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Moment bounds and geometric ergodicity of diffusions with random switching and unbounded transition rates
 Xin T. Tong^{1}Email authorView ORCID ID profile and
 Andrew J. Majda^{2}
https://doi.org/10.1186/s4068701600892
© The Author(s) 2016
 Received: 12 April 2016
 Accepted: 26 October 2016
 Published: 14 November 2016
Abstract
A diffusion with random switching is a Markov process that consists of a stochastic differential equation part \(X_t\) and a continuous Markov jump process part \(Y_t\). Such systems have a wide range of applications, where the transition rates of \(Y_t\) may not be bounded or Lipschitz. A new analytical framework is developed to understand the stability and ergodicity of these processes and allows for genuinely unbounded transition rates. Assuming the averaged dynamics is dissipative, the first part of this paper explicitly demonstrates how to construct a polynomial Lyapunov function and furthermore moment bounds. When the transition rates have multiple scales, this construction comes interestingly as a dual process of the averaging of fast transitions. The coefficients of the Lyapunov function can be seen as the potential dissipation of each regime in different scales, and a comparison principle comes naturally under this interpretation. On the basis of these results, the second part of this paper establishes geometric ergodicity for the joint processes. This can be achieved in two scenarios. If there is a commonly accessible regime that satisfies the minorization condition, the geometric convergence to the ergodic measure takes place in the total variation distance. If there is contraction on average, the geometric convergence takes place in a proper Wasserstein distance and is proved through an application of the asymptotic coupling framework.
1 Background
 1.
For stochastic lattice models in climate science [12, 14, 15, 25, 26, 31, 33, 48, 49], \(X_t\) represents the dry atmosphere, so (1.1) is the spatial discretization of a fluid equation. Meanwhile, \(Y_t\) represents the unresolved behavior of moisture and clouds.
 2.
In material science [21–24] and molecular biology [6, 45], \(X_t\) represents some macroscopic quantities such as the transmembrane electronic potential, and \(Y_t\) stands for the behavior of some particular clusters, proteins, channels and cells.
 3.
As a simulation strategy for complex processes [8, 33], a Markov jump process can be used as a stochastic parameterization of some subgrid scale processes. It reduces the model dimension and preserves most of the statistical quantities. It can be seen as the \(Y_t\) part in our joint process.
 4.
In filtering and predictive modeling [16, 30, 32], diffusions with random switching are used as test beds to quantify the uncertainty from model errors.

When does the joint process \(Z_t=(X_t,Y_t)\) possess an invariant measure? What kind of statistics, for example moments of \(X_t\), is integrable under the invariant measures?

Is the invariant measure unique? How does it attract other statistical states?
In the last decade, a series of works have been devoted for the questions above [1, 3–5, 7, 11, 39, 46]. In the simplest setting, the transition rates are constants, \(\lambda (x,y,y')=\lambda (y,y')\), and \(X_t\) is driven by the linear equation (1.2). Both questions above are relatively well understood in this setting, thanks to an application of Perron–Frobenius theorem [3, 11]. These intuitive results can be extended to nonconstant transition rates through a probabilistic coupling argument [7, 46]. However, for this argument to work, the transition rates have to be globally bounded and Lipschitz. This restriction excludes many important applications [14, 36] or imposes additional nonphysical compact requirements on the model space [6]. This paper intends to bridge this gap by developing a new analytical framework.
 1.
Inspired by the formulation in [48, 49], we assume there is a \(X_t\) controlled multiscale structure in the transition rates \(\lambda \), while the fast averaging procedures induce a dissipation. It is important to note that the multiple scales here are not introduced by an auxiliary variable \(\epsilon \) as in other standard settings [43, 44].
 2.
There is a comparison principle in favor of dissipation.
 1.
If there is a commonly reachable regime that satisfies the minorization condition, Theorem 3.4 proves geometrically ergodicity in the total variation distance.
 2.
If there is contraction on average, and the transition rates and their first derivatives are bounded by the Lyapunov function, Theorem 3.6 shows geometric ergodicity in a proper Wasserstein distance.
The remainder of this paper is arranged as follows. Section 2 discusses criterions that lead to dissipation on average, and how to construct a Lyapunov function in different scenarios. Section 3 gives the precise statements of geometric ergodicity when there is a hypoelliptic regime or there is contraction on average. Conditions leading to the second scenario are briefly discussed and compared with results in Sect. 2. The proofs of geometric ergodicity are contained in Sect. 4, where we also discuss how to verify the accessibility of one regime. Section 5 summarizes the results and discusses some related questions.
