Criteria for transience and recurrence of regime-switching diffusion processes

We provide some on-off type criteria for recurrence and transience of regime-switching diffusion processes using the theory of M-matrix and the Perron-Frobenius theorem. State-independent and state-dependent regime-switching diffusion processes in a finite space and a countable space are both studied. We put forward a finite partition method to deal with switching process in a countable space. As an application, we improve the known criteria for recurrence of linear regime-switching diffusion processes, and provide an on-off type criterion for a kind of nonlinear regime-switching diffusion processes.

and references therein for more details on their application. The regime-switching diffusion process (for short, RSDP) studied in this work can be viewed as a number of diffusion processes modulated by a random switching device or as a diffusion process which lives in a random environment. More precisely, RSDP is a two-component process (X t , Λ t ), where (X t ) describes the continuous dynamics, and (Λ t ) describes the random switching device. (X t ) satisfies the stochastic differential equation (for short, SDE) for δ > 0. The Q-matrix Q x = (q kl (x)) is irreducible and conservative for each x ∈ R d . If the Q-matrix (q kl (x)) does not depend on x, then (X t , Λ t ) is called a state-independent RSDP ; otherwise, it is called a state-dependent one. When N is finite, namely, (Λ t ) is a Markov chain on a finite state space, we call (X t , Λ t ) a RSDP in a finite state space. When N is infinite, we call (X t , Λ t ) a RSDP in an infinite state space. Next, we collect some conditions used later.
(H) There exists constantK > 0 such that (i) x → q ij (x) is a bounded continuous function for each pair of i, j ∈ M.
Here and in the sequel, σ * stands for the transpose of matrix σ, and σ denotes the operator norm. Hypothesis (Hi),(Hii) and (Hiii) guarantee the existence of a unique nonexplosive solution of (1.1) and (1.2) (cf. [16,Theorem 2.1]). Hypothesis (Hiv) is used to ensure that (X t , Λ t ) possesses strong Feller property (cf. [15], [17]), which will be used in the study of exponential ergodicity.
Corresponding to the process (X t , Λ t ), there is a family of diffusion processes defined by dX (i) for each i ∈ M. These processes (X (i) t ) (i ∈ M) are the diffusion processes associated with (X t , Λ t ) in each fixed environment. The recurrent behavior of (X t , Λ t ) is intensively connected with its recurrent behavior in each fixed environment. But this connection is rather complicated as having been noted by [11]. In [11], some examples in [0, ∞) with reflecting boundary at 0 and M = {1, 2} were constructed. They showed that even when (X (1) t ) and (X (2) t ) are both positive recurrent (transient), (X t , Λ t ) could be transient (positive recurrent, respectively) by choosing suitable transition rate (q ij ) between two states. In view of this complicity, it is a challenging work to determine the recurrent property of a regime-switching diffusion process.
There are lots of work having been dedicated to this task. See, for instance, [3,11,10,16,4] and references therein. Except giving the examples we mentioned above, [11] also studied the reversible state-independent RSDP . In [10], the author provided a theoretically complete characterization of recurrence and transience for a class of state-independent RSDP , which we will state more precisely later. In [4], some on-off type criteria were established to justify the exponential ergodicity of state-independent and state-dependent RSDP in a finite state space. The convergence in total variation norm and in Wasserstein distance were both studied in [4]. The cost function used to define the Wasserstein distance in [4] is bounded. All the previously mentioned work considered only the RSDP in a finite state space. Although the general criteria by the Lyapunov functions for Markov processes still work for RSDP , it is well known that finding a suitable Lyapunov function is a difficult task for RSDP due to the coexistence of generators for diffusion process and jump process. So it is better to provide some easily verifiable criteria in terms of the coefficients of diffusion process (X t ) and the Q-matrix of (Λ t ). In this direction, [17] has provided some criteria for a class of state-dependent RSDP (X t , Λ t ) in a finite state space. Precisely, the continuous component (X t ) considered in [17] behaves like a linear one and Q-matrix (q ij (x)) behaves like a state-independent Q-matrix (q ij ) in a neighborhood of ∞. In addition, the recurrent property for geometric Brownian motion in a two-state random environment was studied in [12].
