2. Branching processes and superprocesses — basic notions

 

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[2]    Dynkin, E. B. (1994) An introduction to branching measure-valued processes. CRM Monograph Series, 6. American Mathematical Society, Providence, RI, 134 pp. MR1280712

[3]    Dynkin, E. B. (2001) Branching exit Markov systems and superprocesses. Ann. Probab. 29(4), 1833–1858. MR1880244

[4]    Engländer, J. and Pinsky, R. (1999) On the construction and support properties of measure-valued diffusions on D ⊆ R^d with spatially dependent branching, Ann. Probab. 27(2), 684–730. MR1698955

[5]    Etheridge, A. (2000) An introduction to superprocesses. AMS lecture notes. MR1779100

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[8]    Le Gall, J.-F. (2000) Processus de branchement, arbres et superprocessus. (French) [Branching processes, trees and superprocesses] Development of mathematics 1950–2000, 763–793, Birkhäuser, Basel. MR1796858

[9]    Perkins, E. (2002) Dawson-Watanabe superprocesses and measure-valued diffusions. Lectures on probability theory and statistics (Saint-Flour, 1999), 125–324, Lecture Notes in Math., 1781, Springer, Berlin. MR1915445

[10]   Pinsky, R. G. (1995) Positive Harmonic Functions and Diffusion. Cambridge University Press. MR1326606

 

3. Connection between spatial branching processes and nonlinear partial differential equations; the qualitative behavior of superprocesses

 

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[2]    Delmas, J.-F. (1999) Path properties of superprocesses with a general branching mechanism. Ann. Probab. 27(3), 1099–1134. MR1733142

[3]    Duquesne, T. and Le Gall, J.-F. (2002) Random trees, Lévy processes and spatial branching processes. Astérisque No. 281 MR1954248

[4]    Dynkin, E. B. (1993) Superprocesses and partial differential equations. Ann. Probab. 21(3), 1185–1262. MR1235414

[5]    Dynkin, E. B. (2004) Superdiffusions and positive solutions of nonlinear partial differential equations. (Appendix A by J.-F. Le Gall and Appendix B by I. E. Verbitsky.) University Lecture Series, 34. American Mathematical Society, Providence, RI. MR2089791

[6]    Evans, S. N.; O’Connell, N. (1994) Weighted occupation time for branching particle systems and a representation for the supercritical superprocess. Canad. Math. Bull. 37(2), 187–196. MR1275703

[7]    Engländer, J. (2000) Criteria for the existence of positive solutions to the equation ρ(x)Δu = u² in ^d for all d ≥ 1—a new probabilistic approach. Positivity 4, no. 4, 327–337. MR1795992

[8]    Engländer, J. and Pinsky, R. (1999) On the construction and support properties of measure-valued diffusions on D ⊆ R^d with spatially dependent branching, Ann. Probab. 27(2), 684–730. MR1698955

[9]    Le Gall, J.-F. (2005) Random trees and applications. Probab. Surv. 2, 245–311 (electronic). MR2203728

[10]   Li, C. (1996) Some properties of (α,d,β)-superprocesses. Acta Math. Sci. (English Ed.) 16(2), 234–240. MR1402965

[11]   Mselati, B. (2004) Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation. Mem. Amer. Math. Soc. 168(798). MR2031708

[12]   Pinsky, R. G. (1996) Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions. Ann. Probab. 24(1), 237–267. MR1387634

[13]   Sheu, Y.-C. (1994) Asymptotic behavior of superprocesses. Stochastics Stochastics Rep. 49(3-4), 239–252. MR1785007

[14]   Tribe, R. (1992) The behavior of superprocesses near extinction. Ann. Probab. 20(1), 286–311. MR1143421

 

4. Local extinction versus local exponential growth; the ‘spine’

 

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[3]    Evans, S. N. (1993) Two representations of a superprocess. Proc. Royal. Soc. Edin. 123A 959-971. Git, Y.; Harris, MR1249698

