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January 2019, 24(1): 109-125. doi: 10.3934/dcdsb.2019001

Convergence rates for semistochastic processes

Department of Mathematics, University of Oklahoma, Norman, OK, 73019, USA

* Corresponding author. Present address: Quantitative Reasoning Program and Department of Mathematics, Bowdoin College, Brunswick, ME 04011, USA

J.B. and N.P.P. were partially supported by NSF grant DMS-0807658. A.G. was partially supported NSF grant DMS-1413428. N.P.P. was also generously supported by the Nancy Scofield Hester Presidential Professorship. We thank Martin Oberlack for useful suggestions

Received  April 2017 Revised  September 2018 Published  October 2018

We study processes that consist of deterministic evolution punctuated at random times by disturbances with random severity; we call such processes semistochastic. Under appropriate assumptions such a process admits a unique stationary distribution. We develop a technique for establishing bounds on the rate at which the distribution of the random process approaches the stationary distribution. An important example of such a process is the dynamics of the carbon content of a forest whose deterministic growth is interrupted by natural disasters (fires, droughts, insect outbreaks, etc.).

Citation: James Broda, Alexander Grigo, Nikola P. Petrov. Convergence rates for semistochastic processes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 109-125. doi: 10.3934/dcdsb.2019001
References:
[1]

K. B. AthreyaD. McDonald and P. Ney, Limit theorems for semi-Markov processes and renewal theory for Markov chains, The Annals of Probability, 6 (1978), 788-797. doi: 10.1214/aop/1176995429.

[2]

K. B. Athreya and P. Ney, A new approach to the limit theory of recurrent Markov chains, Transactions of the American Mathematical Society, 245 (1978), 493-501. doi: 10.1090/S0002-9947-1978-0511425-0.

[3]

R. Azaïs and A. Genadot, A new characterization of the jump rate for piecewise-deterministic Markov processes with discrete transitions, Comm. Statist. Theory Methods, 47 (2018), 1812–1829, arXiv:1606.06130v2 [stat.ME] doi: 10.1080/03610926.2017.1327072.

[4]

R. Azaïs and A. Muller-Guedin, Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes, Electronic Journal of Statistics, 10 (2016), 3648-3692. doi: 10.1214/16-EJS1207.

[5]

R. Bartoszyński, On the risk of rabies, Mathematical Biosciences, 24 (1975), 355-377. doi: 10.1016/0025-5564(75)90089-9.

[6]

B. BeckageW. J. Platt and L. J. Gross, Vegetation, fire, and feedbacks: A disturbance-mediated model of savannas, The American Naturalist, 174 (2009), 805-818.

[7]

P. BertailS. Clémençon and J. Tressou, Statistical analysis of a dynamic model for dietary contaminant exposure, Journal of Biological Dynamics, 4 (2010), 212-234. doi: 10.1080/17513750903222960.

[8]

W. Biedrzycka and M. Tyran-Kamínska, Existence of invariant densities for semiflows with jumps, Journal of Mathematical Analysis and Applications, 435 (2016), 61-84. doi: 10.1016/j.jmaa.2015.10.019.

[9]

B. Bond-LambertyS. D. PeckhamD. E. Ahl and S. T. Gower, Fire as the dominant driver of central Canadian boreal forest carbon balance, Nature, 450 (2007), 89-92.

[10]

T. Bourgeron, M. Doumic and M. Escobedo, Estimating the division rate of the growth-fragmentation equation with a self-similar kernel, Inverse Problems, 30 (2014), 025007 (28pp). doi: 10.1088/0266-5611/30/2/025007.

[11]

P. J. BrockwellJ. Gani and S. I. Resnick, Birth, immigration and catastrophe process, Advances in Applied Probability, 14 (1982), 709-731. doi: 10.2307/1427020.

[12]

P. J. BrockwellJ. M. Gani and S. I. Resnick, Catastrophe processes with continuous state-space, Australian Journal of Statistics, 25 (1983), 208-226. doi: 10.1111/j.1467-842X.1983.tb00374.x.

[13]

B. J. Cairns, Evaluating the expected time to population extinction with semi-stochastic models, Mathematical Population Studies, 16 (2009), 199-220. doi: 10.1080/08898480903034843.

[14]

V. Calvez, M. Doumic and P. Gabriel, Self-similarity in a general aggregation-fragmentation problem. Application to fitness analysis, Journal de Mathématiques Pures et Appliquées (9), 98 (2012), 1–27. doi: 10.1016/j.matpur.2012.01.004.

