# Teoria do Problema de Martingale

### Martingale problem and lumpability for Markov processes (I)

The martingale problem is an important tool in probability theory introduced by Stroock and Varadhan as an approach to Markov processes (see [8, 9]). The purpose of this course is to present a detailed introduction to the martingale problem theory as well as a review of recent applications to the study of two different subjects:

1. metastability of zero range processes and
2. Kingman’s process as scaling limit for coalescing random walks.

Interestingly, they both will be tackled as instances of a more general problem: lumpability for Markov processes.

In the first part, we motivate the martingale problem as an analogue of an ordinary differential equation (ode). In the same way that a solution of an ode is a path whose mechanism of evolution is determined by a vector field, a solution of a martingale problem is a random path whose mechanism of evolution is determined by a so-called Markov generator. As it is well known, the flow property for an ode is a consequence of uniqueness for the solutions. Analogously, we get the Markov property as a consequence of uniqueness for the solutions of a martingale problem. As a canonical example, the brownian motion is the solution of the martingale problem corresponding to the Laplace operator. In addition, we shall see that an ode can be seen in fact as an example of a martingale problem whose Markov generator is the derivation determined by the vector field. We shall also introduce the Markov processes that will be studied via the martingale problem.

In the last part of the course we apply the martingale problem theory to get a sort of asymptotic lumpability for a sequence of Markov processes. After discussing some general results, we shall illustrate this approach by getting some interesting results for two models:

1. Zero Range processes can exhibit condensating behaviour for a suitable choice of jump rates, in the sense that the microcanonical invariant measures of the process are asymptotically supported on configurations where a macroscopically large fraction of the particles (condensate) is found on a single site (see e.g. [2, 5, 7]). The location of this site may not be determined by the invariant measure, and in this case one expects the condensate to migrate between sites on a proper time scale. We shall use the martingale problem approach to prove asymptotic results on the motion of the condensate for reversible zero range processes on a (fixed) finite set of sites.
2. The Kingman’s process is a very well known model in coalescent theory (see [3]). It is believed that this process is a universal scaling limit for a system of coalescing random walks on large transitive graphs. We shall present some progress in this direction by using an approach based on the martingale problem. In addition, a system of coalescing random walks for which a scaling limit for the full coalescence time has been extensively studied (see [4, 6] and Chapter 14 of [1]).

### References

1. Aldous, D., & Fill, J. (2002). Reversible Markov chains and random walks on graphs.
2. Beltran, J., & Landim, C. (2012). Metastability of reversible condensed zero range processes on a finite set. Probability Theory and Related Fields, 152(3-4), 781-807.
3. Berestycki, N. (2009). Recent progress in coalescent theory. Ensaios Matematicos, 16(1), 1-193.
4. Cox, J. T. (1989). Coalescing random walks and voter model consensus times on the torus in $\mathbb{Z}^d$. The Annals of Probability, 1333-1366.
5. Evans, M. R., & Hanney, T. (2005). Nonequilibrium statistical mechanics of the zero-range process and related models. Journal of Physics A: Mathematical and General, 38(19), R195.
6. Oliveira, R. I. (2013). Mean field conditions for coalescing random walks. The Annals of Probability, 41(5), 3420-3461.
7. Seo, I. (2018). Condensation of non-reversible zero-range processes.
8. D. W. Stroock, S. R. S. Varadhan (1979). Multidimensional Diffusion Processes. Grundlehren der mathematischen Wissenschaften vol. 233, Springer.
9. D. W. Stroock, S. R. S. Varadhan. Diffusion processes with boundary conditions. Comm. Pure Appl. Math., 24 147–225 (1971).

### Course Program

Week 1
Conditional probability distributions, the martingale problem and the cadlag path space.
Week 2
Uniqueness for a martingale problem, Markov property and examples.
Week 3
Convergence of random paths in Skorokhod topology. Assymptotic lumpability.
Week 4
Kingman’s process and coalescing random walks. Trace process and metastability of zero range process.