In [13], we studied the ergodicity for RSDP in Wasserstein distance. Both state-independent and state-dependent RSDP in finite and infinite state spaces are studied in [13]. The cost function used in [13] is not necessarily bounded. We put forward some new criteria for ergodicity based on the theory of M-matrix and Perron-Frobenius theorem. Our present work is devoted to studying the recurrent property of RSDP in total variation norm. Compared with [4,13], the on-off type criteria given there own only one hand " on ", that is, if the condition holds, then the process is ergodic. Examples can show these criteria are sharp. In the present work, we shall show that these on-off type criteria can own two hands, both " on " and " off ", that is, if the condition holds, then the process is recurrent, and if not, then the process is transient.
Another contribution of this work is that we put forward a finite partition method to study the transience and recurrence for state-dependent RSDP in an infinite state space. Up to our knowledge, there is few result for the recurrent property of RSDP in an infinite state space.
In this work, based on the criteria given by M-matrix theory, we put forward a finite partition approach study the recurrent properties of RSDP in an infinite state space. Its basic idea is to transform the state-dependent RSDP in an infinite state space into a new state-independent RSDP in a finite state space (see Theorem 2.6 for details).
As an application of our criteria, we develop the study in [10] and [17]. In [10], the authors considered the state-independent RSDP (X t , Λ t ) in R d × M with d ≥ 2 and M a finite set.
For each i ∈ M, the associated diffusion (X Let S d−1 denote the d − 1-dimension sphere, and µ be the invariant probability measure for (Λ t ). In [10], they studied the process under the condition thatb(φ, i) ≡ 0,b(φ, i) ∈ C 1 (S d−1 ) for each i ∈ M, and i∈Mb (φ, i)µ i = 0 for each φ ∈ S d−1 . (1.5) Condition (1.5) allows them to transform the problem into studying the recurrent behavior of the generatorL is a first-order operator on S d−1 and L S d−1 is a (possible degenerate) diffusion generator on S d−1 . By posing some further conditions on c 1 (φ) and c 2 (φ), they got a quantity ρ expressed in terms of c 1 (φ), c 2 (φ) and the density of invariant probability measure of the process corresponding toL. They showed that (X t , Λ t ) is recurrent or transient according to whether ρ ≤ 0 or ρ > 0. Theoretically, this result is complete although calculating ρ is a difficult task, which has been pointed out in [10].
In this work, roughly speaking, we consider the processes corresponding to i∈M µ ibi (φ, i) = 0.
In Section 3, we consider the case δ = 1 and in Section 4, after developing the criteria given in Section 2, we consider the case δ ∈ [−1, 1). Some easily verifiable criteria are provided.
The usefulness and sharpness of the criteria established in this work can be seen from the following example. Let (Λ t ) be a continuous time Markov chain on {1, 2, . . . , N }, N < ∞, equipped with an irreducible conservative Q-matrix (q ij ). Let µ be the invariant probability measure of (Λ t ). Let (X t ) be a random diffusion on [0, ∞) with reflecting boundary at 0 satisfying This work is organized as follows. We shall provide some new criteria on transience, recurrence and exponential ergodicity for RSDP in Section 2. In Section 2, we first study the state-independent RSDP in a finite state space, then study the state-dependent RSDP in an infinite state space. In Section 3, we consider the recurrent property of Ornstein-Uhlenbeck process and linear diffusion in random environments. In Section 4, we provide another kind of criteria then apply these criteria to study the recurrent properties of nonlinear regime-switching diffusion processes.
2 Criteria for recurrence and transience: I Let (X t , Λ t ) be defined by (1.1) and (1.2). We first consider the situation that Q-matrix of (Λ t ) is independent of x and N < ∞. For a diffusion process in R d with generator L = For the vector β = (β 1 , . . . , β N ) * , we use diag(β) = diag(β 1 , . . . , β N ) to denote the diagonal matrix generated by vector β as usual. Before stating our results, we introduce some notation and basic properties on M-matrix. We refer the reader to [2] for more discussion on this topic.
Let B be a matrix or vector. By B ≥ 0 we mean that all elements of B are non-negative.
By B > 0 we mean that B ≥ 0 and at least one element of B is positive. By B ≫ 0, we mean that all elements of B are positive. B ≪ 0 means that −B ≫ 0. We cite some conditions equivalent to that A is a nonsingular M-matrix as follows, and refer to [2] for more discussion on this topic.
The following statements are equivalent.

1.
A is a nonsingular n × n M-matrix.
2. All of the principal minors of A are positive; that is, 3. Every real eigenvalue of A is positive.

