On the control of an interacting particle estimation of Schrödinger ground states

Abstract. We consider a general Schrodinger operator L + V on a domain E Rd, and its associated positive ground state h solution to the maximal eigenvalue problem L(h) + V h = h. In this work, an interacting particle model approximating the pair (h; ) is studied. When V 0, a basic version of this particle system consists of N walkers evolving independently according to the Markov generator L, each walker dying at a rate given by the value of the potential jV j at the walker's current location; when a walker dies, any other one splits in two. The long time distribution of the particle system is then an estimator of h. Under some reasonable assumptions (with examples for E = Rd), we get a non-asymptotic control of the Lp deviations (resp. the bias) of this estimator with the genuine rate of convergence in 1= p N (resp.1=N). We also compute explicitly the asymptotic standard deviation of the estimation of , which remains bounded in usual mild situations. Key words. Schrodinger ground states, stochastic particle methods, long time behavior, Quan-tum Monte Carl