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Correlated coalescing Brownian flows on R and the circle

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Hajri, Hatem

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We consider a stochastic differential equation on the real line which is driven by two correlated Brownian motions B+ and B- respectively on the positive half line and the negative half line. We assume vertical bar d < B+, B->(t)vertical bar <= rho dt with rho is an element of [0, 1). We prove it has a unique flow solution. Then, we generalize this flow to a flow on the circle, which represents an oriented graph with two edges and two vertices. We prove that both flows are coalescing. Coalescence leads to the study of a correlated reflected Brownian motion on the quadrant. Moreover, we find the distribution of the hitting time to the origin of a reflected Brownian motion. This has implications for the effect of the correlation coefficient rho on the coalescence time of our flows.

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The International Marine Purchasing Association (IMPA)

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Statistics and probability

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Alea- Latin American Journal of Probability and Mathematical Statistics

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10.30757/ALEA.v15-54

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