4.
A is semipositive; that is, there exists x ≫ 0 in R n such that Ax ≫ 0.
Next result is our first main result for the recurrent property of state-independent regimeswitching diffusion processes in a finite state space. We will use often the following condition for a function V ∈ C 2 (R d ).
(A1) There exist constants r 0 > 0 and Here the constant β i could be negative or positive.
Proof. Denote by A the generator of (X t , Λ t ). Due to [14], where Re γ, where spec(Q p ) denotes the spectrum of Q p .
Let Q (p,t) = e tQp , then the spectral radius Ria(Q (p,t) ) of Q (p,t) equals to e −ηpt . Since all coefficients of Q (p,t) are positive, Perron-Frobenius theorem (see [2,Chapter 2]) yields −η p is a simple eigenvalue of Q p . Moreover, note that the eigenvector of Q (p,t) corresponding to e −ηpt is also an eigenvector of Q p corresponding to −η p . Then Perron-Frobenius theorem ensures that there exists an eigenvector ξ ≫ 0 of Q p associated with the eigenvalue −η p . Now applying Proposition then there exists some p 0 > 0 such that η p > 0 for any 0 < p < p 0 . Fix a p with 0 < p < min{1, p 0 } and an eigenvector ξ ≫ 0, then we obtain Then analogous to the argument of Theorem 2.3, we can conclude the proof.

Remark 2.5
We give a heuristic explanation of the condition (2.3) in previous theorem. As µ is the invariant probability measure of (Λ t ), µ i represents in some sense the time ratio spent by (Λ t ) in the state i. β i represents the recurrent behavior of (X (i) t ). Therefore, the quantity i∈M µ i β i averages the recurrent behavior of (X (i) t ) with respect to µ, which determine the recurrent behavior of (X t , Λ t ) according to previous theorem. Now we go to consider the state-dependent RSDP with (Λ t ) being a Markov chain in a countable space M. Namely, the Q-matrix (q ij (x)) is dependent on x and N = ∞. Let V ∈ C 2 (R d ) such that (A1) holds and M = sup i∈M β i < ∞. As the M-matrix theory is about matrices with finite size, we shall put forward a finite partition method to transform the RSDP in an infinite state space into a new RSDP in a finite state space. Let We assume each F i is nonempty, otherwise, we can delete some points in the partition Γ. Set After doing these preparation, we can get the following result.
As an application of Theorem 2.6, we construct an example of state-independent RSDP in an infinite state space.
By this example, we also want to show that when (Λ t ) is a Markov chain on a countable set, the process (Λ t ) and (X t , Λ t ) may have very different recurrent property. More precisely, if we take b = 2 and a = 1, then (Λ t ) is transient, but for κ < 2 − √ 3, (2.9) holds and hence (X t , Λ t ) is recurrent. If we take b = 1 and a = 2, then (Λ t ) is exponentially ergodic, but for κ > √ 3 − 1, (2.10) holds and hence (X t , Λ t ) is transient.

Recurrent property of Ornstein-Uhlenbeck process and linear diffusion in random environments
In this section, we first consider the Ornstein-Uhlenbeck type process in random environment, that is, the process (X t , Λ t ) satisfies:

is a Brownian motion in R, and (Λ t ) is a continuous Markov chain on the space
is independent of (X t ), and is irreducible and conservative. We assume that d × d matrix σ i is positive definite for every i ∈ M. Let µ = (µ i ) be the invariant probability measure of (q ij ). In [9], the authors showed that when i∈M µ i b i < 0, the process (X t , Λ t ) is ergodic in weak topology, that is, the distribution of (X t , Λ t ) converges weakly to a probability measure ν. In [6,1], the tail behavior of ν was studied. Using the criteria in Section 2, we can get the following result. Proof.
(1) By (3.1), we get the generator L (i) of (X (i) t ) is given by As lim (2) Now we take V (x) = |x| −γ with γ > 0. We have for |x| > r 0 > 0. When r 0 is sufficiently large, it is easy to see that Therefore, we get By Theorem 2.4, as lim Now we consider the linear diffusion in random environments. Let (X t , Λ t ) satisfy mutually independent. The Q-matrix (q ij ) of (Λ t ) is irreducible and conservative. Rewrite (3.3) in component form, Therefore, for i ∈ M, the generator L (i) is given by where a jm (i) = d l=1 σ l (i)x) j (σ l (i)x) m or in matrix form a jm (i) = d l=1 σ l (i)x σ l (i)x * . For every p ∈ R, ∂|x| p ∂x j = p|x| p−2 x j and ∂ 2 |x| p ∂x j ∂xm = p|x| p−2 δ jm + p(p − 2)|x| p−4 x m x j , j, m = 1, . . . , d. By direct calculation, and Then for 0 < p < 1, For p < 0, Consequently, according to Theorem 2.4, and letting p → 0 + or p → 0 − , we obtain the following result.
Proof. It is easy to check that in 1-dimensional case Therefore, we conclude the proof by applying Theorem 3.2.
Define two new conservative Q-matricesQ andQ bỹ Applying Theorem 2.6, we obtain the following result.