[4]    Git, Y., Harris, J. W. and Harris, S. C. (2007) Exponential growth rates in a typed branching diffusion. Ann. Appl. Probab. 17(2), 609–653. MR2308337

[5]    Hardy, R. and Harris, S. C. (2006) A conceptual approach to a path result for branching Brownian motion. Stoc. Proc. Appl. 116(12), 1992–2013. MR2307069

[6]    Lyons, R., Pemantle R. and Peres, Y. (1995) Conceptual proofs of Llog L criteria for mean behavior of branching processes. Ann. Probab. 23, 1125–1138. MR1349164

 

5. The Law of Large Numbers for spatial branching processes and superprocesses

 

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[2]    Chen, Z-Q., Ren, Y. and Wang, H. (2007) An almost sure scaling limit theorem for Dawson-Watanabe superprocesses, Preprint, electronically available at http://www.math.washington.edu/~zchen/paper.html

[3]    Chen, Z-Q. and Shiozawa Y. (2007) Limit theorems for branching Markov processes, J. Funct. Anal. 250(2), 374–399. MR2352485

[4]    Engländer, J. (2007) Law of large numbers for superdiffusions: the non-ergodic case, to appear in Ann. Inst. H. Poincare B.

[5]    Engländer, J. and Turaev, D. (2002) A scaling limit theorem for a class of superdiffusions. Ann. Probab. 30(2) 683–722. MR1905855

[6]    Engländer, J. and Winter, A. (2006) Law of large numbers for a class of superdiffusions. Ann. Inst. H. Poincare B, 42(2), 171–185. MR2199796

[7]    Engländer, J., Harris, S.C. and Kyprianou A. E. (2007) Strong Law of Large Numbers for branching diffusions, Preprint, electronically available at http://www.pstat.ucsb.edu/faculty/englander/publications.html

[8]    Watanabe, S. (1968) A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8, 141–167. MR0237008

 

6. Further topics in superprocesses: compact support property and polar decomposition

 

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[3]    Dynkin, E.B. and Kuznetsov, S. Superdiffusions and removable singularities for quasilinear partial differential equations, Comm. Pure Appl. Math. 49 (1996), 125–176. MR1371926

[4]    Engländer, J., Pinsky, Ross G. (2006) The compact support property for measure-valued processes. Ann. Inst. H. Poincaré Probab. Statist. 42(5), 535–552. MR2259973

[5]    Etheridge, A. and March, P. (1991), A note on superprocesses, Probab. Theory Related Fields, 89(2), 141–147. MR1110534

[6]    Pinsky, R. (2006) Positive Solutions of Reaction diffusion equations with super-linear absorption: universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions, J. Differential Equations 220(2), 407–433. MR2183378

 

7. Classical problems for random media: Brownian motion among Poissonian obstacles and related topics

 

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[5]    Sznitman, A. (1998) Brownian motion, obstacles and random media. Springer Monographs in Mathematics. Springer-Verlag, Berlin. MR1717054

[6]    Sznitman, A. (2004) Topics in random walks in random environment. School and Conference on Probability Theory, 203–266 (electronic), ICTP Lect. Notes, XVII, Abdus Salam Int. Cent. Theoret. Phys., Trieste. MR2198849

[7]    Sznitman, A. (2001) On a class of transient random walks in random environment. Ann. Probab. 29(2), 724–765. MR1849176

[8]    Sznitman, A. and Zerner, M. P. W. (1999) A law of large numbers for random walks in random environment. Ann. Probab. 27(4), 1851–1869. MR1742891

[9]    Sznitman, A. (1999) Slowdown and neutral pockets for a random walk in random environment. Probab. Theory Related Fields 115(3), 287–323. MR1725386

[10]   Varadhan, S. R. S. (2003) Large deviations for random walks in a random environment. Dedicated to the memory of Jürgen K. Moser. Comm. Pure Appl. Math. 56(8), 1222–1245. MR1989232

