Section: New Results

Piecewise deterministic Markov processes

Participants: A. Crudu, A. Debussche, A. Muller-Gueudin, O. Radulescu.

Piecewise deterministic Markov processes are models which feature in a prominent way in Biomedical applications. They appear in two contributions of our team this year.

(1) Convergence of stochastic gene networks. In [24] , [5] , we propose simplified models for the stochastic dynamics of gene network models arising in molecular biology. Those gene networks are classically modeled by Markov jump processes, which are extremely time consuming. To overcome this drawback, we study the asymptotic behavior of multiscale stochastic gene networks using weak limits of Markov jump processes.

We consider a set of chemical reactions $R_r$, $r\in ℛ$; $ℛ$ is supposed to be finite. These reactions involve species indexed by a set $S=1,\cdots ,M$, the number of molecules of the species $i$ is denoted by $n_i$ and $X\in {\mathbb{N}}^M$ is the vector consisting of the $n_i$'s. Each reaction $R_r$ has a rate ${\lambda }_r\left(X\right)$ which depends on the state of the system, described by $X$ and corresponds to a change $X\to X+{\gamma }_r$, ${\gamma }_r\in {\mathbb{Z}}^M$.

Mathematically, this evolution can be described by the following Markov jump process. It is based on a sequence ${\left({\tau }_k\right)}_{k\ge 1}$ of random waiting times with exponential distribution. Setting $T_0=0$, $T_i={\tau }_1+\cdots +{\tau }_i$, $X$ is constant on $[T_{i-1},T_i)$ and has a jump at $T_i$. The parameter of ${\tau }_i$ is given by ${\sum }_{r\in ℛ}{\lambda }_r\left(X\left(T_{i-1}\right)\right)$:

At time $T_i$, a reaction $r\in ℛ$ is chosen with probability ${\lambda }_r\left(X\left(T_{i-1}\right)\right)/{\sum }_{r\in ℛ}{\lambda }_r\left(X\left(T_{i-1}\right)\right)$ and the state changes according to $X\to X+{\gamma }_r$: $X\left(T_i\right)=X\left(T_{i-1}\right)+{\gamma }_r$. This Markov process has the following generator:

In the applications we have in mind, the numbers of molecules have different scales. Some of the molecules are in small numbers and some are in large numbers. Accordingly, we split the set of species into two sets $C$ and $D$ with cardinals $M_C$ and $M_D$. This induces the decomposition $X=(X_C,X_D)$, ${\gamma }_r=({\gamma }_r^C,{\gamma }_r^D)$. For $i\in D$, $n_i$ is of order 1 while for $i\in C$, $n_i$ is proportional to $N$ where $N$ is a large number. For $i\in C$, setting ${\tilde{n}}_i=n_i/N$, ${\tilde{n}}_i$ is of order 1. We define $x_C=X_C/N$ and $x=(x_C,X_D)$.

For this kind of system, we are able to give in [5] some relevant information on the asymptotic regime $N\to \infty$ when different type of reactions are involved. Depending on the time and concentration scales of the system we distinguish four types of limits:

• Continuous piecewise deterministic processes (PDP) with switching.

• PDP with jumps in the continuous variables.

• Averaged PDP.

• PDP with singular switching.

We justify rigorously the convergence for the four types of limits.

(2) Variable length Markov chains. A classical random walk $(S_n,n\in \mathbb{N})$ is defined by $S_n:={\sum }_{k=0}^n{X}_k$, where $\left(X_k\right)$ are i.i.d. When the increments ${\left(X_k\right)}_{k\in \mathbb{N}}$ are a one-order Markov chain, a short memory is introduced in the dynamics of $\left(S_n\right)$. This so-called “persistent” random walk is no longer Markovian and, under suitable conditions, the rescaled process converges towards the integrated telegraph noise (ITN) as the time-scale and space-scale parameters tend to zero (see [70] , [71] , [50] ). The ITN process is effectively non-Markovian too. In [28] our aim has been to consider persistent random walks $\left(S_t\right)$ whose increments are Markov chains with variable order which can be infinite.

Associated with a process $\left(X_n\right)$ which takes its values in a finite set, we consider an integer valued process $\left(M_n\right)$ so that $(X_n,M_n)$ is Markov and $M_n$ measures the size of the memory at time $n$. This variable memory is justified by a one-to-one correspondence between $\left(X_n\right)$ and a suitable Variable Length Markov Chain (VLMC), since for a VLMC the dependency from the past can be unbounded. We prove in [28] that, under a suitable rescaling, $(S_n,X_n,M_n)$ converges in distribution towards a time continuous process $(S^0\left(t\right),X\left(t\right),M\left(t\right))$. The process $\left(S^0\left(t\right)\right)$ is a semi-Markov and Piecewise Deterministic Markov Process whose paths are piecewise linear.

Observe that, though our study in [28] is made at a theoretical level, it leads to potentially interesting applications in growth models for tumors. This kind of link will be developed in the next future.