2 Dissipation on average and Lyapunov functions
3 Geometric ergodicity
In Sect. 2, Lyapunov functions are constructed for diffusions with random switching when the dynamics is dissipative on average. Because these Lyapunov functions are in the form of polynomials, \(\mathbb {E}X_t^m\) is bounded uniformly for a proper m. Then by the Krylov–Bogoliubov theorem [9], there is at least one invariant measure for the joint process \(Z_t=(X_t,Y_t)\). It is natural to ask whether this invariant measure is unique, and how does the law of \(Z_t\) converge to the unique invariant measure \(\pi \).
 1.
in total variation distance if there is a commonly reachable minorization regime;
 2.
in a proper Wasserstein distance if there is contraction on average.
3.1 Convergence in total variation with a minorization regime
Theorem 3.1
When F consists of only one regime, \(X_t\) is simply an SDE on \(\mathcal {R}^d\). In this context, following the arguments in [40], the minorization conditions of Theorem 3.1 can be verified by the hypoellipticity and reachability conditions below:
Assumption 3.2
 1.Hypoellipticity condition: let \(\mathcal {L}\) be the Lie algebra generated by the vector fieldswith \(\sigma _i\) being the columns of \(\sigma \), and \(\mathcal {L}_0\) is the ideal in \(\mathcal {L}\) generated by \(\{\sigma _1,\ldots , \sigma _m\}\), assuming \(\mathcal {L}_0\) spans \(\mathcal {R}^d\) at all points.$$\begin{aligned} \{f,\sigma _1,\ldots , \sigma _m\} \end{aligned}$$
 2.Reachability condition: there is a point \(x_h\in \mathcal {R}^d\) such that for any compact set C and \(\epsilon >0\), there is a \(t_0\) such that from any \(x\in C\) there is a piecewise constant process \(w_t\) such that the solution to the following ODEsatisfies \(x_{t_0}x_h\le \epsilon \).$$\begin{aligned} \hbox {d}x_t=[b(x_t)+\sigma (x_t)w_t] \hbox {d}t,\quad x_0=x, \end{aligned}$$
Theorem 3.4 below indicates that for diffusions with random switching, it suffices to check minorization condition for one particular regime, using say Assumption 3.2, and show this regime is commonly accessible and satisfies a mild growth condition for \(V(Z_t)\). In particular, we define
Definition 3.3
Theorem 3.4
Let \(Z_t=(X_t,Y_t)\) be a diffusion with random switching that admits a Lyapunov function V. Suppose the transition rates satisfy \(\bar{\lambda }(z)\le M V(z)\), moreover there is a regime \(y_h\in F\) such that it is commonly accessible and has polynomial growth for V. Then if the SDE given by \(\hbox {d}X'_t=b(X'_t,y_h)\hbox {d}t+\sigma (X'_t,y_h)\hbox {d}W_t\) satisfies the minorization condition in Theorem 3.1, the diffusion \(Z_t\) has an invariant measure \(\pi \) and is geometrically ergodic under the total variation distance.
The proof is located in Sect. 4, where we will also show a simple way to verify the common accessibility of one regime.
3.2 Wasserstein metric convergence with contraction on average
The total variation norm is often too stringent to capture contraction. For example, consider a trivial deterministic process in \(\mathcal {R}\), \(\hbox {d}X_t=\rho X_t \hbox {d}t\). The invariant measure is obviously \(\delta _0\), a point mass at the origin, and it attracts other points. Yet, starting from any nonzero point, the distribution of \(X_t\) is a point mass at \(e^{\rho t}X_0\), which has total variation distance 2 from \(\delta _0\).
The distance function here can be very flexible. One remarkable discovery in [17, 19] is that by properly incorporating the Lyapunov function into d, the corresponding Wasserstein distance can characterize relatively weak convergence. This is known as the asymptotic coupling framework. As for diffusions with random switching, this framework allows us to generalize geometric ergodicity results to cases where the transition rates and their derivatives are bounded only by the Lyapunov function.
Similar to the situation with dissipation, the notion of contraction on average is essential for our discussion. The precise statement is the following:
Theorem 3.5
 1.V(x, y) has polynomial growth in x, so there are n, M such that$$\begin{aligned} V(x+u, y)\le M(V(x)+u^{n})\quad \text {and}\quad V(x,y)\ge \frac{1}{M}x^{\frac{1}{n}}. \end{aligned}$$
 2.The transition rates and their derivatives are bounded by MV with a constant \(M>0\)$$\begin{aligned} \bar{\lambda }(z)\le MV(z),\quad \sum _{y'}\nabla _x \lambda (x,y,y')\le MV(z). \end{aligned}$$
 3.Each regime admits a contraction rate \(\rho \) in the sense of (3.2), and the averaged dynamics is contractive as there are \(C_\rho ,m,\bar{\rho }>0\):$$\begin{aligned} \mathbb {E}^z \exp \bigg (m\int ^t_0 \rho (Y_s){\text {d}}s\bigg )\le C_\rho \exp (\bar{\rho } t). \end{aligned}$$
 4.
