Random walks provide one of the simplest models of randomness, while Brownian motion stands as a central object in continuous-time stochastic analysis. This talk explores the precise sense in which Brownian motion arises as the scaling limit of random walks. Beginning with simple symmetric random walks, we examine appropriate space–time rescaling and discuss convergence of paths in distribution. The main ideas behind Donsker’s invariance principle are presented, highlighting how discrete stochastic processes converge to a universal continuous limit. Along the way, we emphasise intuition, key probabilistic tools, and the role of independence and finite variance in the emergence of Brownian motion.

Dr Debopriya Mukherjee  is an Assistant Professor in the Department of Mathematics, IIT Indore. She has been awarded Marie Skłodowska-Curie Individual Fellowship in 2020. Her research interest lies in the crossover between probability theory, stochastic analysis, analysis of PDEs, and mathematical biological models. Her research has been widely published in renowned international academic journals.

All are welcome!