Measure-valued processes and related topics

This is an introduction to some research results of the author and his collaborators by the year 2011. Most of the results are related to measure-valued branching processes, a class of infinite-dimensional Markov processes with beautiful mathematical structures and interesting applications. The reader may refer to Dawson [D1] for the backgrounds of the subject. The results summarized in paragraphs 1–4 have been retreated in the monograph [1] of the author. The publications of the author have been cited in papers by K.B. Athreya, D.A. Dawson, J.-F. Delmas, E.B. Dynkin, S.N. Ethier, P.J. Fitzsimmons, L.G. Gorostiza, R.C. Griffiths, K. Handa, T.G. Kurtz, A. Lambert, S. Méléard, K.V. Mitov, E.A. Perkins, M. Röckner, W. Schachermayer, B. Schmuland, A. Shied, A. Wakolbinger, L. Zambotti and many others.

[1]  N. Konno,et al.  Stochastic partial differential equations for some measure-valued diffusions , 1988 .

[2]  Donald A. Dawson,et al.  Measure-Valued processes and renormalization of branching particle systems , 1999 .

[3]  Nonlocal Branching Superprocesses and Some Related Models , 2002, math/0606616.

[4]  Claudia Ceci,et al.  Modelling a Multitype Branching Brownian Motion: Filtering of a Measure-Valued Process , 2006 .

[5]  Critical age-dependent branching Markov processes and their scaling limits , 2007, math/0701661.

[6]  Geometric Analysis for Symmetric Fleming–Viot Operators: Rademacher's Theorem and Exponential Families , 2002 .

[7]  Viet Chi Tran,et al.  Stochastic and deterministic models for age-structured populations with genetically variable traits , 2008, 0809.3767.

[8]  T. Shiga,et al.  A Reversibility Problem for Fleming-Viot Processes , 1999 .

[9]  Laurence Marsalle,et al.  Limit theorems for Markov processes indexed by continuous time Galton--Watson trees , 2009, 0911.1973.

[10]  Amaury Lambert,et al.  Proof(s) of the Lamperti representation of Continuous-State Branching Processes , 2008, 0802.2693.

[11]  Donald A. Dawson,et al.  Measure-valued Markov processes , 1993 .

[12]  M. Barczy,et al.  A Jump type SDE approach to positive self-Similar Markov processes , 2011, 1111.3235.

[13]  W. Schachermayer,et al.  Affine processes are regular , 2009, 0906.3392.

[14]  D. Duffie,et al.  Affine Processes and Application in Finance , 2002 .

[15]  M. Caballero,et al.  A Lamperti-type representation of continuous-state branching processes with immigration , 2010, 1012.2346.

[16]  K. Handa.  Reversible distributions of multi-allelic Gillespie?Sato diffusion models , 2004 .

[17]  Robert C. Griffiths,et al.  The Transition Function of a Measure-Valued Branching Diffusion with Immigration , 1993 .

[18]  V. Bogachev,et al.  Generalized Mehler semigroups and applications , 1996 .

[19]  V. Tran,et al.  Branching Feller diffusion for cell division with parasite infection , 2010, 1004.0873.

[20]  Generalized Mehler Semigroups and Catalytic Branching Processes with Immigration , 2004, math/0606619.

[21]  Donald A. Dawson,et al.  Stochastic equations, flows and measure-valued processes , 2010, 1009.0578.

[22]  Zenghu Li.  Immigration structures associated with Dawson-Watanabe superprocesses , 1996 .

[23]  V. Tran,et al.  Slow and fast scales for superprocess limits of age-structured populations , 2010, 1006.5136.

[24]  Quasi-invariance and reversibility in the Fleming–Viot process , 1999, math/9909189.

[25]  A. Lambert.  Quasi-Stationary Distributions and the Continuous-State Branching Process Conditioned to Be Never Extinct , 2007 .

[26]  E. Dynkin.  Superprocesses and Partial Differential Equations , 1993 .

[27]  Stewart N. Ethier,et al.  Fleming-Viot processes in population genetics , 1993 .

[28]  D. Dawson,et al.  Skew convolution semigroups and affine Markov processes , 2005, math/0505444.

[29]  A. Barbour,et al.  A TRANSITION FUNCTION EXPANSION FOR A DIFFUSION MODEL WITH SELECTION , 2000 .

[30]  J. Taylor.  The Common Ancestor Process for a Wright-Fisher Diffusion , 2007 .

[31]  Poisson representations of branching Markov and measure-valued branching processes. , 2011, 1104.1496.

[32]  D. Dawson,et al.  Construction of immigration superprocesses with dependent spatial motion from one-dimensional excursions , 2002, math/0606618.

[33]  Self-Intersection Local Time for ′(ℝd)-Ornstein-Uhlenbeck Processes Arising from Immigration Systems , 2002 .

[34]  T. Shiga,et al.  Measure-valued branching diffusions: immigrations, excursions and limit theorems , 1995 .