There is a commonly accessible regime \(y_c\), such that \(\rho (y_c)>0\) and V has at most polynomial growth in the sense of Definition 3.3.
3.3 Contraction on average
Given the contraction rates \(\rho \) in each regime, the contraction on average condition (3.3) can be verified using arguments similar to the ones of dissipation on average in Sect. 2. Yet, there is an important difference. When we construct a Lyapunov function, it suffices to consider the transitions for large x, and upper bounds suffice to hold modulo a constant, see Lemma 6.2. This is no longer the case for contraction on average, and we will need the “\(\lesssim \)” inequalities in Sect. 2 to hold with “\(\le \).” As a consequence, the spectrum and comparison arguments still work with a variant, but the scaling argument no longer works.
The following theorem is the contraction version of Theorem 2.2, while the transition rates are allowed to be nonconstant.
Theorem 3.6
Proof
Following the representation (2.10), we can see a(y) in (3.5) as the potential contraction of one regime. Therefore, the comparison principle can be formulated as follows:
Proposition 3.7
The proof is a direct verification and ignored here. On the other hand, contraction on average will require the contraction to hold homogenous inside \(\mathcal {R}^d\), but not just for large enough x. This is probably an intrinsic requirement due to the following example:
Example 3.8
4 Geometric ergodicity through random PDMPs
4.1 Random PDMPs
On the other hand, diffusions with random switching can be viewed as random PDMPs. As noted in Sect. 3.2, the solution of the SDE in a fixed regime can be written as \(X_t=\Psi ^{y,\omega }_{s,t} X_s\), where \(\omega \) denotes the realization of the Wiener process \(W_s\), and we denote the law of \(\omega \) as \(P_W\). Therefore, if we condition on each realization \(\omega \), a diffusion with random switching is simply a PDMP. Following the nomenclature of statistical physics [47], this PDMP will be called a quenched process, as it is the conditioning of the original joint process \(Z_t\), on one realization of random outcome \(\omega \). In contrast, the original process without conditioning will be called the annealed process. We will adopt this simple terminology.
4.2 Accessibility analysis
In both the minorization and contraction on average scenarios, we need a good regime to be commonly accessible. In this section we will discuss the consequence of this assumption and also provide a simple way to verify in Lemma 4.2. Most derivations here are relatively standard and may have a simpler version in [4, 5, 7] when the transition rates are bounded. We provide the complete proofs here to be selfcontained.
Lemma 4.1
 1.
If \(\mathbb {P}^z(Y_t=\tilde{y})>0\), then \(\mathbb {P}^z(Y_{t+s}=\tilde{y})>0\) for any \(s\ge 0\);
 2.For each z and \(t>0\), there exists a neighbor of x, \(O_x\subset \mathbb {R}^d\), such that$$\begin{aligned} \mathbb {P}^{x',y}(Y_t=\tilde{y})\ge \frac{1}{2}\mathbb {P}^z(Y_t=\tilde{y}),\quad \forall x'\in O_x; \end{aligned}$$
 3.If there is a \(\tilde{y}\in F\) that is commonly accessible, then for any fixed compact set C, there exists some \(t_0,m_0>0\) such that$$\begin{aligned} \mathbb {P}^z(Y_{t_0}=\tilde{y})\ge m_0,\quad \forall z\in C. \end{aligned}$$
Proof
The following Lemma provides an easy verification that a regime \(y^*\) is commonly accessible.
Lemma 4.2
Proof
4.3 Ergodicity with a minorization regime
Due to Theorem 3.1, the proof of Theorem 3.4 is a relative standard verification of the small set condition for the full Markov semigroup \(P_t\).
Proof of Theorem 3.4
4.4 Ergodicity with contraction on average
The proof of Theorem 3.5 uses the asymptotic coupling mechanism introduced by [18, 19]. Theorem 4.8 of [19] presented below formulates our application of this mechanism.
Theorem 4.3
 1.\(P_t\) is locally contracting in d:$$\begin{aligned} d(P_t^* \delta _z, P_t^*\delta _{z'})\le \frac{1}{2}d(z,z'),\quad \forall d(z,z')<1. \end{aligned}$$
 2.
Smallness: for any two \(z,z'\) such that \(V(z),V(z')\le K\), \(d(P_t^* \delta _z, P_t^*\delta _{z'})\le 1\epsilon \).
