# Schedule for: 16w2696 - Retreat for Young Researchers in Stochastics

Arriving in Banff, Alberta on Friday, September 23 and departing Sunday September 25, 2016

Friday, September 23 | |
---|---|

16:00 - 19:30 |
Check-in begins (Front Desk – Professional Development Centre - open 24 hours) ↓ Note: the Lecture rooms are available after 16:00. (Front Desk – Professional Development Centre) |

17:30 - 19:30 | Dinner (TCPL 201) |

19:30 - 22:00 |
Informal gathering in 2nd floor lounge, Corbett Hall. ↓ Beverages and a small assortment of snacks are available in the lounge on a cash honour system. (TCPL or Corbett Hall Lounge (CH 2110)) |

Saturday, September 24 | |
---|---|

07:00 - 09:00 |
Breakfast ↓ A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops. (Vistas Dining Room) |

08:50 - 09:00 | Ed Perkins: Welcoming Remarks (TCPL 201) |

09:00 - 09:45 |
Nicholas Beaton: Self-avoiding polygons in confined geometries ↓ I will discuss self-avoiding polygons in a restricted geometry, namely
an infinite $L\times M$ tube in $\mathbb Z^3$. These polygons can be
viewed as a simple model of DNA molecules in viral capsids. The model
can be equipped with a Boltzmann weight corresponding to a stretching
or compressing force, and I will discuss the behaviour of the polygons
as the magnitude of the force becomes large. I will also present some
related results regarding the topological properties of confined
polygons, that is, when and how they can be knotted. As the knotting of
DNA can affect its ability to replicate correctly, this is of great
importance from a biological standpoint. (TCPL 201) |

09:45 - 10:30 |
Raphael Chetrite: Nonequilibrium Markov processes conditioned on large deviations ↓ I will present a general approach for constructing a Markov process
that describes the dynamics of a process when one or more observables
of this process are observed to fluctuate in time away from their
typical values. This will extend some well known notions as bridge and
Q-quasi-stationary process.
Articles : Phys. Rev. Lett. 2013, Ann Henri Poincaré 2014, J. Stat.
Mech 2015 (TCPL 201) |

10:30 - 10:50 | Coffee Break (TCPL Foyer) |

10:50 - 11:35 |
Noah Forman: Stationary diffusions on a space of interval partitions ↓ We construct two diffusions on a space of partitions of the
unit interval. These are stationary with the law of the complement of
the zero sets of Brownian motion and Brownian bridge, respectively. Our
construction is based on decorating the jumps of a spectrally positive
L\'evy process with independent continuous excursions. The processes of
ranked interval lengths of our partitions belong to a two parameter
family of diffusions introduced by Ethier and Kurtz (1981) and Petrov
(2009). These are continuum limits of up-down Markov chains on Chinese
restaurant processes. Our construction works towards building a
diffusion on the space of real trees whose existence has been
conjectured by Aldous. (TCPL 201) |

11:35 - 12:20 |
Simone Brugiapaglia: Sparse Approximation of PDEs based on Compressed Sensing ↓ We present the CORSING (COmpRessed SolvING) method for the numerical
approximation of PDEs. Establishing an analogy between the bilinear
form associated with the weak formulation of a PDE and the signal
acquisition process, CORSING combines the classical Petrov-Galerkin
method with Compressed Sensing.
Considering the advection-diffusion-reaction equation of fluid dynamics
as a case study, we discuss some MATLAB numerical examples and analyze
the method from a theoretical perspective. (TCPL 201) |

12:20 - 12:35 |
Group Photo ↓ Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo! (TCPL Foyer) |

12:35 - 13:30 |
Lunch ↓ A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops. (Vistas Dining Room) |

14:00 - 14:45 |
Zhenan Wang: Some analytic approaches to stochastic partial differential equations ↓ We will start with the De Giorgi iterations scheme on SPDEs
and the a priori Holder continuity results on parabolic SPDEs. We will
then explain the extension of such scheme on a Harnack inequality for
the parabolic SPDE and introduce a strong maximum principle. Some
variants of the method will also be introduced for SPDEs with random
drifts. (TCPL 201) |