[11]   Zeitouni, O. (2004) Random walks in random environment. Lectures on probability theory and statistics, 189–312, Lecture Notes in Math., 1837, Springer, Berlin. MR2071631

[12]   Zerner, M. P. W. and Merkl, F. (2001) A zero-one law for planar random walks in random environment. Ann. Probab. 29(4), 1716–1732. MR1880239

[13]   Zerner, Martin P. W. (2000) Velocity and Lyapounov exponents of some random walks in random environment. Ann. Inst. H. Poincaré Probab. Statist. 36(6), 737–748. MR1797391

 

8. Spatial branching processes among Poissonian obstacles: hard obstacles

 

[1]    Engländer, J. (2000) On the volume of the supercritical super-Brownian sausage conditioned on survival, Stochastic Process. Appl. 88, 225–243. MR1767846

[2]    Engländer, J., den Hollander, F. (2003) Survival asymptotics for branching Brownian motion in a Poissonian trap field, Markov Process. Related Fields 9, no. 3, 363–389. MR2028219

 

9. Spatial branching processes among Poissonian obstacles: ‘mild’ obstacles

 

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[4]    R. S. Cantrell, C. Cosner (2003) Spatial ecology via reaction-diffusion equations. Wiley Series in Mathematical and Computational Biology. John Wiley and Sons. MR2191264

[5]    Engländer, J. (2007) Quenched Law of Large numbers for Branching Brownian motion in a random medium. Ann. Inst. H. Poincaré Probab. Statist., to appear

[6]    M. Freidlin, (1985) Functional integration and partial differential equations. Annals of Mathematics Studies, 109, Princeton University Press. MR0833742

[7]    J. W. Harris, S. C. Harris and A. E. Kyprianou (2006), Further probabilistic analysis of the Fisher-Kolmogorov-Petrovskii-Piscounov equation: one sided travelling-waves, Ann. Inst. H. Poincaré Probab. Statist. 42(1), 125–145. MR2196975

[8]    H. Kesten and V. Sidoravicius (2003) Branching random walk with catalysts. Electron. J. Probab. 8(5) (electronic). MR1961167

[9]    A. E. Kyprianou (2005) Asymptotic radial speed of the support of supercritical branching and super-Brownian motion in R^d. Markov Process. Related Fields. 11(1), 145–156. MR2099406

[10]   T. Y. Lee and F. Torcaso (1998) Wave propagation in a lattice KPP equation in random media, Ann. Probab., 26(3), 1179-1197. MR1634418

[11]   N. M. Shnerb, Y. Louzoun, E. Bettelheim and S. Solomon (2000) The importance of being discrete: Life always wins on the surface, Proc. Nat. Acad. Sciences 97, 10322-10324.

[12]   N. M. Shnerb, E. Bettelheim, Y. Louzoun, O. Agam and S. Solomon (2001) Adaptation of autocatalytic fluctuations to diffusive noise, Phys. Rev. E 63, 21103-21108.

[13]   J. Xin (2000) Front propagation in heterogeneous media, SIAM Rev. 42(2), 161–230 (electronic) MR1778352

 

10. Generalizations and open problems for mild obstacles

 

[1]    Engländer, J. (2007) Quenched Law of Large numbers for Branching Brownian motion in a random medium. Ann. Inst. H. Poincaré Probab. Statist., to appear

 

11. Some catalytic branching problems

 

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[2]    Dawson, D. A., Etheridge, A. M., Fleischmann, K., Mytnik, L., Perkins, E. A. and Xiong, J. (2002) Mutually catalytic branching in the plane: infinite measure states. Electron. J. Probab. 7(15), 61 pp. (electronic). MR1921744

[3]    Dawson, D. A.; Fleischmann, K. (1994) A super-Brownian motion with a single point catalyst. Stochastic Process. Appl. 49(1), 3–40. MR1258279

[4]    Dawson, D. A., Fleischmann, K. (2002) Catalytic and mutually catalytic super-Brownian motions. Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999), 89–110, Progr. Probab., 52, Birkhäuser, Basel. MR1958811

