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Reciprocal Time Relation of Noncolliding Brownian Motion with Drift

We consider an $N$-particle system of noncolliding Brownian motion starting from $x_1 \leq x_2 \leq ... \leq x_N$ with drift coefficients $ν_j, 1 \leq j \leq N$ satisfying $ν_1 \leq ν_2 \leq ... \leq ν_N$. When all of the initial points are degenerated to be zero, $x_j=0, 1 \leq j \leq N$, the equivalence is proved between a dilatation with factor $1/t$ of this drifted process and the noncolliding Brownian motion starting from $ν_1 \leq ν_2 \leq ... \leq ν_N$ without drift observed at reciprocal time $1/t$, for arbitrary $t > 0$. Using this reciprocal time relation, we study the determinantal property of the noncolliding Brownian motion with drift having finite and infinite numbers of particles.