On a family of coupled diffusions that can never change their initial order. (English) Zbl 1509.60176
Summary: We introduce a real-valued family of interacting diffusions where their paths can meet but cannot cross each other in a way that would alter their initial order. Any given interacting pair is a solution to coupled stochastic differential equations with time-dependent coefficients satisfying certain regularity conditions with respect to each other. These coefficients explicitly determine whether these processes bounce away from each other or stick to one another if/when their paths collide. When all interacting diffusions in the system follow a martingale behaviour, and if all these paths ultimately come into collision, we show that the system reaches a random steady-state with zero fluctuation thereafter. We prove that in a special case when certain paths abide to a deterministic trend, the system reduces down to the topology of captive diffusions. We also show that square-root diffusions form a subclass of the proposed family of processes. Applications include order-driven interacting particle systems in physics, adhesive microbial dynamics in biology and risk-bounded quadratic optimization solutions in control theory.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
References:
| [1] | Itô, K., On stochastic differential equations on a differentiable manifold I, Nagoya Math. J., 1, 35-47 (1950) · Zbl 0039.35103 · doi:10.1017/S0027763000022819 |
| [2] | Skorokhod, A. V., Stochastic equations for diffusion processes in a bounded region I, Theory Probab. Appl., 6, 264-74 (1961) · doi:10.1137/1106035 |
| [3] | Skorokhod, A. V., Stochastic equations for diffusion processes in a bounded region II, Theory Probab. Appl., 7, 3-23 (1962) · Zbl 0201.49302 · doi:10.1137/1107002 |
| [4] | Durrett, R. T.; Iglehart, D. L., Functionals of Brownian meander and Brownian excursion, Ann. Probab., 5, 130-5 (1977) · Zbl 0356.60035 |
| [5] | Pitman, J.; Yor, M.; Williams, D., Bessel Processes and Infinitely Divisible Laws, Stochastic Integrals, vol 851 (1980), Berlin: Springer, Berlin · Zbl 0469.60076 |
| [6] | Harrison, J. M.; Reiman, M. I., On the distribution of multidimensional reflected Brownian motion, SIAM J. Appl. Math., 41, 345-61 (1981) · Zbl 0464.60081 · doi:10.1137/0141030 |
| [7] | Göing-Jaeschke, A.; Yor, M., A survey and some generalizations of bessel processes, Bernoulli, 9, 313-49 (2003) · Zbl 1038.60079 · doi:10.3150/bj/1068128980 |
| [8] | Deuschel, J-D; Zambotti, L., Bismut-Elworthy’s formula and random walk representation for SDEs with reflection, Stoch. Process. Appl., 115, 907-25 (2005) · Zbl 1071.60044 · doi:10.1016/j.spa.2005.01.002 |
| [9] | Linetsky, V., On the transition densities for reflected diffusions, Ann. Appl. Probab., 37, 435-60 (2005) · Zbl 1073.60074 · doi:10.1239/aap/1118858633 |
| [10] | Costantini, C.; Gobet, E.; El Karoui, N., Boundary sensitivities for diffusion processes in time dependent domains, Appl. Math. Optim., 54, 159-87 (2006) · Zbl 1109.49043 · doi:10.1007/s00245-006-0863-4 |
| [11] | Karl, L.; Dong, W., Nonintersecting Brownian motions on the unit circle, Ann. Probab., 44, 1134-211 (2016) · Zbl 1342.60138 · doi:10.1214/14-AOP998 |
| [12] | Nechaev, S.; Polovnikov, K.; Shlosman, S.; Valov, A.; Vladimirov, A., Anomalous one-dimensional fluctuations of a simple two-dimensional random walk in a large deviation regime, Phys. Rev. E, 99 (2019) · doi:10.1103/PhysRevE.99.012110 |
| [13] | Dyson, F. J., A Brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys., 3, 1191-8 (1962) · Zbl 0111.32703 · doi:10.1063/1.1703862 |
