Deterministic dynamic models with delayed feedback and state constraints arise in a variety of applications in science and engineering. There is interest in understanding what effect noise has on the behavior of such models. Here we consider a multidimensional stochastic delay differential equation with normal reflection as a noisy analogue of a deterministic system with delayed feedback and positivity constraints. We obtain sufficient conditions for existence and uniqueness of stationary distributions for such equations. The results are applied to an example from Internet rate control and a simple biochemical reaction system.
References
Anderson, D. F. A modified next reaction method for simulating chemical systems with time dependent propensities and delays. J. Chem. Phys. 127 (2007) 214107.Anderson, D. F. A modified next reaction method for simulating chemical systems with time dependent propensities and delays. J. Chem. Phys. 127 (2007) 214107.
Bakhtin, Yuri; Mattingly, Jonathan C. Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations. Commun. Contemp. Math. 7 (2005), no. 5, 553-582. 1098.34063 10.1142/S0219199705001878Bakhtin, Yuri; Mattingly, Jonathan C. Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations. Commun. Contemp. Math. 7 (2005), no. 5, 553-582. 1098.34063 10.1142/S0219199705001878
Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9 MR1700749Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9 MR1700749
Brauer, Fred; Castillo-Chávez, Carlos. Mathematical models in population biology and epidemiology. Texts in Applied Mathematics, 40. Springer-Verlag, New York, 2001. xxiv+416 pp. ISBN: 0-387-98902-1Brauer, Fred; Castillo-Chávez, Carlos. Mathematical models in population biology and epidemiology. Texts in Applied Mathematics, 40. Springer-Verlag, New York, 2001. xxiv+416 pp. ISBN: 0-387-98902-1
Chung, K. L.; Williams, R. J. Introduction to stochastic integration. Second edition. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1990. xvi+276 pp. ISBN: 0-8176-3386-3Chung, K. L.; Williams, R. J. Introduction to stochastic integration. Second edition. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1990. xvi+276 pp. ISBN: 0-8176-3386-3
Da Prato, G.; Zabczyk, J. Ergodicity for infinite-dimensional systems. London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. xii+339 pp. ISBN: 0-521-57900-7 0849.60052Da Prato, G.; Zabczyk, J. Ergodicity for infinite-dimensional systems. London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. xii+339 pp. ISBN: 0-521-57900-7 0849.60052
Deb, Supratim; Srikant, R. Global stability of congestion controllers for the Internet. IEEE Trans. Automat. Control 48 (2003), no. 6, 1055-1060. 1364.93560 10.1109/TAC.2003.812809Deb, Supratim; Srikant, R. Global stability of congestion controllers for the Internet. IEEE Trans. Automat. Control 48 (2003), no. 6, 1055-1060. 1364.93560 10.1109/TAC.2003.812809
Dupuis, Paul; Ishii, Hitoshi. On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics Stochastics Rep. 35 (1991), no. 1, 31-62. 0721.60062 10.1080/17442509108833688Dupuis, Paul; Ishii, Hitoshi. On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics Stochastics Rep. 35 (1991), no. 1, 31-62. 0721.60062 10.1080/17442509108833688
Filippov, A. F. Differential equations with discontinuous righthand sides. Translated from the Russian. Mathematics and its Applications (Soviet Series), 18. Kluwer Academic Publishers Group, Dordrecht, 1988. x+304 pp. ISBN: 90-277-2699-XFilippov, A. F. Differential equations with discontinuous righthand sides. Translated from the Russian. Mathematics and its Applications (Soviet Series), 18. Kluwer Academic Publishers Group, Dordrecht, 1988. x+304 pp. ISBN: 90-277-2699-X
Hairer, M.; Ohashi, A. Ergodic theory for SDEs with extrinsic memory. Ann. Probab. 35 (2007), no. 5, 1950-1977. 1129.60052 10.1214/009117906000001141 euclid.aop/1189000933Hairer, M.; Ohashi, A. Ergodic theory for SDEs with extrinsic memory. Ann. Probab. 35 (2007), no. 5, 1950-1977. 1129.60052 10.1214/009117906000001141 euclid.aop/1189000933
Hairer, M.; Mattingly, J. C.; Scheutzow, M. Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations. Probability Theory and Related Fields (2009) DOI 10.1007/s00440-009-0250-6. 1238.60082 10.1007/s00440-009-0250-6Hairer, M.; Mattingly, J. C.; Scheutzow, M. Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations. Probability Theory and Related Fields (2009) DOI 10.1007/s00440-009-0250-6. 1238.60082 10.1007/s00440-009-0250-6