[5]    Dawson, D. A. and Fleischmann, K. (2000) Catalytic and mutually catalytic branching. Infinite dimensional stochastic analysis (Amsterdam, 1999), 145–170, Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., 52, R. Neth. Acad. Arts Sci., Amsterdam. MR1831416

[6]    Dawson, D. A., Fleischmann, K., Mytnik, L., Perkins, E. A. and Xiong, J. (2003) Mutually catalytic branching in the plane: uniqueness. Ann. Inst. H. Poincar Probab. Statist. 39(1), 135–191. MR1959845

[7]    Dawson, D. A., Perkins, E. A. (1998) Long-time behavior and coexistence in a mutually catalytic branching model. Ann. Probab. 26(3), 1088–1138. MR1634416

[8]    Dawson, D. A. and Perkins, E. A. (2006) On the uniqueness problem for catalytic branching networks and other singular diffusions. Illinois J. Math. 50(1-4), 323–383 (electronic). MR2247832

[9]    Dawson, D. A., Li, Z., Schmuland, B. and Sun, W. (2004) Generalized Mehler semigroups and catalytic branching processes with immigration. Potential Anal. 21(1), 75–97. MR2048508

[10]   Dynkin, E. B. (1995) Branching with a single point catalyst. Stochastic analysis (Ithaca, NY, 1993), 423–425, Proc. Sympos. Pure Math., 57, Amer. Math. Soc., Providence, RI. MR1335486

[11]   Etheridge, A. M. and Fleischmann K. (1998) Persistence of a two-dimensional super-Brownian motion in a catalytic medium. Probab. Theory Related Fields 110(1), 1–12. MR1602028

[12]   Fleischmann, K. and Le Gall, J.-F. (1995) A new approach to the single point catalytic super-Brownian motion. Probab. Theory Related Fields 102(1), 63–82. MR1351711

[13]   Fleischmann, K. and Mueller, C. (2000) Finite time extinction of catalytic branching processes. Stochastic models (Ottawa, ON, 1998), 125–139, CMS Conf. Proc., 26, Amer. Math. Soc., Providence, RI. MR1765007

[14]   Fleischmann, K. and Xiong, J. (2001) A cyclically catalytic super-Brownian motion. Ann. Probab. 29(2), 820–861. MR1849179

[15]   Kesten, H., Sidoravicius, V. (2003) Branching random walk with catalysts. Electron. J. Probab. 8, no. 5. (electronic). MR1961167

[16]   Klenke, A. (2000) A review on spatial catalytic branching. Stochastic models (Ottawa, ON, 1998), 245–263, CMS Conf. Proc., 26, Amer. Math. Soc., Providence, RI. MR1765014

[17]   Klenke, A. (2003) Catalytic branching and the Brownian snake. Stochastic Process. Appl. 103(2), 211–235. MR1950764

[18]   Mörters, P., Vogt, P. (2005) A construction of catalytic super-Brownian motion via collision local time. Stochastic Process. Appl. 115(1), 77–90. MR2105370

[19]   Mytnik, L. (1998) Uniqueness for a mutually catalytic branching model. Probab. Theory Related Fields 112(2), 245–253. MR1653845

[20]   Topchii, V. and Vatutin, V. (2004) Two-dimensional limit theorem for a critical catalytic branching random walk. Mathematics and computer science. III, 387–395, Trends Math., Birkhäuser, Basel. MR2090528

[21]   Topchii, V. and Vatutin, V. (2003) Individuals at the origin in the critical catalytic branching random walk. Discrete random walks (Paris, 2003), 325–332 (electronic), Discrete Math. Theor. Comput. Sci. Proc., AC, Assoc. Discrete Math. Theor. Comput. Sci., Nancy. MR2042398

[22]   Vatutin, V. and Xiong, J. (2007) Some limit theorems for a particle system of single point catalytic branching random walks. Acta Math. Sin. (Engl. Ser.) 23(6), 997–1012. MR2319610

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