[15]

J. S. Clark, Ecological disturbance as a renewal process: theory and application to fire history, Oikos, 56 (1989), 17-30.

[16]

J. N. Corcoran and R. L. Tweedie, Perfect sampling from independent Metropolis-Hastings chains, Journal of Statistical Planning and Inference, 104 (2002), 297-314. doi: 10.1016/S0378-3758(01)00243-9.

[17]

M. H. A. Davis, Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models, Journal of the Royal Statistical Society B, 46 (1984), 353-388.

[18]

M. H. A. Davis, Markov Models and Optimization, Chapman & Hall, London, 1993. doi: 10.1007/978-1-4899-4483-2.

[19]

J. I. Doob, Stochastic Processes, Wiley, New York, 1953.

[20]

A. Economou and D. Fakinos, Alternative approaches for the transient analysis of Markov chains with catastrophes, Journal of Statistical Theory and Practice, 2 (2008), 183-197. doi: 10.1080/15598608.2008.10411870.

[21]

G. Gripenberg, A stationary distribution for the growth of a population subject to random catastrophes, Journal of Mathematical Biology, 17 (1983), 371-379. doi: 10.1007/BF00276522.

[22]

G. Gripenberg, Extinction in a model for the growth of a population subject to catastrophes, Stochastics: An International Journal of Probability and Stochastic Processes, 14 (1985), 149-163. doi: 10.1080/17442508508833336.

[23]

F. B. Hanson and D. Ryan, Optimal harvesting with exponential growth in an environment with random disasters and bonanzas, Mathematical Biosciences, 74 (1985), 37-57. doi: 10.1016/0025-5564(85)90024-0.

[24]

F. B. Hanson and D. Ryan, Optimal harvesting of a logistic population in an environment with stochastic jumps, Journal of Mathematical Biology, 24 (1986), 259-277. doi: 10.1007/BF00275637.

[25]

F. B. Hanson and H. C. Tuckwell, Persistence times of populations with large random fluctuations, Theoretical Population Biology, 14 (1978), 46-61. doi: 10.1016/0040-5809(78)90003-5.

[26]

F. B. Hanson and H. C. Tuckwell, Logistic growth with random density independent disasters, Theoretical Population Biology, 19 (1981), 1-18. doi: 10.1016/0040-5809(81)90032-0.

[27]

F. B. Hanson and H. C. Tuckwell, Population growth with randomly distributed jumps, Journal of Mathematical Biology, 36 (1997), 169-187. doi: 10.1007/s002850050096.

[28]

S. KapodistriaT. Phung-Duc and J. Resing, Linear birth/immigration-death process with binomial catastrophes, The stationary distribution of a stochastic clearing process, Probability in the Engineering and Informational Sciences, 30 (2016), 79-111. doi: 10.1017/S0269964815000297.

[29]

R. Lande, Risks of population extinction from demographic and environmental stochasticity and random catastrophes, The American Naturalist, 142 (1993), 911-927.

[30]

P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Communications in Mathematical Sciences, 7 (2009), 503-510. doi: 10.4310/CMS.2009.v7.n2.a12.

[31]

M. C. A. LeiteN. P. Petrov and E. Weng, Stationary distributions of semistochastic processes with disturbances at random times and with random severity, Nonlinear Analysis: Real World Applications, 13 (2012), 497-512. doi: 10.1016/j.nonrwa.2011.02.025.

[32]

F. Malrieu, Some simple but challenging Markov processes, Annales de la Faculté des Sciences de Toulouse. Mathématiques (6), 24 (2015), 857–883. doi: 10.5802/afst.1468.

[33]

S. P. Meyn and R. L. Tweedie, Computable bounds for geometric convergence rates of Markov chains, Annals of Applied Probability, 4 (1994), 981-1011. doi: 10.1214/aoap/1177004900.

[34]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag, London, 1993. doi: 10.1007/978-1-4471-3267-7.

[35]

E. Nummelin, A splitting technique for Harris recurrent Markov chains, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 43 (1978), 309-318.

[36]

E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators, Cambridge University Press, Cambridge, 1984. doi: 10.1017/CBO9780511526237.

[37]

A. G. PakesA. C. Trajstman and P. J. Brockwell, A stochastic model for a replicating population subjected to mass emigration due to population pressure, Mathematical Biosciences, 45 (1979), 137-157. doi: 10.1016/0025-5564(79)90099-3.

