## On the Asymptotics of an Infinite-Dimensional Stochastic Dynamical System

*13 Nov 2017 11:43*

I am working on an idea where I need to show that, in the long run, a certain infinite-dimensional discrete-time stochastic dynamical system has a stable limiting distribution, and calculate certain properties of that distribution. This notebook is for storing related references.

I'm a little paranoid about being scooped, so I won't say much more, only
that there are certain connections with genetic
algorithms, and that the finite-dimensional analog is related to the
replicator equation of evolutionary biology. In
that representation, the fitness function would show a particular kind of
frequency-dependence, with random fluctuations which are *not*
necessarily independent over time, though I'd be willing to ignore serial
dependence if the alternatives were simply intractable. Now, the replicator
equation can be re-written as a linear system (or so Nihat tells me), which may
be helpful...

**Update**: Actually, the project
was this paper, and I ended
up being able to solve the problem *without* gong through the more
direct route I had in mind here. Which I still think is interesting!

See also: Ergodic Theory; Filtering (since related equations occur in nonlinear filtering); Stochastic Processes

- Definitely useful for this project:
- Nihat Ay and Ionas Erb, "On the Linearity of Replicator Equations",
SFI
Working Paper 2003-10-053 [I
*think*this is applicable to my problem, but I need to carefully re-read it to make sure. Of course, I have a stochastic fitness function, whereas this is all about deterministic ones, but an averaging argument should take care of that.] - Michel Benaïm, "Dynamics of stochastic approximation
algorithms", Séminaire de probabilités (Strasbourg)
**33**(1999): 1--68 [Link to full text, bibliography, etc.] - Nicolas Champagnat, "Large Deviations for Singular and Degenerate Diffusion Models in Adaptive Evolution", arxiv:0903.2345
- Robin Pemantle, "A Survey of Random Processes with Reinforcement", arxiv:math/0610076

- To read:
- Arash A. Amini, XuanLong Nguyen, "Bayesian inference as iterated random functions with applications to sequential inference in graphical models", arxiv:1311.0072
- Siva R. Athreya, Richard F. Bass, Maria Gordina, and Edwin A. Perkins, "Infinite dimensional stochastic differential equations of Ornstein-Uhlenbeck type", math.PR/0503165
- Yuri Bakhtin and Jonathan C. Mattingly, "Malliavin Calculus for Infinite-Dimensional Systems with Additive Noise", math.PR/0610754
- J. Bérard and A. Bienvenüe, "Sharp asymptotic results
for simplified mutation-selection algorithms", The Annals of Applied
Probability
**13**(2003): 1534--1568 - Àngel Calsina and Sílvia Cuadrado, "Small mutation
rate and evolutionarily stable strategies in infinite-dimensional adaptive
dynamics",
Mathematical
Biology
**48**(2004): 135--159 - Alain Cercuiel and Olivier François, "Sharp Asymptotics for
Fixation Times in Stochastic Population Genetics Models at Low Mutation
Probabilities", Journal of Statistical
Physics
**110**(2003): 311--332 - Fabio A. C. C. Chalub and Max O. Souza, "The continuous limit of the Moran process and the diffusion of mutant genes in infinite populations", math.AP/0602530
- Igor Chueshov, Jinqiao Duan and Björn Schmalfuss, "Determining
functionals for random partial differential equations", Nonlinear
Differential Equations and Applications
**10**(2003): 431--454 = math.DS/0409481 - Giuseppe Da Prato and Jerzy Zabczyk [apparently mostly
continuous time, which is more complicated than I need]
- Ergodicity for Infinite Dimensional Systems
- Stochastic Equations in Infinite Dimensions

