CS41: Asymptotic Behavior of Selected Markov Random Dynamical Systems

Organizer: Hanna Wojewódka-Ściążko (University of Silesia in Katowice)

Law of the iterated logarithm for Markov semigroups with exponential mixing property in the Wasserstein distance

Dawid Czapla

The main goal of this talk is to present our recent result concerning the law of the iterated logarithm (LIL) for non-stationary, continuous-time Markov processes on Polish spaces. Specifically, our theorem applies to additive functionals of processes governed by stochastically continuous Markov-Feller semigroups that exhibit exponential mixing and non-expansiveness in the Wasserstein metric.

To state the result more precisely, let us first introduce some notation. We work on a complete, separable metric space $(X,\rho)$, endowed with its Borel $\sigma$-field $\mathcal{B}(X)$. By $\mathcal{M}_1(X)$ we denote the family of all Borel probability measures on $X$, and, given any $r>0$, we write $\mathcal{M}_{1,r}(X)$ for the subset of $\mathcal{M}_1(X)$ consisting of those measures $\mu$ for which $\int_X \rho(x_0,x)^r\,\mu(dx)<\infty$ with some $x_0\in X$. Furthermore, let us recall that the Wasserstein distance on the set $\mathcal{M}_{1,1}(X)$ is given by \[d_W(\mu,\nu)=\sup\left\{\left|\int_X f\,d(\mu-\nu) \right|:\; f:X\to\mathbb{R},\;\,\operatorname{Lip}f \leq 1\right\}\quad\text{for}\quad \mu,\nu\in\mathcal{M}_{1,1}(X),\] where $\operatorname{Lip}f:=\sup_{x\neq y} |f(x)-f(y)|/\rho(x,y)$.

The object of our study is a continuous-time Markov process $\Phi:=\{\Phi_t\}_{t\geq 0}$, evolving on the space $(X,\rho)$, with associated transition semigroup $\{P_t\}_{t\geq 0}$, consisting of Markov operators acting on $\mathcal{M}_1(X)$. Concretely, this means that $\Phi$ is a family of $X$-valued random variables on a measurable space $(\Omega,\mathcal{F})$, equipped with a collection of probability measures $\{\mathbb{P}_x\}_{x\in X}$, such that, for any $s,t\geq 0$, $x\in X$, and $A\in\mathcal{B}(X)$, \[\mathbb{P}_x(\Phi_0=x)=1,\quad \mathbb{P}_x(\Phi_{s+t}\in A\,|\,\mathcal{F}_s)=\mathbb{P}_{\Phi_s}(\Phi_t\in A),\quad\text{and}\quad \mathbb{P}_x(\Phi_t\in A)=P_t\delta_x(A),\] where $\{\mathcal{F}_t\}_{t\geq 0}$ is the natural filtration of $\Phi$, and $\delta_x$ denotes the Dirac measure at $x$. Additionally, as usual, for every $\mu\in\mathcal{M}_1(X)$, we define \[\mathbb{P}_{\mu}(F):=\int_X \mathbb{P}_x(F)\,\mu(dx)\quad\text{for}\quad F\in\mathcal{F}_{\infty}:=\sigma(\Phi_t:\;s\geq 0).\] Clearly, $\Phi$ can be then regarded as a Markov process on $(\Omega,\mathcal{F}_{\infty},\mathbb{P}_{\mu})$ with initial distribution $\mu$. Our result relies on the following assumptions:

$\{P_t\}_{t\geq 0}$ is stochastically continuous, i.e., for every $x\in X$ and each bounded continuous function $f:X\to\mathbb{R}$, the map $[0,\infty) \ni t\mapsto \int_X f\,d(P_t\delta_x)$ is continuous at $t=0$.
$\{P_t\}_{t\geq 0}$ is Feller, i.e., the map $X \ni x\mapsto \int_X f\,d(P_t\delta_x)$ is continuous for every bounded continuous function $f:X\to\mathbb{R}$.
There exists $\gamma\in (0,\infty)$ such that \[d_W(P_t\delta_x,\,P_t\delta_y)\leq e^{-\gamma t}\rho(x,y)\quad\text{for any}\quad t\geq 0,\; x,y\in X.\]
For some $\mu_0\in\mathcal{M}_1(X)$, there exist $x_0\in X$ and $\zeta\in (2,\infty)$ such that \[\sup_{t\geq 0} \int_X \rho(x_0,x)^{\zeta}\,P_t\mu_0(dx)<\infty.\]

Under conditions (A2)-(A4), it can be shown that $\{P_t\}_{t\geq 0}$ admits a unique invariant probability measure $\mu_*$, which is a member of $\mathcal{M}_{1,\zeta}(X)$.

