On a family of coupled diffusions that can never change their initial order
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Type :
Article
Publication Status :
Published
Access :
openAccess
https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
Abstract
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.
Source :
Journal of Physics A: Mathematical and Theoretical
Date :
2022-11-18
Volume :
55
Issue :
46
Publisher :
IOP Publishing
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