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# A new criterion on existence and uniqueness of stationary distribution for diffusion processes

- Zhenzhong Zhang
^{1, 2}Email author and - Dayue Chen
^{2}

**2013**:13

https://doi.org/10.1186/1687-1847-2013-13

© Zhang and Chen; licensee Springer 2013

**Received:**23 November 2012**Accepted:**28 December 2012**Published:**17 January 2013

## Abstract

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.

**MSC:**60H10, 37A25.

## Keywords

- diffusion processes
- stationary distribution
- Feller property

## 1 Introduction

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 [1], Pinsky [2], Prato and Zabcyk [3], Yin and Zhu [4], Zhang [5]. 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.

*f*in ${C}_{0}^{2}(\mathbb{S})$ with infinitesimal generator

- (i)
$a=\{{a}_{ij}(x)\}$ and $b=\{{b}_{i}(x)\}$ are locally bounded and measurable functions on

*D*; - (ii)
*a*is continuous on*D*; - (iii)
there exits a unique strong solution $X(t)$;

- (iv)
the solution $X(t)$ does not explode at any finite time.

*f*on $\mathbb{S}$. A diffusion process $X(t)$ is said to have the (weak) Feller property if for any bounded and continuous function

*f*, ${P}^{t}f$ is a bounded continuous function. Besides, if ${a}_{ij}(x)\equiv 0$, $1\le i,j\le n$, then the processes become deterministic stochastic processes. In this case, for any bounded continuous function

*f*,

*t*, the random variable $X(t)$ can take on at least a countable number of values with positive probabilities. More precisely, the diffusion $X(t)$ whose semigroup should be irreducible (see, for example, Cerrai [6]). Hence, we impose the following assumption on the diffusion semigroup:

- (v)
the semigroup ${P}^{t}$ 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:

${A}^{T}$: the transpose of any matrix or vector *A*;

$|A|$: the trace norm of matrix *A*, *i.e.*, $|A|=\sqrt{trace({A}^{T}A)}$;

$B(0,r)=\{x\in \mathbb{S}:|x|<r\}$ and ${B}^{c}(0,r):=\{x\in \mathbb{S}:|x|\ge r\}$;

*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) $E|X(t)|<+\mathrm{\infty}$ for each $t>0$;

(A2) ${sup}_{t\ge 0}E|X(t)|<+\mathrm{\infty}$.

To begin with, we cite a known result from Bhattacharya and Waymire [7] as a lemma.

**Lemma 2.1** [[7], pp.643-645]

*Let*${P}_{{x}_{0}}^{t},{P}_{{x}_{1}}^{t},\dots ,{P}_{{y}_{0}}^{t}$

*be probability measures on*$\mathbb{S}$

*for every*$t\ge 0$.

*The following are equivalent statements*.

- (a)
$\{{P}_{{x}_{i}}^{t}:i=0,1,2,\dots \}$

*converge weakly to*${P}_{{y}_{0}}^{t}$. - (b)
*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* $X(t)$ *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.* [8]. □

**Theorem 2.1** *If assumptions* (i)-(v) *and* (A2) *hold*, *then the diffusion process* $X(t)$ *has a unique stationary distribution*.

*Proof* The proof of this theorem is divided into two steps as follows.

*i.e.*, given $\epsilon >0$, there exists $r<\mathrm{\infty}$ such that ${P}^{t}({x}_{0},{B}^{c}(0,r))\le \epsilon $. It will follow from the Prohorov theorem that there exist ${t}_{n}\to \mathrm{\infty}$ and a probability measure

*π*, perhaps depending on ${x}_{0}$, such that

Therefore, the tightness of the family $\{{P}^{t}({x}_{0},dy):t\ge 0\}$ follows from assumption (A2).

*r*is large enough. Therefore, $\{{\mu}_{T},T>0\}$ is tight, namely, there exists a subsequence ${t}_{n}\uparrow +\mathrm{\infty}$, the sequence ${\mu}_{{t}_{n}}$ converges weakly to a measure

*π*. By the Krylov-Boyoliubov theorem(see,

*e.g.*, Prato and Zabczyk [3], Corollary 3.1.2, p.22), one can follow

*π*is an invariant measure for ${P}^{t}$, $t\ge 0$. In other words,

*f*is bounded and continuous, then ${P}^{t}f$ is a bounded continuous function. Therefore, applying (2.6) to ${P}^{t}f$ yields that

*i.e.*, if $X(0)$ has a distribution *π*, then $Ef(X(t))=Ef(X(0))$ for all $t\ge 0$. Namely, *π* is a stationary distribution.