Let $g:X\to\mathbb{R}$ be an arbitrary bounded Lipschitz continuous function such that $\int_X g\,d\mu_*=0$, and define \[I_t(g):=\int_0^t g(\Psi_s)\,ds\quad\text{for}\quad t\geq 0.\vspace{-0.2cm}\] Moreover, put \[\chi_g(x)=\int_0^{\infty} P_t g(x)\,dt\quad\text{for}\quad x\in X\quad\text{and}\quad \sigma_g^2:=\mathbb{E}_{\mu_*}\left[\left(\chi_g(\Phi_1)-\chi_g(\Phi_0)+I_1(g) \right)^2\right].\]

The announced result can be now formulated as follows.
Theorem. Suppose that conditions (A1)-(A4) hold. Then $\sigma_g^2$ is finite, and whenever it is positive, the process $\{I_t(g)\}_{t\geq 0}$ satisfies the LIL under $\mathbb{P}_{\mu_0}$, with $\sigma_g^2$ serving as its asymptotic variance, i.e., \[\liminf_{t\to\infty}\frac{I_t(g)}{\sqrt{2\sigma_g^2\, t\ln\ln t}}=-1 \quad \text{and}\quad \limsup_{t\to\infty}\frac{I_t(g)}{\sqrt{2\sigma_g^2 \,t\ln\ln t}}=1\;\;\;\mathbb{P}_{\mu_0}-\text{a.s.}\] Moreover, $\sigma_g^2$ can also be expressed as $\sigma_g^2=2\int_X g\chi_g\,d\mu_*=\lim_{t\to \infty}\mathbb{E}_{\mu_0}\left[I_t^2(g) \right]/t.$
In the talk, we will discuss the techniques employed in the proof of this theorem and, if time permits, outline some of its potential applications. Basically, the proof strategy involves a reduction to the discrete-time case and a martingale-type decomposition of $\Phi$, enabling the use of $[3, \text{Theorem}~1]$ (a LIL criterion for martingales). The most challenging task, however, lies in transitioning from the stationary to the non-stationary setting, which we accomplish by leveraging certain ideas from $[1]$. The result applies, for instance, to solution process of stochastic differential equation with dissipative drift considered in $[4]$, as well as to diffusion processes on $\mathbb{R}^d$ discussed in $[\text{§4.1}, 2]$.

Bibliography

$[1]$ Bołt, W., Majewski, A. and Szarek, T. “An invariance principle for the law of the iterated logarithm for some Markov chains”. Studia Mathematica, vol. 212, 2012, pp. 41—53.

$[2]$ Cloez, B. and Hairer, M. “Exponential ergodicity for Markov processes with random switching”. Bernoulli, vol. 21, no. 1, 2015, pp. 505–536.

$[3]$ Heyde, C. and Scott, D. “Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments”. The Annals of Probability, vol. 1, no. 3, 1973, pp. 428–436.

$[4]$ Komorowski, T. and Walczuk, A. “Central limit theorem for Markov processes with spectral gap in the Wasserstein metric”. Stoch. Proc. Appl., vol. 122, no. 5, 2012, pp. 2155—2184.

Hybrid stochastic particle model of proliferating cells with chemotaxis.

Radosław Wieczorek

Stochastic particle models are very common in modern mathematical modelling, especially in biology and chemistry. In the talk we consider a situation when one of the population is abundant enough to use macroscopic approximation, while the other consist of a few particles and is described by a stochastic particle system. Such a scale separation leads to the so called hybrid models, where a stochastic particle system is coupled to partial differential equation.

A hybrid stochastic individual-based model of proliferating cells with chemotaxis $[2]$ will be presented. The model is expressed by a branching diffusion process coupled to a partial differential equation describing concentration of chemotactic factor. It is shown that in the hydrodynamic limit when number of cells goes to infinity the model converges to the solution of a nonconservative Patlak-Keller-Segel-type system. A nonlinear mean-field stochastic particle model is defined and it is proven that the movement of descendants of a single cell in the individual model converges to this mean-field process when the number of particles goes to infinity.

