- Open Access
A new criterion on existence and uniqueness of stationary distribution for diffusion processes
© Zhang and Chen; licensee Springer 2013
Received: 23 November 2012
Accepted: 28 December 2012
Published: 17 January 2013
In this paper, we provide a new criterion on the existence and uniqueness of stationary distribution for diffusion processes. An example is given to illustrate our results.
In many applications such as finance and biology, we often wish to replace the time dependent instantaneous measure by a stationary (or ergodic) measure. Thus, we face the following questions: Do the systems possess ergodic properties? Under what conditions do the systems have the desired properties of ergodicity? There are many criteria for the existence and uniqueness of stationary distribution for diffusion processes. See, for example, Hasminskii , Pinsky , Prato and Zabcyk , Yin and Zhu , Zhang . However, most criteria suppose the infinitesimal operators satisfy the uniform ellipticity condition. An interesting question is: What happens if the infinitesimal operators can be degenerate? Is there a simple and general sufficiency condition? In this paper, we will give a new sufficient condition to verify the existence and uniqueness of stationary distribution for general diffusion processes.
and are locally bounded and measurable functions on D;
a is continuous on D;
there exits a unique strong solution ;
the solution does not explode at any finite time.
the semigroup is irreducible.
The aim of the present paper is twofold. First, we aim to give a new criterion for general diffusions, especially when the coefficients of diffusions are non-Lipschitz or coefficients of diffusions are degenerate. This is highly non-trivial because when the diffusion matrix is singular, the corresponding infinitesimal operator L is a class of non-elliptic operators, which the maximum principle on an elliptic operator fails. Our second aim is to give general sufficient conditions on the stationary distribution for population dynamical systems.
To proceed, we list several notions:
: the transpose of any matrix or vector A;
: the trace norm of matrix A, i.e., ;
K: a generic positive constant whose values may vary at its different appearances.
2 Main results
Before we show the main result, we first impose the following assumptions:
(A1) for each ;
To begin with, we cite a known result from Bhattacharya and Waymire  as a lemma.
Lemma 2.1 [, pp.643-645]
converge weakly to .
Equation (1.3) holds for all infinitely differentiable functions vanishing outside a bounded set.
Lemma 2.2 Let assumptions (i)-(v) and (A1) hold, the diffusion process has the weak Feller property.
Proof The proof of this lemma is essentially the same as that of Lemma 3.2 of Tong et al. . □
Theorem 2.1 If assumptions (i)-(v) and (A2) hold, then the diffusion process has a unique stationary distribution.
Proof The proof of this theorem is divided into two steps as follows.
Therefore, the tightness of the family follows from assumption (A2).
i.e., if has a distribution π, then for all . Namely, π is a stationary distribution.
But by (2.6), the left-hand side of equality (2.9) converges to . Therefore, , which implies . □
3 An example
In this section, we give an example to illustrate our conditions and results.
where , , , , . Besides, we suppose that and are independent and for , are nonnegative constants, and for some i. If , , , , then the model (3.2) is termed the facultative Lotka-Volterra model. If , , , , then the model (3.2) is termed the competitive Lotka-Volterra model. Both Lotka-Volterra models have been extensively studied by many authors (see, e.g., Bao et al. [10, 11]). For the competitive model (3.2) with Poisson jumps, Bao et al. [10, 11] show that Eq. (3.2) has some nice results such as global positive solution, existence of an invariant measure, some asymptotic properties. To obtain our results, we impose the following assumptions:
(H1) −A is a nonsingular M-matrix;
(H2) , , .
Theorem 3.1 If one of assumptions (H1) and (H2) holds, then the model (3.2) has a unique stationary distribution.
Proof According to the results obtained by Mao , Tong et al. , and Bao et al. , it is not hard to check that the model (3.2) satisfies all the conditions of Theorem 2.1 together with Lemma 3.1. Hence, the uniqueness of stationary distribution follows immediately. □
Remark 3.1 Assumption (H1) relaxes the sufficient conditions obtained by Tong et al. . Assumption (H2) means that if population dynamics is competitive, then it has a unique stationary distribution, which implies ergodic properties of the model (3.2). This gives a new method to estimate parameters for competitive population dynamics.
The authors are grateful to the anonymous referees for their valuable comments and suggestions which led to improvements in this manuscript. The research of Z. Zhang was partially supported by the National Natural Science Foundation of China (Nos. 11071037, 11171062, 11126253 and 11201062), the Fundamental Research Funds for the Central Universities, and the Innovation Program of Shanghai Municipal Education Commission (No. 12ZZ063).
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