Metastability: A Brief Introduction Through Three Examples
Metastability is a very frequent phenomenon in nature. It also finds many applications in science and engineering. A noticeable basic feature is the presence of “quasi-equilibria states” and relatively sudden transitions between them. The goal of this short expository note is to discuss some aspects of the stochastic modeling of metastability, usually done through the consideration of special stochastic processes. This includes a “pathwise approach” developed since the 1980s. Thought as an invitation to the readership, three examples are quickly reviewed, starting with a class of reaction-diffusion equations subject to a small stochastic noise, for which the theory of large deviations has been a very useful tool, and further precision achieved through the help of potential theoretical techniques. We present then brief summaries of results on the Harris contact process on suitable finite graphs, and a quick discussion of stochastic dynamics for the well-known Ising model. The first can be thought as an oversimplified model for the propagation of an infection, and the second has been used in the context of magnetization. From a probabilistic analysis and technical viewpoint, the Ising model enjoys time-reversibility, which provides useful tools, while the contact process is non-reversible.
M. E. Vares acknowledges support of CNPq (grant 305075/2016-0) and FAPERJ (grant E-26/203.048/2016).
- 8.A. Bianchi, A. Gaudillière, P. Milanesi: On soft capacities, quasi-stationary distributions and the pathwise approach to metastability. arXiv:1807.11233Google Scholar
- 10.A. Bovier, F. den Hollander: Metastability: A potential theoretic approach. Springer (2015)Google Scholar
- 13.V. H. Can. Metastability for the contact process on the preferential attachment graph. Internet Math. 45pp. (2017)Google Scholar
- 21.A. Debussche, M. Högele, and P. Imkeller: The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise, Lecture Notes in Mathematics 2085, Springer (2013)Google Scholar
- 23.R. Durrett: Random Graph Dyamics. Cambridge Univ. Press, Cambridge (2007)Google Scholar
- 26.J. Farfan, C. Landim, K. Tsunoda: Static large deviations for a reaction-diffusion model. arXiv:1606.07227 (2016)Google Scholar
- 28.M. I. Freidlin and A. D. Wentzell: Random Perturbations of Dynamical Systems. Grundlehren der mathematischen Wissenschaften. Springer, Berlin- Heidelberg (1998)Google Scholar
- 30.A. Gaudillière, P. Milanesi, M. E. Vares. Asymptotic exponential law for the transition time to equilibrium of the metastable kinetic Ising model with vanishing magnetic field. arXiv:1809.07044Google Scholar
- 32.D. Henry: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics 840, Berlin-Heidelberg-New York: Springer-Verlag., (1981)Google Scholar
- 47.E. Olivieri, M. E. Vares: Large deviations and metastability. Cambridge University Press (2005)Google Scholar
- 50.M. Salzano: The contact process on graphs. PhD thesis, UCLA, (2000). (Reprinted Publicações Matemáticas. IMPA, 2003.)Google Scholar