Asymptotic results for super-Brownian motions and semilinear differential equations
Author:
Tzong-Yow Lee
Journal:
Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 56-62
MSC (1991):
Primary 60B12, 60F10; Secondary 60F05, 60J15
DOI:
https://doi.org/10.1090/S1079-6762-98-00048-1
Published electronically:
September 14, 1998
MathSciNet review:
1641127
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Abstract: Limit laws for three-dimensional super-Brownian motion are derived, conditioned on survival up to a large time. A large deviation principle is proved for the joint behavior of occupation times and their difference. These are done via analyzing the generating function and exploiting a connection between probability and differential/integral equations.
- Maury Bramson, J. Theodore Cox, and David Griffeath, Occupation time large deviations of the voter model, Probab. Theory Related Fields 77 (1988), no. 3, 401–413. MR 931506, DOI 10.1007/BF00319297
- J. Theodore Cox and David Griffeath, Occupation times for critical branching Brownian motions, Ann. Probab. 13 (1985), no. 4, 1108–1132. MR 806212
- D. A. Dawson, The critical measure diffusion process, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40 (1977), no. 2, 125–145. MR 478374, DOI 10.1007/BF00532877
- Donald A. Dawson and Jürgen Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics 20 (1987), no. 4, 247–308. MR 885876, DOI 10.1080/17442508708833446
- Deuschel, J.-D. and Rosen, J. (1998) Occupation time large deviations for critical branching Brownian motion, super-Brownian motion and related processes. Ann. Probab. 26, no. 2, 602-643.
- E. B. Dynkin, Superprocesses and their linear additive functionals, Trans. Amer. Math. Soc. 314 (1989), no. 1, 255–282. MR 930086, DOI 10.1090/S0002-9947-1989-0930086-7
- I. Iscoe, A weighted occupation time for a class of measure-valued branching processes, Probab. Theory Relat. Fields 71 (1986), no. 1, 85–116. MR 814663, DOI 10.1007/BF00366274
- I. Iscoe, Ergodic theory and a local occupation time for measure-valued critical branching Brownian motion, Stochastics 18 (1986), no. 3-4, 197–243. MR 861108, DOI 10.1080/17442508608833409
- Ian Iscoe and Tzong-Yow Lee, Large deviations for occupation times of measure-valued branching Brownian motions, Stochastics Stochastics Rep. 45 (1993), no. 3-4, 177–209. MR 1306931, DOI 10.1080/17442509308833861
- T.-Y. Lee, Some limit theorems for critical branching Bessel processes, and related semilinear differential equations, Probab. Theory Related Fields 84 (1990), no. 4, 505–520. MR 1042063, DOI 10.1007/BF01198317
- Tzong-Yow Lee and Bruno Remillard, Large deviations for the three-dimensional super-Brownian motion, Ann. Probab. 23 (1995), no. 4, 1755–1771. MR 1379167
- Bramson, M., Cox, J.T. and Griffeath, D. (1988) Occupation time large deviations of the voter model. Probab. Th. Rel. Fields 77, 401-413.
- Cox, J.T. and Griffeath, D. (1985) Occupation times for critical branching Brownian motions. Ann. Probab. 13, 1108-1132.
- Dawson, D. (1977) The critical measure diffusion process. Z. Wahrsch. Verw. Gebiete 40, 125-145.
- Dawson, D., and Gartner, J. (1987) Large deviations for McKean-Vlasov limit of weakly interacting diffusions. Stochastics, 20, 247-308.
- Deuschel, J.-D. and Rosen, J. (1998) Occupation time large deviations for critical branching Brownian motion, super-Brownian motion and related processes. Ann. Probab. 26, no. 2, 602-643.
- Dynkin, E. B. (1989) Superprocesses and their linear additive functionals. Trans. Amer. Math. Soc., 314, 255-282.
- Iscoe, I. (1986) A weighted occupation time for a class of measure-valued branching processes. Z. Wahr. verw. Gebiete. 71, 85-116.
- Iscoe, I. (1986) Ergodic theory and a local occupation time for measure-valued critical branching Brownian motion. Stochastics, 18, 197-243.
- Iscoe, I. and Lee, T.-Y. (1993) Large deviations for occupation times of measure-valued branching Brownian motions. Stochastics and Stochastic Reports, 45, 177-209.
- Lee, T.-Y. (1990) Some limit theorems for critical branching Bessel processes and related semilinear differential equations. Probab. Th. Rel. Fields, 84, 505-520.
- Lee, T.-Y. and Remillard, B. (1995) Large deviations for the three-dimensional super-Brownian motion. Ann. Probab. 23, 1755-1771.
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Additional Information
Tzong-Yow Lee
Affiliation:
University of Maryland, College Park, MD
Email:
tyl@math.umd.edu
Keywords:
Large deviations,
occupation time,
measure-valued process,
branching Brownian motion,
semilinear PDE,
asymptotics
Received by editor(s):
April 15, 1998
Published electronically:
September 14, 1998
Communicated by:
Mark Freidlin
Article copyright:
© Copyright 1998
American Mathematical Society