Vicious walkers in a potential

We consider N vicious walkers moving in one dimension in a one-body potential v(x). Using the backward Fokker-Planck equation we derive exact results for the asymptotic form of the survival probability Q(x, t) of vicious walkers initially located at (x(1),...,x(N)) = x, when v(x) is an arbitrary attractive potential. Explicit results are given for a square-well potential with absorbing or reflecting boundary conditions at the walls, and for a harmonic potential with an absorbing or reflecting boundary at the origin and the walkers starting on the positive half line. By mapping the problem of N vicious walkers in zero potential onto the harmonic potential problem, we rederive the results by Fisher (1984 J. Stat. Phys. 34 667) and Krattenthaler et al (2000 J. Phys. A: Math. Gen. 33 8835) respectively for vicious walkers on an infinite line and on a semi-infinite line with an absorbing wall at the origin. This mapping also gives a new result for vicious walkers on a semi-infinite line with a reflecting boundary at the origin: Q(x, t) similar to t - N(N-1)/2.