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We prove non-convergence theorems towards an unstable equilibrium (or a trap)for stochastic processes. The processes we consider are continuous-time ordiscrete-time processes and can be pertubations of the flow generated by avector field. Our results extend previous results given for discrete-timeprocesses by O. Brandi{\`e}re and M. Duflo in [BD96; Duf96], by R. Pemantle in[Pem90] and by P. Tarr{\`e}s in [Tar00]. We correct and give a correctformulation to some theorems stated in [BD96; Duf96]. The method used to provesome of our theorems follow a method introduced by P. Tarr{\`e}s in [Tar00].Finally our non-convergence theorems are applied to give correct proofs of thenon-convergence towards traps for the empirical measure of vertex reinforcedrandom walks in [BRS13] and for non-backtracking vertex reinforced random walksin [LR18].

Non-convergence to unstable equilibriums for continuous-time and discrete-time stochastic processes