In this talk we show almost sure convergence of the gradient flow trajectory of a particle on an energy landscape which is exposed to stochastic noise. Assuming that the particle does not escape to infinity and the energy-function satisfies locally Lojasiewicz-inequalities we give sufficient convergence rates for the noise decreasing in time in order to guarantee convergence. We compare our results with the classical ODE statement from Lojasiewicz and point out the main difficulties in the proof when adding stochastic noise.