Probability Seminar
Probability Seminar

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Probability Seminar - Spring 2016

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Thursdays 3:30-4:30 PM in Ungar 406.

Abstract: The ability to control quantum systems is becoming an essential step towards emerging technologies such as quantum computation, quantum cryptography and high precision metrology. In this talk, we consider a controlled quantum system whose finite dimensional state is governed by a discrete-time nonlinear Markov process. By assuming the quantum non-demolition (QND) measurements in open-loop, we construct a strict control Lyapunov function which is based on the open-loop stationary states. We propose a measurement-based feedback scheme which ensures the almost sure convergence towards a target state. Moreover, I discuss the estimation and filtering problem for continuous-time quantum systems which are described by continuous-time stochastic master equations.

Chemical reactions within cells involve sequences of random events among small numbers of interacting molecules. As a consequence, biochemical reaction networks are extremely noisy. These reactions are also non-linear, making analytical treatment of these systems difficult. I will present a method for approximating the statistics of molecular species in arbitrarily connected networks of non-linear biochemical reactions in small volumes, which I validate with stochastic simulations. I demonstrate that noise slow flux through biochemical networks with nonlinear reaction kinetics, with implications for the evolution of robustness in living cells.

Abstract: In this paper, we resolve a longstanding open statistical problem. The problem is to analytically determine the second moment of the empirical correlation coefficient
\beqn
\theta := \frac{\int_0^1W_1(t)W_2(t) dt - \int_0^1W_1(t) dt \int_0^1 W_2(t) dt}{\sqrt{\int_0^1 W^2_1(t) dt - \parens{\int_0^1W_1(t) dt}^2} \sqrt{\int_0^1 W^2_2(t) dt - \parens{\int_0^1W_2(t) dt}^2}}
\eeqn
of two {\em independent} Wiener processes, $W_1,W_2$. Using tools from Fredholm integral equation theory, we successfully calculate the second moment of $\theta$ to be .240522. This gives a value for the standard deviation of $\theta$ of nearly .5. As such, we are the first to offer formal proof that
two Brownian motions may be independent and yet can also be
highly correlated with significant probability. This spurious correlation, unrelated to a third variable, is induced because each Wiener process is
``self-correlated'' in time. This is because a Wiener process is an integral
of pure noise and thus its values at different time points are correlated. In addition to providing an explicit formula for the second moment of $\theta$, we offer implicit formulas for higher moments of $\theta$.

Abstract: We study locally-interacting birth-and-death processes on nodes of a finite connected graph; the model which is motivated by modelling interactions between populations, adsorption-desorption processes, and is related to interacting particle systems, Gibbs models, and interactive urn models.
Alongside with general results, we obtain a more detailed description of the asymptotic behaviour in the case of certain special graphs.
Based on a joint work with Vadim Scherbakov (Royal Holloway, University of London).

Abstract: We investigate a manager's decision to restrict the contribution of an agent with private information. We explore the link between this problem and contribution restrictions in global games with a continuum of agents and continuous actions.

Abstract: How should a gambler place bets in order to maximize the probability of reaching a goal. If the goal can be reached for sure,what strategy will minimize the expected time to get there?
Problems in discrete and continuous time will be considered as well as games in which the gambler has an opponent.

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Probability Seminar - Fall 2015

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Tuesdays 4:00PM in Ungar 406

Abstract:

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In 1993, Bak and Sneppen proposed a model aiming to describe an ecosystem of interacting species that evolve by mutation and natural selection. Thereafter various mathematical attempts have been made to study the model in its equilibrium. In this talk we'll investigate a variant of the Bak-Sneppen model and its hydrodynamic limit. The solution solves a heat equation with mass creation at a source inside the domain, normalized to have mass one. We discuss its representation as the average of the empirical measure of an auxiliary branching system with mass growing exponentially fast and the relationship between the stationary measure and quasi-stationarity for the auxiliary semigroup.
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Abstract:
We study the impact of random exponential edge weights on the distances in a random graph and,
in particular, on its diameter. Our main result consists of a precise asymptotic expression for the
maximal weight of the shortest weight paths between
all vertices (the weighted diameter) of sparse random graphs, when the edge weights are iid
exponential random variables. This is based on a joint work with Marc Lelarge.

Abstract:
We discuss a general class of stochastic processes obtained from a
given Markov process whose behavior is modified upon contact with
a catalyst, from the perspective of a particle system that undergoes
branching
with conservation of mass (Fleming-Viot mechanism).
We explain the relation of the process and its scaling limit
to the existence of quasi-stationary
distributions and their simulation. Non-explosion and large deviations
for the soft catalyst case will be discussed if time permits.
Joint work with Min Kang.

Abstract: We propose a dynamic model that explains the build-up of short
term debt when the creditors are strategic and have different beliefs about
the prospects of the borrowers' fundamentals. We define a dynamic game
among creditors, whose outcome is the short term debt. As common in the
literature, this game features multiple Nash equilibria. We give a
refinement of the Nash equilibrium concept that leads to a unique
equilibrium.
For the resulting debt-to-asset process of the borrower we define a
notion of stability and find the debt ceiling which marks the point when
the borrower becomes illiquid. We show existence of early warning signals
of bank runs: a bank run begins when the debt-to-asset process leaves the
stability region and becomes a mean-fleeing sub-martingale with tendency
to reach the debt ceiling. Our results are robust across a wide variety of
specifications for the distribution of the capital across creditors'
beliefs. (joint with J Wissel)

Abstract: We show how Super-Brownian motion is obtained from a particle system under
fast scaling of the branching mechanism.

Abstract: Tightness for continuous time processes and convergence in distribution with applications.

Abstract:

We review examples of limit theorems where scaling plays an important role. The invariance principle,
urn models,
Feller and Wright - Fisher diffusions and the associated martingale problems are discussed. Part 1.

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Probability Seminar - Fall 2014

### 11/6
William Sudderth
University of Minnesota

Some Remarks on Finitely-additive Probability