- P. Del Moral, "Measure-Valued Processes and Interacting Particle
Systems. Application to Nonlinear Filtering Problems", The Annals of
Applied Probability
**8**(1998): 438--495 [JSTOR] - Pierre Del Moral and Arnaud Doucet, "Interacting Markov chain Monte Carlo methods for solving nonlinear measure-valued equations", Annals of Applied Probability
**20**(2010): 593--639 - Iva Dosálková and Pavel Kindlmann, "Evolutionarily
stable strategies for stochastic processes", Theoretical Population
Biology
**65**(2004): 205--210 - Jinqiao Duan, Kening Lu and Bjorn Schmalfuss, "Smooth stable and unstable manifolds for stochastic partial differential equations", math.DS/0409483
- Stefano Favaro, Alessandra Guglielmi, and Stephen G. Walker, "A class of measure-valued Markov chains and Bayesian nonparametrics",
Bernoulli
**18**(2012): 1002--1030 - Wendell H. Fleming and Michel Viot, "Some Measure-Valued Markov Processes in Population Genetics Theory", Indiana University Mathematics Journal
**28**(1979): 817--843 [JSTOR] - Tobias Galla
- "Dynamics of random replicators with Hebbian interactions", Journal of Statistical Mechanics: Theory and Experiment (2005) P11005, cond-mat/0507473
- "Random replicators with asymmetric couplings", cond-mat/0508174 ["The dynamics of random replicators is studied using generating functional techniques... We first discuss in detail how dynamical theories can be formulated for general replicator models in terms of an effective single-species process, and how persistent order parameters of the ergodic stationary states can be extracted from this process. We then detail how different types of dynamical phase transitions can be identified and related to each other. As an application of the general theory we address replicator models with Gaussian couplings of arbitrary symmetry between pairs and triples of species, respectively. Numerical simulations verify our theory, and also indicate regimes in which only a finite number of species survives, even when the thermodynamic limit is considered."]

- Mats Gyllenberg and Géza Meszéna, "On the
impossibility of coexistence of infinitely many strategies", Journal of
Mathematical Biology
**50**(2005): 133--160 [This would be*very*useful to me, if the result generalizes to my setting.] - Dirk Helbing and Nicole J. Saam, "Analytical Investigation of Innovation Dynamics Considering Stochasticity in the Evaluation of Fitness", cond-mat/051217
- David Hochberg, M.-P. Zorzano, Federico Moran, "Complex noise in
diffusion-limited reactions of replicating and competing
species", cond-mat/0606378
= Physical
Review E
**73**(2006): 066109 - Lorens A. Imhof, "The long-run behavior of the stochastic
replicator dynamics", Annals of Applied
Probability
**15**(2005): 1019--1045 = math.PR/0503529 - Vassili N. Kolokoltsov, "Nonlinear Markov Semigroups and
Interacting Lévy Type
Processes", Journal
of Statistical Physics
**126**(2007): 585-642 - Kai Liu, "Uniform stability of autonomous linear stochastic
functional differential equations in infinite dimensions", Stochastic Processes
and Their Applications
**115**(2005): 1131--1165 - A.G. Munoz, J. Ojeda, D. Sierra and T. Soldovieri, "Variational and Potential Formulation for Stochastic Partial Differential Equations", nlin.SI/0502010
- Marcello Pelillo, "Replicator Equations, Maximal Cliques, and Graph
Isomorphism", Neural Computation
**11**(1999): 1933--1955 - A. J. Roberts, "Resolve the multitude of microscale interactions to model stochastic partial differential equations", math.DS/0506533
- Wilhelm Stannat, "On the convergence of genetic algorithms --- a
variational approach", Probability Theory and
Related Fields
**129**(2004): 113--132 - Mark Stegeman and Paul Rhode, "Stochastic Darwinian equilibria
in small and large populations", Games and Economic Behavior
**49**(2004): 171--214 - Joe Suzuki, "A Markov chain analysis of genetic algorithms: large deviation principle approach", Journal of Applied
Probability
**47**(2010): 967--975 - Anatoly V. Swishchuk and Jianhong Wu, Evolution of Biological Systems in Random Media: Limit Theorems and Stability
- Marcel O. Vlad, Stefan E. Szedlacsek, Nader Pourmand, L. Luca
Cavalli-Sforza, Peter Oefner and John Ross, "Fisher's theorems for
multivariable, time- and space-dependent systems, with applications in
population genetics and chemical kinetics", PNAS
**102**(2005): 9848--9853