행사 상세 정보 :

2025.06.17. (Tue.) 16:00 ~ 17:00

Title: When Probability Meets Functional Analysis (and Vice Versa)

• Speaker: Hyunchul Park (SUNY New Paltz)
• Education Convergence Building, 205
• Abstract
  • This talk explores the rich interplay between probability theory and functional analysis through accessible examples and applications, aiming to pro vide a broad perspective for advanced undergraduate and beginning graduate students.
  • We begin with a classical problem in functional analysis: understanding the space of continuous functions on an interval, C[ab], equipped with the uniform norm. Using tools from elementary probability, speci cally the binomial distribution, we show that the set of polynomials is dense in C[ab] . This leads to a simple and elegant proof that C[ab] is separable.
  • Next, we reverse the perspective and apply functional analytic ideas to prob ability. We de ne the conditional expectation of a random variable using Hilbert space projections a technique grounded in functional analysis. We then present two important applications of conditional expectation:
    • (a) The Levy-Ito decomposition, which describes the structure of sample paths of Levy processes random processes with independent and stationary in crements.
    • (b) The construction of stochastic integrals for continuous semimartingales, in cluding Brownian motion. Using this stochastic integrals, we introduce stochastic dynamical systems driven by Levy processes and analyze exit time problems in systems perturbed by both Brownian motion and jump processes.