Università Roma Tre

1. Brownian motion I. Gaussian multivariate distribution. Processes with incremental and independent increments. Definition and continuity properties of Brownian motion. Non-differentiability of the trajectories. Property of Markov. Strong Markov property and reflection principle. 2. Brownian motorcycle II. Brownian motorcycle in multiple dimensions. Harmonic functions and Dirichlet problem. Solution of the Dirichlet problem through Brownian motion for regular domains. Poisson's problem and its solution for regular domains. Law of the iterated log arithm. Skorohod embedding. Donsker invariance principle. Applications: arcosine laws and the law of the maximum of random walks. 3. Stochastic integration. Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. Itˆo formula and applications. Local weather and Tanaka formula. Ito formula in more dimensions and for general stochastic differential. 4. Stochastic differential equations. Stochastic differential linear equations: examples of solutions. Existence and uniqueness theorem for stochastic differential equations. Diffusion process within zero noise limit. Infinitesimal generator of a diffusion and partial differential equations. Feynman-Kac formula and applications.

[1]P. M ̈orters and Y. Peres,Brownian Motion.Cambridge University Press, (2010).
[2]L.C. Evans,An Introduction to Stochastic Differential Equations.AMS bookstore, (2013).
[3]T.M.Liggett,Continuous time Markov processes.AMS, (2010).