A class of degenerate diffusion processes occurring in population genetics

Optimal control problems for degenerate two-dimensional diffusion processes are considered. The processes could serve as models for wear processes. The objective is to make the controlled process leave the continuation region through a given part of the boundary. Explicit and exact solutions are obtained for important processes such as the geometric Brownian motion and the Ornstein-Uhlenbeck process.

LetY(t) be a one-dimensional diffusion process anddX(t)=Y(t)dt. The process (X(t), Y(t)) is considered in the second quadrant. First, the probability that (X(t), Y(t)) will hit thex-axis before they-axis is computed explicitly whenY(t) is a standard Brownian motion or a particular case of the Bessel process. Next, an exact expression is obtained for the average time it takes (X(t), Y(t)) to exit the regionC={(x, y) ∈ ℝ2 :x < 0,0 <d 1 <y <d 2 whenY(t) is a standard Brownian motion. The solution is expressed as a generalized Fourier series.

In this chapter we express the classical Wigner function by a degenerate diffusion process [164] solving a randomly perturbed Newton equation. We discuss first some estimates on the solution of these equations and the long time behaviour of the diffusion process. We discuss a relation between the classical Wigner function and the quantum one. We approximate the large time behaviour of the density matrix by the classical Wigner function. In this way we obtain some estimates on the decoherence in space expressed by the correlation functions of the diffusion process.

This study formulates an open problem on spatial localization for quasilinear parabolic equations of pLaplacian type on the half-space R+N, with smooth, nonnegative initial data and boundary conditions that grow unbounded over time. While localization has been widely studied under homogeneous or decaying boundary conditions, its persistence under unbounded boundary input remains unresolved. We introduce two forms of spatial localization: effective localization, where the solution remains within a finite spatial domain, and strict localization, where it vanishes beyond a fixed boundary. The problem examines how the structure of the nonlinearities A(v) and B(v), and the growth rate of the boundary function ϕ influence solution confinement or spread. The study extends classical localization theory to degenerate and singular diffusion processes with diffusion exponent p &gt; 1, offering new insight into spatial confinement under persistent external forcing.

We study the sensitivity of the densities of non degenerate diffusion processes and related Markov Chains with respect to a perturbation of the coefficients. Natural applications of these results appear in models with misspecified coefficients or for the investigation of the weak error of the Euler scheme with irregular coefficients.

We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in population genetics, the so-called generalized Kimura diffusion operators Epstein and Mazzeo [SIAM J. Math. Anal. 42 (2010) 568–608; Degenerate Diffusion Operators Arising in Population Biology (2013) Princeton University Press; Applied Mathematics Research Express (2016)]. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients and the proof of the scale-invariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on Feynman–Kac formulas concerning the assumptions on the degeneracy of the diffusion matrix, the boundedness of the drift coefficients and the a priori regularity of the weak solutions.

The direct Lyapunov method for stability of dynamical systems is extended to the almost sure stability of Ito stochastic differential equations by means of semicontinuous Lyapunov functions satisfying a suitable system of partial differential inequalities in viscosity sense. The a.s. stabilizability of controlled degenerate diffusion processes is also treated. The key tools are some geometric characterizations of the invariance and viability properties of closed sets for controlled diffusions obtained by PDE-viscosity methods.

A CLT for degenerate diffusions with periodic coefficients, and application to homogenization of linear PDEs