Hale, Jack K.; Verduyn Lunel, Sjoerd M. Introduction to functional-differential equations. Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. x+447 pp. ISBN: 0-387-94076-6 MR1243878 0787.34002Hale, Jack K.; Verduyn Lunel, Sjoerd M. Introduction to functional-differential equations. Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. x+447 pp. ISBN: 0-387-94076-6 MR1243878 0787.34002
Harrison, J. Michael; Reiman, Martin I. Reflected Brownian motion on an orthant. Ann. Probab. 9 (1981), no. 2, 302-308. 0462.60073 10.1214/aop/1176994471 euclid.aop/1176994471Harrison, J. Michael; Reiman, Martin I. Reflected Brownian motion on an orthant. Ann. Probab. 9 (1981), no. 2, 302-308. 0462.60073 10.1214/aop/1176994471 euclid.aop/1176994471
Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1988. xxiv+470 pp. ISBN: 0-387-96535-1 0638.60065Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1988. xxiv+470 pp. ISBN: 0-387-96535-1 0638.60065
Kushner, Harold J. Numerical methods for controlled stochastic delay systems. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 2008. xx+281 pp. ISBN: 978-0-8176-4534-2Kushner, Harold J. Numerical methods for controlled stochastic delay systems. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 2008. xx+281 pp. ISBN: 978-0-8176-4534-2
Liptser, R. S.; Shiryayev, A. N. Statistics of random processes. I. General theory. Translated by A. B. Aries. Applications of Mathematics, Vol. 5. Springer-Verlag, New York-Heidelberg, 1977. x+394 pp. ISBN: 0-387-90226-0 0364.60004Liptser, R. S.; Shiryayev, A. N. Statistics of random processes. I. General theory. Translated by A. B. Aries. Applications of Mathematics, Vol. 5. Springer-Verlag, New York-Heidelberg, 1977. x+394 pp. ISBN: 0-387-90226-0 0364.60004
Mao, Xuerong. Exponential stability of stochastic differential equations. Monographs and Textbooks in Pure and Applied Mathematics, 182. Marcel Dekker, Inc., New York, 1994. xii+307 pp. ISBN: 0-8247-9080-4 * 0806.60044Mao, Xuerong. Exponential stability of stochastic differential equations. Monographs and Textbooks in Pure and Applied Mathematics, 182. Marcel Dekker, Inc., New York, 1994. xii+307 pp. ISBN: 0-8247-9080-4 * 0806.60044
Mao, Xuerong. Razumikhin-type theorems on exponential stability of stochastic functional-differential equations. Stochastic Process. Appl. 65 (1996), no. 2, 233-250. 0889.60062 10.1016/S0304-4149(96)00109-3Mao, Xuerong. Razumikhin-type theorems on exponential stability of stochastic functional-differential equations. Stochastic Process. Appl. 65 (1996), no. 2, 233-250. 0889.60062 10.1016/S0304-4149(96)00109-3
Mather, W.; Bennett, M. R.; Hasty, J.; Tsiing L. S. Delay-induced degrade-and-fire oscillations in small genetic circuits. Physical Review Letters. 102 068105, pp. 1-4.Mather, W.; Bennett, M. R.; Hasty, J.; Tsiing L. S. Delay-induced degrade-and-fire oscillations in small genetic circuits. Physical Review Letters. 102 068105, pp. 1-4.
Mohammed, S. E. A. Stochastic functional differential equations. Research Notes in Mathematics, 99. Pitman (Advanced Publishing Program), Boston, MA, 1984. vi+245 pp. ISBN: 0-273-08593-XMohammed, S. E. A. Stochastic functional differential equations. Research Notes in Mathematics, 99. Pitman (Advanced Publishing Program), Boston, MA, 1984. vi+245 pp. ISBN: 0-273-08593-X
Mohammed, S. E. A. Stability of linear delay equations under a small noise. Proc. Edinburgh Math. Soc. (2) 29 (1986), no. 2, 233-254. MR847877 0571.34038 10.1017/S0013091500017612Mohammed, S. E. A. Stability of linear delay equations under a small noise. Proc. Edinburgh Math. Soc. (2) 29 (1986), no. 2, 233-254. MR847877 0571.34038 10.1017/S0013091500017612
Orey, Steven. Lecture notes on limit theorems for Markov chain transition probabilities. Van Nostrand Reinhold Mathematical Studies, No. 34. Van Nostrand Reinhold Co., London-New York-Toronto, Ont., 1971. viii+108 pp. 0295.60054Orey, Steven. Lecture notes on limit theorems for Markov chain transition probabilities. Van Nostrand Reinhold Mathematical Studies, No. 34. Van Nostrand Reinhold Co., London-New York-Toronto, Ont., 1971. viii+108 pp. 0295.60054
Paganini, F.; Doyle, J.; Low, S. Scalable laws for stable network congestion control. Proceedings of the 40th IEEE Conference on Decision and Control, 2001, 185-190.Paganini, F.; Doyle, J.; Low, S. Scalable laws for stable network congestion control. Proceedings of the 40th IEEE Conference on Decision and Control, 2001, 185-190.
Paganini, F.; Wang, Z. Global stability with time-delay in network congestion control. Proceedings of the 41st IEEE Conference on Decision and Control, 2002, 3632-3637.Paganini, F.; Wang, Z. Global stability with time-delay in network congestion control. Proceedings of the 41st IEEE Conference on Decision and Control, 2002, 3632-3637.