[38]

K. S. Pregitzer and E. S. Euskirchen, Carbon cycling and storage in world forests: Biome patterns related to forest age, Global Change Biology, 10 (2004), 2052-2077.

[39]

D. H. ReedJ. J. O'GradyJ. D. Ballou and R. Frankham, The frequency and severity of catastrophic die-offs in vertebrates, Animal Conservation, 6 (2003), 109-114.

[40]

G. O. Roberts and J. S. Rosenthal, Quantitative bounds for convergence rates of continuous time Markov processes, Electronic Journal of Probability, 1 (1996), approx. 21 pp. doi: 10.1214/EJP.v1-9.

[41]

G. O. Roberts and R. L. Tweedie, Rates of convergence of stochastically monotone and continuous time Markov models, Journal of Applied Probability, 37 (2000), 359-373. doi: 10.1239/jap/1014842542.

[42]

W. H. RommeE. H. EverhamL. E. FrelichM. A. Moritz and R. E. Sparks, Sparks, Are large, infrequent disturbances qualitatively different from small, frequent disturbances?, Ecosystems, 1 (1998), 524-534.

[43]

J. S. Rosenthal, Minorization conditions and convergence rates for Markov chain Monte Carlo, Journal of the American Statistical Association, 90 (1995), 558–566 [corr.: 90 (1995), 1136] doi: 10.1080/01621459.1995.10476548.

[44]

S. W. Running, Ecosystem disturbance, carbon, and climate, Science, 321 (2008), 652-653.

[45]

A. R. TeelA. Subbaramana and A. Sferlazza, Stability analysis for stochastic hybrid systems: A survey, Automatica, 50 (2014), 2435-2456. doi: 10.1016/j.automatica.2014.08.006.

[46]

P. E. ThorntonB. E. LawH. L. GholzK. L. ClarkE. FalgeD. S. EllsworthA. H. GoldsteinR. K. MonsonD. HollingerM. FalkJ. Chen and J. P. Sparks, Modeling and measuring the effects of disturbance history and climate on carbon and water budgets in evergreen needleleaf forests, Agricutural and Forest Meteorology, 113 (2002), 185-222.

[47]

W. Whitt, The stationary distribution of a stochastic clearing process, Operations Research, 29 (1981), 294-308. doi: 10.1287/opre.29.2.294.

show all references

References:
[1]

K. B. AthreyaD. McDonald and P. Ney, Limit theorems for semi-Markov processes and renewal theory for Markov chains, The Annals of Probability, 6 (1978), 788-797. doi: 10.1214/aop/1176995429.

[2]

K. B. Athreya and P. Ney, A new approach to the limit theory of recurrent Markov chains, Transactions of the American Mathematical Society, 245 (1978), 493-501. doi: 10.1090/S0002-9947-1978-0511425-0.

[3]

R. Azaïs and A. Genadot, A new characterization of the jump rate for piecewise-deterministic Markov processes with discrete transitions, Comm. Statist. Theory Methods, 47 (2018), 1812–1829, arXiv:1606.06130v2 [stat.ME] doi: 10.1080/03610926.2017.1327072.

[4]

R. Azaïs and A. Muller-Guedin, Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes, Electronic Journal of Statistics, 10 (2016), 3648-3692. doi: 10.1214/16-EJS1207.

[5]

R. Bartoszyński, On the risk of rabies, Mathematical Biosciences, 24 (1975), 355-377. doi: 10.1016/0025-5564(75)90089-9.

[6]

B. BeckageW. J. Platt and L. J. Gross, Vegetation, fire, and feedbacks: A disturbance-mediated model of savannas, The American Naturalist, 174 (2009), 805-818.

[7]

P. BertailS. Clémençon and J. Tressou, Statistical analysis of a dynamic model for dietary contaminant exposure, Journal of Biological Dynamics, 4 (2010), 212-234. doi: 10.1080/17513750903222960.

[8]

W. Biedrzycka and M. Tyran-Kamínska, Existence of invariant densities for semiflows with jumps, Journal of Mathematical Analysis and Applications, 435 (2016), 61-84. doi: 10.1016/j.jmaa.2015.10.019.

[9]

B. Bond-LambertyS. D. PeckhamD. E. Ahl and S. T. Gower, Fire as the dominant driver of central Canadian boreal forest carbon balance, Nature, 450 (2007), 89-92.

