Title:

Some problems in stochastic analysis : Itô's formula for convex functions, interacting particle systems and Dyson's Brownian motion

This thesis consists of two main parts: Chapter 1 is concerned with studying an extension of the Itô lemma to the convex functions. We prove that the local martingale part of the decomposition of a convex function f of a continuous semimartingale can be expressed in a similar way to the classical formula with the gradient of f replaced with its subgradient. The result itself is not new, however, our approach via Brownian perturbation is. The second, and the largest, part of the thesis focusses on the study of a certain family of bivariate diffusions Z(Θ,μ) = (X,R) in a wedge W = {(x, r) ∈ R x R+ : x ≤ rg, parameterised by Θ ∈ (0,∞) and μ ≥ 0, with the property that X is distributed as a Brownian motion with drift μ and R is the socalled 3dimensional Bessel process of drifting Brownian motion. By letting parameter Θ tend to ∞ and 0 we can recover the two wellknown couplings of the two processes coming from the Pitman’s theorem and by considering radial part of the 3dimensional BM (with drift μ ≥ 0) respectively. This family of continuous processes is obtained as a diffusion approximation in Chapter 3 of a certain family of twodimensional Markov chains arising in representation theory and is characterised, for each Θ ∈ (0,∞) and μ ≥ 0, as a unique solution to a certain martingale problem in Chapter 4. Moreover, we show that the process Z(μ,Θ) together with the marginal Rprocess provide an example of intertwined processes. Finally, in Chapter 5 we consider a family of certain Markov chains in the GelfandCetlin cone of depth n. We show that for n = 2 the Markov chains of Chapter 3 can be recovered. We identify several intertwining relationships and make a step towards linking the diffusion limit of the chain to a certain Markov function of the GUE minor process of random matrix theory, which consists of two interlaced Dyson’s Brownian motions.