Papachristodoulou, A. Global stability analysis of a TCP/AQM protocol for arbitrary networks with delay. Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, 1029-1034.Papachristodoulou, A. Global stability analysis of a TCP/AQM protocol for arbitrary networks with delay. Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, 1029-1034.
Papachristodoulou, A.; Doyle J.; Low, S. Analysis of nonlinear delay differential equation models of TCP/AQM protocols using sums of squares. Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, 4684-4689.Papachristodoulou, A.; Doyle J.; Low, S. Analysis of nonlinear delay differential equation models of TCP/AQM protocols using sums of squares. Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, 4684-4689.
Peet, Matthew; Lall, Sanjay. Global stability analysis of a nonlinear model of Internet congestion control with delay. IEEE Trans. Automat. Control 52 (2007), no. 3, 553-559. 1366.93448 10.1109/TAC.2007.892379Peet, Matthew; Lall, Sanjay. Global stability analysis of a nonlinear model of Internet congestion control with delay. IEEE Trans. Automat. Control 52 (2007), no. 3, 553-559. 1366.93448 10.1109/TAC.2007.892379
Reiß, M.; Riedle, M.; van Gaans, O. Delay differential equations driven by Lévy processes: stationarity and Feller properties. Stochastic Process. Appl. 116 (2006), no. 10, 1409-1432. 1109.60045 10.1016/j.spa.2006.03.002Reiß, M.; Riedle, M.; van Gaans, O. Delay differential equations driven by Lévy processes: stationarity and Feller properties. Stochastic Process. Appl. 116 (2006), no. 10, 1409-1432. 1109.60045 10.1016/j.spa.2006.03.002
Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 0917.60006Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 0917.60006
Rogers, L. C. G.; Williams, David. Diffusions, Markov processes, and martingales. Vol. 1. Foundations. Reprint of the second (1994) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. xx+386 pp. ISBN: 0-521-77594-9Rogers, L. C. G.; Williams, David. Diffusions, Markov processes, and martingales. Vol. 1. Foundations. Reprint of the second (1994) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. xx+386 pp. ISBN: 0-521-77594-9
Rogers, L. C. G.; Williams, David. Diffusions, Markov processes, and martingales. Vol. 2. Itô calculus. Reprint of the second (1994) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. xiv+480 pp. ISBN: 0-521-77593-0Rogers, L. C. G.; Williams, David. Diffusions, Markov processes, and martingales. Vol. 2. Itô calculus. Reprint of the second (1994) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. xiv+480 pp. ISBN: 0-521-77593-0
Scheutzow, M. Qualitative behaviour of stochastic delay equations with a bounded memory. Stochastics 12 (1984), no. 1, 41-80. 0539.60061 10.1080/17442508408833294Scheutzow, M. Qualitative behaviour of stochastic delay equations with a bounded memory. Stochastics 12 (1984), no. 1, 41-80. 0539.60061 10.1080/17442508408833294
Scheutzow, Michael. Stability and instability of routings through a queueing network. Queueing Systems Theory Appl. 35 (2000), no. 1-4, 117-128. 0966.90017 10.1023/A:1019137910086Scheutzow, Michael. Stability and instability of routings through a queueing network. Queueing Systems Theory Appl. 35 (2000), no. 1-4, 117-128. 0966.90017 10.1023/A:1019137910086
Scheutzow, Michael. Exponential growth rates for stochastic delay differential equations. Stoch. Dyn. 5 (2005), no. 2, 163-174. MR2147280 1079.60058 10.1142/S0219493705001468Scheutzow, Michael. Exponential growth rates for stochastic delay differential equations. Stoch. Dyn. 5 (2005), no. 2, 163-174. MR2147280 1079.60058 10.1142/S0219493705001468
Srikant, R. The mathematics of Internet congestion control. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 2004. xii+164 pp. ISBN: 0-8176-3227-1Srikant, R. The mathematics of Internet congestion control. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 2004. xii+164 pp. ISBN: 0-8176-3227-1
Vinnicombe, G. On the stability of end-to-end congestion control for the Internet. Univ. Cambridge Tech. Rep. CUED/F-INFENG/, TR.398 [Online], 2002. Available at http://www.eng.cam.ac.uk/~gvVinnicombe, G. On the stability of end-to-end congestion control for the Internet. Univ. Cambridge Tech. Rep. CUED/F-INFENG/, TR.398 [Online], 2002. Available at http://www.eng.cam.ac.uk/~gv
Williams, R. J. An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Systems Theory Appl. 30 (1998), no. 1-2, 5-25. 0911.90170 10.1023/A:1019156702875Williams, R. J. An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Systems Theory Appl. 30 (1998), no. 1-2, 5-25. 0911.90170 10.1023/A:1019156702875
Ying, L.; Dullerud, G.; Srikant, R. Global stability of Internet congestion controllers with heterogeneous delays. IEEE/ACM Transactions on Networking, 14 (2006), 579-591.Ying, L.; Dullerud, G.; Srikant, R. Global stability of Internet congestion controllers with heterogeneous delays. IEEE/ACM Transactions on Networking, 14 (2006), 579-591.