[10]

T. Bourgeron, M. Doumic and M. Escobedo, Estimating the division rate of the growth-fragmentation equation with a self-similar kernel, Inverse Problems, 30 (2014), 025007 (28pp). doi: 10.1088/0266-5611/30/2/025007.

[11]

P. J. BrockwellJ. Gani and S. I. Resnick, Birth, immigration and catastrophe process, Advances in Applied Probability, 14 (1982), 709-731. doi: 10.2307/1427020.

[12]

P. J. BrockwellJ. M. Gani and S. I. Resnick, Catastrophe processes with continuous state-space, Australian Journal of Statistics, 25 (1983), 208-226. doi: 10.1111/j.1467-842X.1983.tb00374.x.

[13]

B. J. Cairns, Evaluating the expected time to population extinction with semi-stochastic models, Mathematical Population Studies, 16 (2009), 199-220. doi: 10.1080/08898480903034843.

[14]

V. Calvez, M. Doumic and P. Gabriel, Self-similarity in a general aggregation-fragmentation problem. Application to fitness analysis, Journal de Mathématiques Pures et Appliquées (9), 98 (2012), 1–27. doi: 10.1016/j.matpur.2012.01.004.

[15]

J. S. Clark, Ecological disturbance as a renewal process: theory and application to fire history, Oikos, 56 (1989), 17-30.

[16]

J. N. Corcoran and R. L. Tweedie, Perfect sampling from independent Metropolis-Hastings chains, Journal of Statistical Planning and Inference, 104 (2002), 297-314. doi: 10.1016/S0378-3758(01)00243-9.

[17]

M. H. A. Davis, Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models, Journal of the Royal Statistical Society B, 46 (1984), 353-388.

[18]

M. H. A. Davis, Markov Models and Optimization, Chapman & Hall, London, 1993. doi: 10.1007/978-1-4899-4483-2.

[19]

J. I. Doob, Stochastic Processes, Wiley, New York, 1953.

[20]

A. Economou and D. Fakinos, Alternative approaches for the transient analysis of Markov chains with catastrophes, Journal of Statistical Theory and Practice, 2 (2008), 183-197. doi: 10.1080/15598608.2008.10411870.

[21]

G. Gripenberg, A stationary distribution for the growth of a population subject to random catastrophes, Journal of Mathematical Biology, 17 (1983), 371-379. doi: 10.1007/BF00276522.

[22]

G. Gripenberg, Extinction in a model for the growth of a population subject to catastrophes, Stochastics: An International Journal of Probability and Stochastic Processes, 14 (1985), 149-163. doi: 10.1080/17442508508833336.

[23]

F. B. Hanson and D. Ryan, Optimal harvesting with exponential growth in an environment with random disasters and bonanzas, Mathematical Biosciences, 74 (1985), 37-57. doi: 10.1016/0025-5564(85)90024-0.

[24]

F. B. Hanson and D. Ryan, Optimal harvesting of a logistic population in an environment with stochastic jumps, Journal of Mathematical Biology, 24 (1986), 259-277. doi: 10.1007/BF00275637.

[25]

F. B. Hanson and H. C. Tuckwell, Persistence times of populations with large random fluctuations, Theoretical Population Biology, 14 (1978), 46-61. doi: 10.1016/0040-5809(78)90003-5.

[26]

F. B. Hanson and H. C. Tuckwell, Logistic growth with random density independent disasters, Theoretical Population Biology, 19 (1981), 1-18. doi: 10.1016/0040-5809(81)90032-0.

[27]

F. B. Hanson and H. C. Tuckwell, Population growth with randomly distributed jumps, Journal of Mathematical Biology, 36 (1997), 169-187. doi: 10.1007/s002850050096.

[28]

S. KapodistriaT. Phung-Duc and J. Resing, Linear birth/immigration-death process with binomial catastrophes, The stationary distribution of a stochastic clearing process, Probability in the Engineering and Informational Sciences, 30 (2016), 79-111. doi: 10.1017/S0269964815000297.

[29]

R. Lande, Risks of population extinction from demographic and environmental stochasticity and random catastrophes, The American Naturalist, 142 (1993), 911-927.

[30]

P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Communications in Mathematical Sciences, 7 (2009), 503-510. doi: 10.4310/CMS.2009.v7.n2.a12.