Criteria for transience and recurrence: II
According to Foster-Lyapunov drift condition for diffusion processes, if there exists a V ∈ C 2 (R d ) satisfying (A1) with β i ≤ 0 and lim |x|→∞ V (x) = ∞, then the diffusion process (X (i) t ) is exponentially ergodic. When there is no diffusion process (X (i) t ), i ∈ M, being exponentially ergodic, we can not find suitable function V ∈ C 2 (R d ) satisfying (A1), so the criteria introduced in Section 2 are useless for this kind of RSDP . For example, the diffusion process corresponding to L (i) = 1 2 ∆ + |x| δb (x/|x|, i) · ∇ with δ ∈ [0, 1) is not exponentially ergodic. Therefore, to deal with this kind of processes, we need to extend our method introduced in Section 2. Let (X t , Λ t ) be defined by (1.1) and (1.2) and (X (i) t ) be the corresponding diffusion process in the fixed environment i ∈ M with the generator L (i) . Instead of finding a function V satisfying condition (A1), we need to find two functions h, g ∈ C 2 (R d ) satisfying the following condition: (A2) There exists some constant r 0 > 0 such that for each i ∈ M, Theorem 4.1 Let (X t , Λ t ) be state-independent RSDP defined by (1.1) and (1.2) with N < ∞.
Assume that (Hi), (Hii), (Hiii) hold. Let µ be the invariant probability measure of the process (Λ t ). Suppose that there exist two functions h, g ∈ C 2 (R d ) such that (A2) holds and Then Proof. As N i=1 µ i β i < 0, by the Fredholm alternative we obtain that there exist a constant κ > 0 and a vector ξ such that . We obtain . (4.1) By (4.1) and condition (A2), we get As N < ∞, ξ is bounded. Since lim |x|→∞ h(x) )h(x) for |x| > r 0 , it is easy to see that there exists r 1 > 0 such that f (x, i) > 0 for |x| > r 1 . In addition, if Now we consider the following state-independent RSDP in a finite state space, Brownian motion. Let (Λ t ) be a continuous time Markov chain on M with irreducible conservative Q-matrix (q ij ), which is also independent of (B t ). Let µ be the invariant probability measure of (Λ t ). Set a (i) (x) = σ(x, i)σ(x, i) * . Suppose conditions (Hi), (Hii), (Hiii) are satisfied.
In [10], the authors considered the recurrent property of (X t , Λ t ) under the condition In this section, we shall study the case i∈M µ ib (φ, i) = 0. Proof.
When δ = −1, it holds lim sup Therefore, if i∈M µ i β i < 0, by choosing γ > 0 sufficiently small and r 0 > 0 sufficiently large, we can use Theorem 4.1 to show that (X t , Λ t ) is recurrent.
When the dimension d is equal to 1, we can obtain more explicit and complete criteria presented as follows. Proof. By taking h(x) and g(x) as in the Theorem 4.2, it is easy to check that β i =β i = b i . So according to Theorem 4.2, (X t , Λ t ) is recurrent if i∈M µ i b i < 0 and is transient if i∈M µ i b i > 0. Therefore, we only need to consider the case i∈M µ i b i = 0. To deal with this situation, we have to consider it separatively according to the range of δ. Note that it holds i∈M µ i b i (Q −1 b)(i) < 0 as i∈M µ i b i = 0 (cf. [10]). Case 1: δ ∈ (0, 1). For p > 0, set where the vector (c i ) would be determined later. By noting that δ ∈ (0, 1), we obtain Take p ∈ (0, 1− δ), then i∈M p(p − 1+ δ)µ i b i (Q −1 b)(i) > 0. By the Fredholm alternative, there exist a constant β > 0 and a vector (c i ) such that Qc(i) = p(p − 1 + δ)b i (Q −1 b)(i) − β. Choosing these p and (c i ), we have A f (x, i) = −βx p−2+2δ + o(x p−2+2δ ). As lim |x|→∞ f (x, i) = ∞ for each i ∈ M, we obtain that (X t , Λ t ) is recurrent when i∈M µ i b i = 0 and δ ∈ (0, 1).