Bibliography

$[1]$ V. Capasso, R. Wieczorek: A hybrid stochastic model of retinal angiogenesis, Math. Methods Appl. Sci. 43(18) 10578–10592 (2020).

$[2]$ R. Wieczorek: Hydrodynamic limit of a stochastic model of proliferating cells with chemotaxis, Kinetic and Related Models 16 373–393 (2023).

Limit theorems for a general class of Markov processes on Polish spaces: with applications to PDMPs with random flow switching.

Hanna Wojewódka-Ściążko

This talk addresses the challenge of proving the Central Limit Theorem (CLT) and the Law of the Iterated Logarithm (LIL) for a general class of continuous-time Markov processes evolving on Polish metric spaces, where the initial distribution is not assumed to be stationary.

An important motivating example is the class of piecewise deterministic Markov processes (PDMPs), whose deterministic parts follow continuous semiflows that are randomly switched at the jump times of a Poisson process (for a detailed description, see Section $7$ of $[1]$). Such models arise naturally in applications, including autoregulated gene expression (cf. $[2]$).

Although limit theorems for stochastic processes have been extensively studied, only a few versions are available for Markov processes on general Polish spaces, and even fewer address the non-stationary case. Regarding the CLT, perhaps the most general result to date is Theorem $2.1$ in $[3]$. An analogous version of the LIL, albeit under slightly stronger assumptions, is given in Theorem $2.1$ in $[4]$. Unfortunately, we have not been able to apply either of these results to the class of PDMPs with random flow switching discussed above. The main obstacle lies in our inability to verify exponential mixing in the Wasserstein distance, as required by both theorems. Instead, we are only able to establish exponential mixing in the bounded-Lipschitz (Fortet–Mourier) distance.

To overcome this limitation, we have developed new versions of limit theorems that apply under weaker assumptions, specifically:

a version of the CLT for nonstationary Markov-Feller processes (governed by semigroups $\{P(t)\}_{t\geq 0}$) with a Polish state space $E$, which are exponentially mixing in the bounded-Lipschitz distance, meaning that \[\begin{gathered} \text{there exist a continuous function $V:E\to[0,\infty)$}\\ \text{and constants $\gamma, \beta>0$, $\delta\in(0,1)$ such that,}\\ \text{for any Borel probability measures $\mu$, $\nu$ and every $t\geq 0$}\\ d_{\text{FM}}\left(\mu P(t),\nu P(t)\right)\leq \beta e^{-\gamma t}\left(\int_E V\;d(\mu+\nu)+1\right)^{\delta},\end{gathered}\] and satisfy a continuous Foster-Lyapunov condition, i.e. \[\begin{gathered} \text{there exist $A,B\geq 0$ and $\Gamma>0$ such that}\\ P(t)V^2(x)\leq Ae^{-\Gamma t}V^2(x)+B\quad \text{for all}\quad x\in E,\;t\geq 0\end{gathered}\] (cf. Theorem $3.1$ in $[1]$);
a version of the LIL for a similar, though slightly narrower, class of processes. The main difference lies in replacing the exponential mixing condition with the assumption of the existence of an appropriate Markovian coupling. This type of assumption originates from $[5]$, and it naturally implies the exponential mixing property of the semigroup $\{P(t)\}_{t\geq 0}$.

Both of these results will be discussed in detail during the talk.

Bibliography

$[1]$ D. Czapla, K. Horbacz, H. Wojewódka-Ściążko. “The central limit theorem for Markov processes that are exponentially ergodic in the bounded-Lipschitz norm”. Qual. Theory Dyn. Syst., vol. 23, no. 7, 2023.

$[2]$ S.C. Hille, K. Horbacz, T. Szarek. “Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene”. Ann. Math. Blaise Pascal, vol. 23, no. 2, 2016, pp. 171–217.

$[3]$ T. Komorowski, A. Walczuk. “Central limit theorem for Markov processes with spectral gap in the Wasserstein metric”. Stoch. Proc. Appl., vol. 122, no. 5, 2012, pp. 2155–2184.

$[4]$ D. Czapla, S.C. Hille, K. Horbacz, H. Wojewódka-Ściążko. “Law of the iterated logarithm for Markov semigroups with exponential mixing in the Wasserstein distance”. Submitted, 2025.

$[5]$ M. Hairer. “Exponential mixing properties of stochastic PDEs through asymptotic coupling”. Probab. Theory Related Fields, vol. 124, 2002.