Title: TBA

Abstract: .

Title: Brownian motion conditioned to spend limited time outside a bounded interval

Abstract: A classic result by Knight (1969) states that if one conditions Brownian motion on remaining inside the interval [-1,1], the resulting limit process is the so-called taboo process, which also has a description by a simple SDE. In our work, we condition Brownian motion on spending at most a total of s > 0 time units outside [-1,1]. Surprisingly, the conditioned process is again the taboo process. In other words, Brownian motion conditioned on having limited time outside [-1,1] will not leave the interval at all. This can be seen as an extreme example of entropic repulsion. Along the way, we explicitly determine the exact asymptotic behaviour of the probability that Brownian motion on [0,T] spends limited time outside [-1,1], as T \to \infty. This is joint work with Martin Kolb (Paderborn) and Dominic Schickentanz (Paderborn/Technion).

Title: TBA

Abstract:

Title: TBA

Abstract:

Title: Critical branching processes with immigration: scaling limits of local extinction sets

Abstract: This talk will describe the joint scaling limit of a critical Bienaym\'e-Galton-Watson process with immigration (BGWI) and its (counting) local time at zero to the corresponding self-similar continuous-state branching process with immigration (CBI) and its (Markovian) local time at zero for balanced offspring and immigration laws in stable domains of attraction. Using a general framework for invariance principles of local times~[Mijatovic, Uribe Bravo, 2022], the problem reduces to the analysis of the structure of excursions from zero and positive levels, together with the weak convergence of the hitting times of points of the BGWI to those of the CBI. A key step in the proof of our main limit theorem is a novel Yaglom limit for the law at time $t$ of an excursion with lifetime exceeding $t$ of a scaled infinite-variance critical BGWI. Our main result implies a joint septuple scaling limit of BGWI $Z_1$, its counting local time at $0$, the random walks $X_1$ and $Y_1$ associated to the reproduction and immigration mechanisms, respectively, the counting local time at $0$ of $X_1$, an additive functional of $Z_1$ and $X_1$ evaluated at this functional. In the septuple limit, four different scaling sequences are identified and given explicitly in terms of the offspring generating function (modulo asymptotic inversion), the local extinction probabilities of the BGWI and the tails of return times to zero of $X_1$. This is joint work with Ben Povar and Geronimo Uribe Bravo

Title: TBA

Abstract:

Title: Stochastic averaging principle for enzyme kinetic reactions: Functional law of large numbers and Gaussian approximations

Abstract: We consider a stochastic model of the Michaelis--Menten (MM) enzyme kinetic reactions in terms of Stochastic Differential Equations (SDEs) driven by Poisson Random Measures (PRMs). It has been argued that among various Quasi-Steady State Approximations (QSSAs) for the deterministic model of such chemical reactions, the total QSSA (tQSSA) is the most accurate approximation, and it is valid for a wider range of parameter values than the standard QSSA (sQSSA). While the sQSSA for this model has been rigorously derived from a probabilistic perspective at least as early as 2006 in Ball et al. (2006), a rigorous study of the tQSSA for the stochastic model appears missing. We fill in this gap by deriving it as a Functional Law of Large Numbers (FLLN), and also studying the fluctuations around this approximation as a Functional Central Limit Theorem (FCLT).