[31]

M. C. A. LeiteN. P. Petrov and E. Weng, Stationary distributions of semistochastic processes with disturbances at random times and with random severity, Nonlinear Analysis: Real World Applications, 13 (2012), 497-512. doi: 10.1016/j.nonrwa.2011.02.025.

[32]

F. Malrieu, Some simple but challenging Markov processes, Annales de la Faculté des Sciences de Toulouse. Mathématiques (6), 24 (2015), 857–883. doi: 10.5802/afst.1468.

[33]

S. P. Meyn and R. L. Tweedie, Computable bounds for geometric convergence rates of Markov chains, Annals of Applied Probability, 4 (1994), 981-1011. doi: 10.1214/aoap/1177004900.

[34]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag, London, 1993. doi: 10.1007/978-1-4471-3267-7.

[35]

E. Nummelin, A splitting technique for Harris recurrent Markov chains, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 43 (1978), 309-318.

[36]

E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators, Cambridge University Press, Cambridge, 1984. doi: 10.1017/CBO9780511526237.

[37]

A. G. PakesA. C. Trajstman and P. J. Brockwell, A stochastic model for a replicating population subjected to mass emigration due to population pressure, Mathematical Biosciences, 45 (1979), 137-157. doi: 10.1016/0025-5564(79)90099-3.

[38]

K. S. Pregitzer and E. S. Euskirchen, Carbon cycling and storage in world forests: Biome patterns related to forest age, Global Change Biology, 10 (2004), 2052-2077.

[39]

D. H. ReedJ. J. O'GradyJ. D. Ballou and R. Frankham, The frequency and severity of catastrophic die-offs in vertebrates, Animal Conservation, 6 (2003), 109-114.

[40]

G. O. Roberts and J. S. Rosenthal, Quantitative bounds for convergence rates of continuous time Markov processes, Electronic Journal of Probability, 1 (1996), approx. 21 pp. doi: 10.1214/EJP.v1-9.

[41]

G. O. Roberts and R. L. Tweedie, Rates of convergence of stochastically monotone and continuous time Markov models, Journal of Applied Probability, 37 (2000), 359-373. doi: 10.1239/jap/1014842542.

[42]

W. H. RommeE. H. EverhamL. E. FrelichM. A. Moritz and R. E. Sparks, Sparks, Are large, infrequent disturbances qualitatively different from small, frequent disturbances?, Ecosystems, 1 (1998), 524-534.

[43]

J. S. Rosenthal, Minorization conditions and convergence rates for Markov chain Monte Carlo, Journal of the American Statistical Association, 90 (1995), 558–566 [corr.: 90 (1995), 1136] doi: 10.1080/01621459.1995.10476548.

[44]

S. W. Running, Ecosystem disturbance, carbon, and climate, Science, 321 (2008), 652-653.

[45]

A. R. TeelA. Subbaramana and A. Sferlazza, Stability analysis for stochastic hybrid systems: A survey, Automatica, 50 (2014), 2435-2456. doi: 10.1016/j.automatica.2014.08.006.

[46]

P. E. ThorntonB. E. LawH. L. GholzK. L. ClarkE. FalgeD. S. EllsworthA. H. GoldsteinR. K. MonsonD. HollingerM. FalkJ. Chen and J. P. Sparks, Modeling and measuring the effects of disturbance history and climate on carbon and water budgets in evergreen needleleaf forests, Agricutural and Forest Meteorology, 113 (2002), 185-222.

[47]

W. Whitt, The stationary distribution of a stochastic clearing process, Operations Research, 29 (1981), 294-308. doi: 10.1287/opre.29.2.294.

Figure 1.  Schematic for pre- and post- disturbance levels
Figure 2.  On the construction of the minorizing measure in Theorem 2.1
Figure 5.  Plots of $(1 - \epsilon_{\Delta t, \kappa})^{r / \Delta t }$ vs. $\Delta t$ for selected $\kappa$
Figure 6.  Plots of $(1 - \epsilon_{\Delta t, \kappa})^{r / \Delta t }$ vs. $\kappa$ for selected $\Delta t$
Figure 3.  Plot of $(1 - \epsilon_{\Delta t})^{1/\Delta t}$ vs. $\Delta t$
Figure 4.  Plots of $(1 - \epsilon_{\Delta t})^{\lfloor t/\Delta t \rfloor}$ vs. $t$ for selected values of $\Delta t$
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