But by (2.6), the left-hand side of equality (2.9) converges to $\overline{f}$. Therefore, ${\int}_{\mathbb{S}}f(z)\pi (dz)={\int}_{\mathbb{S}}f(z){\pi}^{\prime}(dz)$, which implies ${\pi}^{\prime}=\pi $. □

## 3 An example

In this section, we give an example to illustrate our conditions and results.

**Example 3.1**Recently, Mao [9] has considered the stationary distribution of stochastic population dynamics. They assumed that population sizes follow the following stochastic differential equations:

*i*at time

*t*, ${b}_{i}$ is the intrinsic growth rate of species

*i*, ${a}_{ij}$ represents the effect of interspecies (if $i\ne j$) or intraspecies (if $i=j$) interaction. Here ${W}_{j}(t)$ is an independent one-dimensional Brownian motion. Let $W(t)={({W}_{1}(t),\dots ,{W}_{n}(t))}^{T}$ be an

*n*-dimensional Brownian motion. Then Eq. (3.1) can be rewritten as

where $X(t)={({X}_{1}(t),\dots ,{X}_{n}(t))}^{T}$, $b={({b}_{1},{b}_{2},\dots ,{b}_{n})}^{T}$, $A={({a}_{ij})}_{n\times n}$, $\sigma ={({\sigma}_{ij})}_{n\times n}$, $\gamma (u)={({\gamma}_{1}(u),\dots ,{\gamma}_{n}(u))}^{T}$. Besides, we suppose that $W(t)$ and $N(t)$ are independent and for $i,j=1,\dots ,n$, ${\sigma}_{ij}$ are nonnegative constants, and ${\sigma}_{ii}>0$ for some *i*. If ${a}_{ii}<0$, ${a}_{ij}>0$, $1\le i,j\le n$, $i\ne j$, then the model (3.2) is termed the facultative Lotka-Volterra model. If ${a}_{ii}<0$, ${a}_{ij}\le 0$, $1\le i,j\le n$, $i\ne j$, 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) ${a}_{ii}<0$, ${a}_{ij}\le 0$, $i=1,\dots ,n$.

**Lemma 3.1**

*If one of assumptions*(H1)

*and*(H2)

*holds*,

*then there is a positive constant*

*K*

*such that for any initial value*${x}_{0}\in {\mathbb{R}}_{+}^{n}$,

*Proof* The proof is essentially the same as the proof of Theorem 3.1 of Mao [9] and Theorem 3.1 of Bao [11]. We omit the proof. □

**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 [9], Tong *et al.* [8], and Bao *et al.* [11], 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.* [8]. 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.

## Declarations

### Acknowledgements

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).

## Authors’ Affiliations

## References

- Hasminskii RZ:
*Stochastic Stability of Differential Equations*. Sijthoff & Noordhoff, Rockville; 1980:71–155.View ArticleGoogle Scholar - Pinsky RG:
*Positive Harmonic Function and Diffusion*. Cambridge University Press, Cambridge; 1995:235–282.View ArticleGoogle Scholar - Da Prato G, Zabczyk J:
*Ergodicity for Infinite Dimensional Systems*. Cambridge University Press, Cambridge; 1996:11–168.MATHView ArticleGoogle Scholar - Yin G, Zhu C:
*Hybrid Switching Diffusions*. Springer, Berlin; 2010:27–133.MATHGoogle Scholar - Zhang XC: Exponential ergodicity of non-Lipschitz stochastic differential equations.
*Proc. Am. Math. Soc.*2009, 137: 329–337.MATHView ArticleGoogle Scholar - Cerrai S Lecture Notes in Mathematics 1762. In
*Second Order PDE’s in Finite and Infinite Dimension. A Probabilistic Approach*. Springer, Berlin; 2001:65–77.View ArticleGoogle Scholar - Bhattacharya RN, Waymire EC:
*Stochastic Processes with Applications*. Wiley, New York; 1990:643–646.Google Scholar - Tong JY, Zhang ZZ, Bao JH: The stationary distribution of the facultative population model with a degenerate noise.
*Stat. Probab. Lett.*2013, 83: 655–664. 10.1016/j.spl.2012.11.003MATHMathSciNetView ArticleGoogle Scholar - Mao XR: Stationary distribution of stochastic population systems.
*Syst. Control Lett.*2011, 60: 398–405. 10.1016/j.sysconle.2011.02.013MATHView ArticleGoogle Scholar - Bao JH, Yuan CG: Stochastic population dynamics driven by Lévy noise.
*J. Math. Anal. Appl.*2012, 391: 363–375. 10.1016/j.jmaa.2012.02.043MATHMathSciNetView ArticleGoogle Scholar - Bao JH, Mao XR, Yin G, Yuan CG: Competitive Lotka-Volterra population dynamics with jumps.
*Nonlinear Anal.*2011, 74: 6601–6616. 10.1016/j.na.2011.06.043MATHMathSciNetView ArticleGoogle Scholar

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