We construct a stochastic process whose drift is a function of the process's local time at a reflecting barrier. The process arose as a model of the interactions of a Brownian particle and an inert particle in a paper by Knight [7]. We construct and give asymptotic results for two different arrangements of inert particles and Brownian particles, and construct the analogous process in higher dimensions.
References
Barlow, Martin; Burdzy, Krzysztof; Kaspi, Haya; Mandelbaum, Avi. Variably skewed Brownian motion. Electron. Comm. Probab. 5 (2000), 57–66 (electronic). 0949.60090 10.1214/ECP.v5-1018Barlow, Martin; Burdzy, Krzysztof; Kaspi, Haya; Mandelbaum, Avi. Variably skewed Brownian motion. Electron. Comm. Probab. 5 (2000), 57–66 (electronic). 0949.60090 10.1214/ECP.v5-1018
Bass, R.; Burdzy, K.; Chen, Z. Q.; Hairer, M. Stationary distributions for diffusions with inert drift, (forthcoming paper). 1210.60058 10.1007/s00440-008-0182-6Bass, R.; Burdzy, K.; Chen, Z. Q.; Hairer, M. Stationary distributions for diffusions with inert drift, (forthcoming paper). 1210.60058 10.1007/s00440-008-0182-6
Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8
Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. Translated from the Russian. Sixth edition. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. Academic Press, Inc., San Diego, CA, 2000. xlvii+1163 pp. ISBN: 0-12-294757-6 MR1773820 0981.65001Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. Translated from the Russian. Sixth edition. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. Academic Press, Inc., San Diego, CA, 2000. xlvii+1163 pp. ISBN: 0-12-294757-6 MR1773820 0981.65001
Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8 0734.60060Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8 0734.60060
Knight, Frank B. On the path of an inert object impinged on one side by a Brownian particle. Probab. Theory Related Fields 121 (2001), no. 4, 577–598. 1015.60072 10.1007/s004400100160Knight, Frank B. On the path of an inert object impinged on one side by a Brownian particle. Probab. Theory Related Fields 121 (2001), no. 4, 577–598. 1015.60072 10.1007/s004400100160
Lions, P.-L.; Sznitman, A.-S. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984), no. 4, 511–537. MR745330 0598.60060 10.1002/cpa.3160370408Lions, P.-L.; Sznitman, A.-S. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984), no. 4, 511–537. MR745330 0598.60060 10.1002/cpa.3160370408
Ramasubramanian, S. A subsidy-surplus model and the Skorokhod problem in an orthant. Math. Oper. Res. 25 (2000), no. 3, 509–538. 1073.91610 10.1287/moor.25.3.509.12215Ramasubramanian, S. A subsidy-surplus model and the Skorokhod problem in an orthant. Math. Oper. Res. 25 (2000), no. 3, 509–538. 1073.91610 10.1287/moor.25.3.509.12215
Royden, H. L. Real analysis. Third edition. Macmillan Publishing Company, New York, 1988. xx+444 pp. ISBN: 0-02-404151-3. 0704.26006Royden, H. L. Real analysis. Third edition. Macmillan Publishing Company, New York, 1988. xx+444 pp. ISBN: 0-02-404151-3. 0704.26006
Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233. Springer-Verlag, Berlin-New York, 1979. xii+338 pp. ISBN: 3-540-90353-4 MR532498Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233. Springer-Verlag, Berlin-New York, 1979. xii+338 pp. ISBN: 3-540-90353-4 MR532498