| [14] | Bru, M. F., Wishart process, J. Theor. Probab., 4, 725-51 (1991) · Zbl 0737.60067 |
| [15] | Grabiner, D. J., Brownian motion in a weyl chamber, non-colliding particles and random matrices, Ann. Inst. H. Poincaré Probab. Stat., 35, 177-204 (1999) · Zbl 0937.60075 |
| [16] | Katori, M.; Tanemura, H., Symmetry of matrix-valued stochastic processes and non-colliding diffusion particle systems, J. Math. Phys., 45, 3058-85 (2004) · Zbl 1071.82045 · doi:10.1063/1.1765215 |
| [17] | Katori, M., Determinantal martingales and noncolliding diffusion processes, Stoch. Process. Appl., 124, 3724-68 (2014) · Zbl 1296.60213 · doi:10.1016/j.spa.2014.06.002 |
| [18] | Mengütürk, L. A.; Mengütürk, M. C., Captive diffusions and their applications to order-preserving dynamics, Proc. R. Soc. A, 476 (2020) · Zbl 1472.82027 · doi:10.1098/rspa.2020.0294 |
| [19] | Pal, S.; Pitman, J., One-dimensional Brownian particle systems with rank-dependent drifts, Ann. Appl. Probab., 18, 2179-207 (2008) · Zbl 1166.60061 · doi:10.1214/08-AAP516 |
| [20] | Shkolnikov, M., Large systems of diffusions interacting through their ranks, Stoch. Process. Appl., 122, 1730-47 (2012) · Zbl 1276.60087 · doi:10.1016/j.spa.2012.01.011 |
| [21] | Pal, S.; Shkolnikov, M., Concentration of measure for Brownian particle systems interacting through their ranks, Ann. Appl. Probab., 24, 1482-508 (2014) · Zbl 1297.82023 · doi:10.1214/13-AAP954 |
| [22] | Mengütürk, L AMengütürk, M C2022From Loewner-captive Hermitian diffusions to risk-captive efficient-frontiersWorking paper (in preparation) |
| [23] | Bonilla, F. A.; Cushman, J. H., Effect of α-stable sorptive waiting times on microbial transport in microflow cells, Phys. Rev. E, 66 (2002) · doi:10.1103/PhysRevE.66.031915 |
| [24] | Bonilla, F. A.; Kleinfelter, N.; Cushman, J. H., Microfluidic aspects of adhesive microbial dynamics: a numerical exploration of flow-cell geometry, Brownian dynamics and sticky boundaries, Adv. Water Resour., 30, 1680-95 (2007) · doi:10.1016/j.advwatres.2006.05.028 |
| [25] | Parashar, R.; Cushman, J. H., Scaling the fractional advective-dispersive equation for numerical evaluation of microbial dynamics in confined geometries with sticky boundaries, J. Comput. Phys., 227, 6598-611 (2008) · Zbl 1145.92003 · doi:10.1016/j.jcp.2008.03.021 |
| [26] | Rogers, L. C G.; Williams, D., Diffusions, Markov Processes and Martingales Vol II (2000), New York: Cambridge Mathematical Library, New York · Zbl 0949.60003 |
| [27] | Øksendal, B. K., Stochastic Differential Equations (2003), Berlin: Springer, Berlin · Zbl 1025.60026 |
| [28] | Adler, S. L.; Brody, D. C.; Brun, T. A.; Hughston, L. P., Martingale models for quantum state reduction, J. Phys. A, 34, 8795-820 (2001) · Zbl 1024.81003 · doi:10.1088/0305-4470/34/42/306 |
| [29] | Brody, D. C.; Hughston, L. P., Efficient simulation of quantum state reduction, J. Math. Phys., 43, 5254-61 (2002) · Zbl 1060.81541 · doi:10.1063/1.1512975 |
| [30] | Mengütürk, L. A., Stochastic Schrödinger evolution over piecewise enlarged filtrations, J. Math. Phys., 57 (2016) · Zbl 1373.93350 · doi:10.1063/1.4944626 |
| [31] | Markowitz, H. M., Portfolio selection, J. Finance, 7, 77-91 (1952) · doi:10.2307/2975974 |
| [32] | Markowitz, H. M., The optimization of a quadratic function subject to linear constraints, Nav. Res. Logist. Q., 3, 111-33 (1956) · doi:10.1002/nav.3800030110 |
| [33] | Cox, J. C.; Ingersoll, J. E.; Ross, S. A., A theory of the term structure of interest rates, Econometrica, 53, 385 (1985) · Zbl 1274.91447 · doi:10.2307/1911242 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