Previous article Next article Haploid Populations Subject to Varying Environment: The Characteristic Values and the Rate of Loss of AllelesKeith GladstienKeith Gladstienhttps://doi.org/10.1137/0132065PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA class of stochastic matrices, arising in genetics, are considered. The characteristic values are obtained and related to the genetic process.[1] C. Cannings, The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models, Advances in Appl. Probability, 6 (1974), 260–290 MR0343949 0284.60064 CrossrefGoogle Scholar[2] C. Cannings, The latent roots of certain Markov chains arising in genetics: a new approach. II. Further haploid models, Advances in Appl. Probability, 7 (1975), 264–282 MR0371430 0339.60067 CrossrefISIGoogle Scholar[3] A. B. Chia, Random mating in a population of cyclic size, J. Appl. Probability, 5 (1968), 21–30 MR0225583 0162.51901 CrossrefISIGoogle Scholar[4] William Feller, Diffusion processes in genetics, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, 227–246 MR0046022 0045.09302 Google Scholar[5] Joseph Felsenstein, The rate of loss of multiple alleles in finite haploid populations, Theoret. Population Biology, 2 (1971), 391–403 10.1016/0040-5809(71)90028-1 MR0332215 0234.92010 CrossrefGoogle Scholar[6] Keith Gladstien, Loss of alleles in a haploid population with varying environment, Theoret. Population Biology, 10 (1976), 383–394 10.1016/0040-5809(76)90025-3 MR0496833 0346.92014 CrossrefISIGoogle Scholar[7] Keith Gladstien, The characteristic values and vectors for a class of stochastic matrices arising in genetics, SIAM J. Appl. Math., 34 (1978), 630–642 10.1137/0134050 MR0475977 0386.92004 LinkISIGoogle Scholar[8] Samuel Karlin, Equilibrium behavior of population genetic models with non-random mating. II. Pedigrees, homozygosity and stochastic models, J. Appl. Probability, 5 (1968), 487–566 MR0246657 0205.49601 CrossrefISIGoogle Scholar[9] S. Karlin, Rates of approach to homozygosity for finite stochastic models with variable population size, Amer. Naturalist, 102 (1968), 443–455 10.1086/282557 CrossrefISIGoogle Scholar[10] S. Wright, Evolution in Mendelain populations, Genetics, 16 (1931), 97–159 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Looking Forwards and Backwards in the Multi-Allelic Neutral Cannings Population ModelJournal of Applied Probability, Vol. 47, No. 03 | 14 July 2016 Cross Ref Looking Forwards and Backwards in the Multi-Allelic Neutral Cannings Population ModelJournal of Applied Probability, Vol. 47, No. 3 | 14 July 2016 Cross Ref The time back to the most recent common ancestor in exchangeable population modelsAdvances in Applied Probability, Vol. 36, No. 01 | 1 July 2016 Cross Ref The time back to the most recent common ancestor in exchangeable population modelsAdvances in Applied Probability, Vol. 36, No. 1 | 1 July 2016 Cross Ref Robustness results for the coalescentJournal of Applied Probability, Vol. 35, No. 02 | 14 July 2016 Cross Ref Robustness results for the coalescentJournal of Applied Probability, Vol. 35, No. 2 | 14 July 2016 Cross Ref Line-of-descent and genealogical processes, and their applications in population genetics modelsTheoretical Population Biology, Vol. 26, No. 2 | 1 Oct 1984 Cross Ref The Characteristic Values and Vectors for a Class of Stochastic Matrices Arising in GeneticsKeith GladstienSIAM Journal on Applied Mathematics, Vol. 34, No. 4 | 12 July 2006AbstractPDF (1212 KB) Volume 32, Issue 4| 1977SIAM Journal on Applied Mathematics707-895 History Submitted:29 April 1976Published online:12 July 2006 InformationCopyright © 1977 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0132065Article page range:pp. 778-783ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

Article Communication structure of discretized degenerate diffusion processes and approximation of Lyapunov exponents was published on January 1, 2001 in the journal Monte Carlo Methods and Applications (volume 7, issue 3-4).

In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied in this paper: the weakly degenerate case, that is, if the reciprocal of the diffusion coefficient is summable, and the strongly degenerate case, that is, if that reciprocal isn’t summable. In our main result we show that the above systems can be steered from an initial continuous state that admits a finite number of points of sign change to a target state with the same number of changes of sign in the same order. Our method uses a recent technique introduced for uniformly parabolic equations employing the shifting of the points of sign change by making use of a finite sequence of initial-value pure diffusion problems. Our interest in degenerate reaction-diffusion equations is motivated by the study of some energy balance models in climatology (see,e.g., the Budyko-Sellers model) and some models in population genetics (see,e.g., the Fleming-Viot model).

A diffusion model in population genetics with dynamic fitness

A certain class of infinite dimensional diffusion process arising in population genetics : Tokuzo Shiga, Tokyo Institute of Technology, Japan

Ideas, behaviors, and opinions spread through social networks. If the probability of spreading to a new individual is a nonlinear function of the fraction of the individuals' affected neighbors, such a spreading process becomes a "complex contagion." This nonlinearity does not typically appear with physically spreading infections, but instead can emerge when the concept that is spreading is subject to game theoretical considerations (e.g., for choices of strategy or behavior) or psychological effects such as social reinforcement and other forms of peer influence (e.g., for ideas, preferences, or opinions). Here we study how the stochastic dynamics of such complex contagions are affected by the underlying network structure. Motivated by simulations of complex contagions on real social networks, we present a framework for analyzing the statistics of contagions with arbitrary nonlinear adoption probabilities based on the mathematical tools of population genetics. The central idea is to use an effective lower-dimensional diffusion process to approximate the statistics of the contagion. This leads to a tradeoff between the effects of "selection" (microscopic tendencies for an idea to spread or die out), random drift, and network structure. Our framework illustrates intuitively several key properties of complex contagions: stronger community structure and network sparsity can significantly enhance the spread, while broad degree distributions dampen the effect of selection compared to random drift. Finally, we show that some structural features can exhibit critical values that demarcate regimes where global contagions become possible for networks of arbitrary size. Our results draw parallels between the competition of genes in a population and memes in a world of minds and ideas. Our tools provide insight into the spread of information, behaviors, and ideas via social influence, and highlight the role of macroscopic network structure in determining their fate.

We analyze the large time behavior of a stochastic model for the lay down of fibers on a moving conveyor belt in the production process of nonwovens. It is shown that under weak conditions this degenerate diffusion process has a unique invariant distribution and is even geometrically ergodic. This generalizes results from previous works [M. Grothaus and A. Klar, SIAM J. Math. Anal., 40 (2008), pp. 968--983; J. Dolbeault et al., arXiv:1201.2156] concerning the case of a stationary conveyor belt, in which the situation of a moving conveyor belt has been left open.