The reason that (1) is called a local contraction, is that in most applications, \(\hbox {d}(z,z')=1\) unless z and \(z'\) are very close. Theorem 4.3 essentially extends a local contraction to a global one.
4.4.1 Contracting distance
For the construction of a contracting distance, we have the following lemma. It is a variant of Lemma 4.13 in [19] which uses a Lyapunov function instead of a super Lyapunov function. The proof goes very similar.
Proposition 4.4
Before we move on to the proof of Proposition 4.4, we need two pieces of arguments. The first one indicates \(d(z,z')\approx 1_{y\ne y'}+ 1_{y=y'}\wedge \delta ^{1}(V(z)xx')^r.\)
Lemma 4.5
Proof
The second lemma gives a bound on the perturbation of measures caused by perturbation on the initial condition:
Lemma 4.6
Proof
\(\square \)
We are finally at the position to prove Proposition 4.4:
Proof of Proposition 4.4
By the definition of contracting metric, it suffices for us to show that \(\hbox {d}(P^*_T\delta _z,P^*_T\delta _{z'})\le \frac{1}{2}\hbox {d}(z,z')\) when \(\hbox {d}(z,z')<1\). This implies that \(y=y'\), and \(xx'\le \frac{1}{2}\).
4.4.2 Small set verification
Verification of condition (2) of Theorem 4.3 is given by the following
Lemma 4.7
Proof
4.4.3 Proof of Theorem 3.5
With the conditions of Theorem 4.3 verified, it is rather elementary to show Theorem 3.5.
Proof
5 Conclusion and discussion
Diffusions with random switching are a general class of stochastic processes with applications in many different areas. Such system consists of a diffusion process part \(X_t\) and a Markov jump part \(Y_t\), and their dynamics can be fully coupled. The stability and ergodicity of such processes remain open questions if the transition rates are not bounded or Lipschitz. This paper closes this gap by developing a new analytical framework.
The first part of this paper constructs polynomialtype Lyapunov functions when there is dissipation on average. These functions can be used to derive moment bounds on the diffusion part. The incorporation of potential dissipation of each regime is found to be an efficient way to capture averaged dissipation. This idea can be easily applied to the classical case where the transition rates are constants as Theorem 2.2. It also leads to a simple illustration of comparison principles, Theorem 2.9. Moreover, with a Fredhlom alternative argument, we demonstrate how can the Lyapunov function be inductively constructed, as a dual process of the averaging procedure, Theorems 2.4, 2.7, assuming the transition rates have a multiscale structure.
The second part of this paper is devoted to the geometric ergodicity of diffusions with random switching, assuming a Lyapunov function exists. If there is a commonly accessible regime that satisfies the minorization condition, Theorem 3.4 proves geometric ergodicity under total variation distance. When there is contraction on average, using the asymptotic coupling framework of [17, 19], Theorem 3.6 demonstrates geometric ergodicity under a proper Wasserstein distance.
 1.
The authors conjecture that the results here hold in a similar form if the regime process \(Y_t\) is instead a continuous stochastic process. One way to see this is taking the limit of a jump process on grid points of vanishing size. But the authors suspect that an independent mechanism can be set up for these processes without little change of the proofs. Such theory will be applicable to many nonlinear models that exhibits intermittency, for example [34].
 2.
The attraction or contraction rates used in this paper provides a uniform control over each component of \(X_t\). A more general situation is that each component of \(X_t\) has a different regimebased attraction rates; in other words, the attraction rate is given by a matrix. A simple example will be \(\hbox {d}X_t=A(Y_t)X_t \hbox {d}t\), where A is a matrixvalued function. It is known such system is very sensitive to the switching even if A has all eigenvalues of negative real parts [28]. It will be interesting if our results can be generalized to this case.
 3.
Theorem 3.4 can probably be generalized. Bakhtin and Hurth [1] show that a PDMP is geometric ergodic in total variation despite the fact that each regime is degenerate. The proofs there only require a Hörmander type of condition on the vector fields generated by different regimes. The authors conjecture that for diffusions with random regime switching, in order to have geometric ergodicity, it suffices to have the Lie algebra generated by stochastic flows of all regimes spanning \(\mathcal {R}^d\). But this requires a completely different set of techniques, and Assumption 3.2 should be general enough to cover most applications.
Declarations
Author's contributions
XTT proposed the problems and drafted the proof. AJM proposed the problems and supervised the proof. Both authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank R. van Handel for his suggestions over the presentation of this article. This research is supported by the MURI Award Grant N000141210912, where A.J.M. is the principal investigator, while X.T.T. was supported as a postdoctoral fellow. This research is also partly supported by the NUS Grant R146000226133, where X.T.T. is the principal investigator.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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