July  2006, 6(4): 867-880. doi: 10.3934/dcdsb.2006.6.867

Brownian flow on a finite interval with jump boundary conditions


Department of Mathematics, Box B6-230, Baruch College - CUNY, One Bernard Baruch Way, New York, NY 10010, United States

Received  February 2005 Revised  November 2005 Published  April 2006

We consider a stochastic flow in an interval $[-a,b]$, where $a,b>0$. Each point of the interval is driven by the same Brownian path and jumps to zero when it reaches the boundary of the interval. Assuming that $a/b$ is irrational we study the long term behavior of a random measure $\mu_t$, the image of a finite Borel measure $\mu_0$ under the flow. We show that if $\mu_0$ is absolutely continuous with respect to the Lebesgue measure then the time averages of the variance of $\mu_t$ converge to zero almost surely. We also prove that for an arbitrary finite Borel measure $\mu_0$ the Lebesgue measure of the support of $\mu_t$ decreases to zero as $t\to\infty$ with probability one.
Citation: Elena Kosygina. Brownian flow on a finite interval with jump boundary conditions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 867-880. doi: 10.3934/dcdsb.2006.6.867

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363


Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374


Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $ (n, m) $-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117


Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050


Sushil Kumar Dey, Bibhas C. Giri. Coordination of a sustainable reverse supply chain with revenue sharing contract. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020165


Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056


Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342


Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011


Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217


Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020167

2019 Impact Factor: 1.27


  • PDF downloads (38)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]