g(r)= 1∫ r C(s)−1 ds 1∫ s C(u) α(u) du, r > 0, f (∞)= lim r→∞f (r), g(0)= limr→0g(r). Theorem 2.9. If α(r) > 0 on (0,∞), f (∞) =∞ and g(0) <∞, then the cor- responding coupling is successful; that is, P (x,k,y,l)(T <∞)= 1, (x, k) �= (y, l). Proof. We first prove that P (x,k,y,k)(T <∞)= 1. (2.12) To do so, following [5], set Sm = inf { t � 0: |Xt − Yt |>m } , m > 1, Tn = inf { t � 0: |Xt − Yt |< 1 n } , n > 1, Tn,m = Tn ∧ Sm, Fn,m(ρ)=− ρ∫ 1/n C(s)−1 ds m∫ s C(u) α(u) du, 1 n � ρ �m, n,m> 1. 466 F. Xi / J. Math. Anal. Appl. 272 (2002) 458–472 Then we can verify that 2L(k)Fn,m ( ρ(x, y) ) � 1 for all those x, y ∈ Rd with 1/n � ρ(x, y) � m and all k ∈ N , where L(k) is defined as in the proof of Theorem 2.7. Put r = ρk(x, y). Using [11, Lemma 3 in Chapter II], we have E(x,k,y,k)Fn,m ( ρ(Xt∧Tn,m, Yt∧Tn,m) )− Fn,m(r) =E(x,k,y,k) t∧Tn,m∫ 0 L ( Z(u) ) Fn,m ( ρ(Xu,Yu) ) du� 1 2 E(x,k,y,k)(t ∧ Tn,m). Then, using the proof of [5, Theorem 4.2], we obtain (2.12). Next, using Lemma 2.1, we have P (x,k,y,l)(T <∞)= P (x,k,y,l)(T <∞, S <∞) = P (x,k,y,l)(S = T ,S <∞)P (x,k,y,l)(T <∞ | S = T ,S <∞) + P (x,k,y,l)(S < T,S <∞)P (x,k,y,l)(T <∞ | S < T,S <∞). Obviously, the first conditional probability equals 1. By virtue of strong Markov property and (2.12), we know that the second conditional probability also equals 1. Finally, by Lemma 2.1 again, we prove Theorem 2.9. ✷ 3. Stability In this section, we use the coupling results obtained in the previous section to investigate the stability of the Markov process {(X(t),Z(t))}. We now introduce the notion of stability in total variation norm. Definition 3.1. The process {(X(t),Z(t))} is said to be stable in total variation norm if there exists a probability measure π(·) such that its transition probability P(t, (x, k), ·) converges to π(·) in total variation norm as t → 0 for every (x, k) ∈ Rd ×N . Remark 3.2. If the above convergence in total variation norm is replaced by the weak convergence, then one gets the notion of stability in distribution which was studied in [3]. Since the convergence in total variation norm leads to the weak convergence, so the stability in total variation norm implies the stability in distribution. In order to formulate our stability result precisely, we should find the lim- iting probability measure π(·). In fact, we will prove that the Markov process F. Xi / J. Math. Anal. Appl. 272 (2002) 458–472 467 {(X(t),Z(t))} has a unique invariant measure which just is the limiting probabil- ity π(·). Now, we first discuss the Feller continuity of {(X(t),Z(t))}. Lemma 3.3. The family of transition probabilities {P(t, (x, k), ·): (x, k) ∈ Rd × N} of the strong Markov process {(X(t),Z(t))} is Feller continuous. Proof. Since N has the discrete metric, we need only to prove that, for any t � 0, x, y ∈Rd and k ∈N , P(t, (x, k), ·) converges weakly to P(t, (y, k), ·) as x→ y . To this end, by means of [4, Theorem 5.6], it suffices to prove that W2 ( P ( t, (x, k), ·),P (t, (y, k), ·))→ 0 as x→ y. (3.1) In order to prove (3.1), we construct a suitable coupling of {(X(t),Z(t))} and itself. To do so, according to the construction method given in the previous section, we now only need to choose a suitable family of {c(x, y, k): k ∈ N} satisfying Assumption 2.2. Let c(x, y, k)= σ(x, k)σ (x, k)∗ for (x, k) ∈ Rd ×N . Then the family of {c(x, y, k): k ∈ N} satisfies Assumption 2.2. Therefore, via Eq. (2.2), we get a specific coupling {(X(t),Z(t), Y (t),Z′(t))}. Similarly to the proof of Theorem 2.7, set Sm = inf { t � 0: |Xt − Yt |>m } , TR = inf { t � 0: |Xt |2 + |Yt |2 >R } , V = TR ∧ Sm. Using [11, Lemma 3 in Chapter II], we have E(x,k,y,k)ρ2(Xt∧V ,Yt∧V ) = ρ2(x, y)+E(x,k,y,k) t∧V∫ 0 [ trA(Xu∧V ,Yu∧V ,Zu∧V ) + 2〈Xu∧V − Yu∧V , b(Xu∧V ,Zu∧V )− b(Xu∧V ,Zu∧V )〉]du, (3.2) where A(x,y, k)= a(x, k)+ a(y, k)− 2σ(x, k)σ (x, k)∗. On the other hand, by some elementary computation and (1.2), we have trA(x,y, k)= ∣∣σ(x, k)− σ(x, k)∣∣2 �H 2|x − y|2 and 〈 x − y, b(x, k)− b(y, k)〉�H |x − y|2. Combining these two estimates with (3.2), we get E(x,k,y,k)ρ2(Xt∧V ,Yt∧V ) � ρ2(x, y)+ (H 2 + 2H) t∫ 0 E(x,k,y,k)ρ2(Xu∧V ,Yu∧V ) du. 468 F. Xi / J. Math. Anal. Appl. 272 (2002) 458–472 Thus, by the Gronwall inequality (cf. [6, Lemma 1.1 in Chapter 2]), we obtain E(x,k,y,k)ρ2(Xt∧V ,Yt∧V )� ρ2(x, y) exp [ (H + 1)2t]. Finally, letting R ↑∞, m ↑∞, we arrive at E(x,k,y,k)ρ2(Xt , Yt )� ρ2(x, y) exp [ (H + 1)2t]. Clearly, this is equivalent to E(x,k,y,k)λ2 ( (Xt ,Zt ), (Yt ,Z ′ t ) ) � ρ2(x, y) exp [ (H + 1)2t], which implies (3.1). The proof is finished. ✷ For A ∈ B(Rd ×N) and τ � 1, set ντ (A)= 1 τ τ∫ 0 P ( t, (x, k),A ) dt, where (x, k) denotes the initial point of {(X(t),Z(t))}. Then {ντ (·): τ � 1} is a family of probability measures on (Rd ×N,B(Rd ×N)). Lemma 3.4. Assume that there exist a family of {Ck: k ∈ N} satisfying Assump- tion 2.6 and constants q > 0, B � 0 and β > 0 such that tr ( a(x, k)Ck )− (2− q)〈Ckx, a(x, k)Ckx〉/D2k (x)+ 2〈Ckx, b(x, k)〉 + 2 q D 2−q k (x) ∑ l �=k qkl ( D q l (x)−Dqk (x) ) � B − βD2k (x) (3.3) for (x, k) ∈ Rd × N . Then {(X(t),Z(t))} has an invariant probability measure π(·). Proof. Let L1(k) denote the differential operator defined by a(x, k) and b(x, k). Then L1(k)D q k (x)= q 2 D q−2 k (x) [ tr ( a(x, k)Ck ) − (2− q)〈Ckx, a(x, k)Ckx〉/Dqk (x)+ 2〈Ckx, b(x, k)〉]. Let Q denote the operator defined by the Q-matrix (qkl) as in [4]. Then QD q k (x)= ∑ l �=k qkl ( D q l (x)−Dqk (x) ) . Combining these with (3.3), we have( L1(k)+Q ) D q k (x)� B q 2 D q−2 k (x)− β q 2 D q k (x) F. Xi / J. Math. Anal. Appl. 272 (2002) 458–472 469 for (x, k) ∈ Rd ×N . Essentially, this is equivalent to( L1(k)+Q ) D q k (x)� B − βDqk (x) for (x, k) ∈ Rd ×N , where constants B � 0 and β > 0 may be different from the previous ones. From this, it is easy to get that (βRq −B)I{Dk(x)>R}(x) � sup{B − βRq : R � 0} − (L1(k)+Q)Dqk (x) (3.4) for (x, k) ∈ Rd ×N . From (2.5), we have{ DZ(s) ( X(s) ) >R }⊃ {√λmin ∣∣X(s)∣∣>R}. It follows from this and (3.4) that (βRq −B)I{√λmin |X(s)|>R}(x) � sup{B − βRq : R � 0} − (L1(Z(s))+Q)DqZ(s)(X(s)) for sufficiently large R. On the other hand, using [11, Lemma 3 in Chapter II] we get that E(x,k)D q Z(s) ( X(s) )−Dqk (x)=E(x,k) t∫ 0 ( L1 ( Z(s) )+Q)DqZ(s)(X(s)) ds, where the superscript (x, k) means the initial condition (X(0),Z(0)) = (x, k). Combining these two facts, we can further prove that for any t � 1, 1 t t∫ 0 P ( s, (x, k),U ( R/ √ λmin )C ×N)ds � ( sup{B − βRq : R � 0} +Dqk (x)/t )/( βRq −B), where U(R/ √ λmin)C := {x ∈ Rd : |x|>R/√λmin}. This implies that the family of probability measures {ντ (·): τ � 1} is tight. So {ντ (·): τ � 1} contains a subsequence {ντ(m)(·): m � 1} such that τ (m)→∞ (as m→∞) and ντ(m)(·) converging weakly t