14:45 - 15:30 |
Khoa Le: Strong law of large numbers for super-critical branching Gaussian processes ↓ We consider a super-critical branching particle system in
which the combined trajectory of each particle and its ancestors
follows a path Gaussian process in $\mathbb{R}^d$. Unlike branching
diffusion systems, such model is not necessary Markov. Assuming that
the offspring distribution has finite second moment and some mild
conditions on the underlying Gaussian process, we show a strong law of
large numbers with the limit object characterized in terms of
asymptotic behavior between the mean and variance of the Gaussian
process. Long memory processes, like fractional Brownian motions and
fractional Ornstein-Uhlenbeck processes with Hurst parameter greater
than 1/2, as well as rough processes, like fractional processes with
Hurst parameter smaller than 1/2 are included as important examples.
Our techniques include moment computations and conditional
probabilities. This is a joint work with Mike Kouritzin and Deniz
Sezer. (TCPL 201) |

15:30 - 16:30 |
Open Problem Session ↓ Graduate student presentations and presentations of open problems for discussion.
Chaired by Omer Angel. (TCPL 201) |

16:30 - 18:15 | Free time to wander and do mathematics. (TCPL 201) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops. (Vistas Dining Room) |

Sunday, September 25 | |
---|---|

07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 09:45 |
Katharina Cera: General Semi-Markov Model for Limit Order Books: Theory, Implementation and Numerics ↓ Presenters are Katharine Cera and Julia Schmidt.
Our talk gives an introduction to Limit Order Books and
summarizes the model proposed by Cont and de Larrard in their paper
“Price Dynamics in a Markovian Limit Order Market” (SIAM J. Finan. Math
(2013)). We present the generalizations done by Swishchuk and Vadori in
“A Semi-Markovian Modeling of Limit Order Markets” (e.g., arXiv (2016))
and the evidence found in our data that justifies their approach. The
talk closes with highlighting our own work on extending this model to
the general semi-Markov model for stock price process in the Limit
Order Book including gained numerical results for diffusion limits. (TCPL 201) |

09:45 - 10:30 |
Jonathan Chavez Casillas: A one level Limit Order Book model with time-dependent rates ↓ In this talk, a model for a level-1 Limit Order Book, in the
same spirit as the model discussed by Cont and deLarrard (2012), will be
presented. However, as empirical evidence shows, the intensity function
of the price process driving the arrivals of orders is non-constant. To
account for that fact, the assumption of homogeneous Poisson arrivals is
relaxed and more general non-homogeneous Poisson arrivals are
considered. This becomes part of an effort to understand the macroscopic
long-run price dynamics and the different limiting price processes when
the orderbook is analyzed from the market micro-structure. (TCPL 201) |

10:30 - 10:50 | Coffee Break (TCPL Foyer) |

10:50 - 11:35 |
Richard Balka: Fractional Brownian motion and new fractal dimensions ↓ Kaufman's dimension doubling theorem states that for a planar Brownian motion
$\{\mathbf{B}(t): t\in [0,1]\}$ we have
$$P(\dim \mathbf{B}(A)=2\dim A \textrm{ for all } A\subset [0,1])=1,$$
where $\dim$ may denote Hausdorff and packing dimension, too. Our goal is to prove similar uniform dimension results in the one-dimensional case. Let $B\colon [0,1]\to \mathbb{R}$
be a fractional Brownian of Hurst index $\alpha$. We try to characterize the Borel sets $D\subset [0,1]$ for which
\[P(\dim B(A)=(1/\alpha) \dim A \textrm{ for all } A\subset D)=1.\]
In order to do so, we introduce a new concept of dimension, the modified Assouad dimension. This is a joint work with Yuval Peres.
\bigskip
Let $B\colon \mathbb{R}^n\to \mathbb{R}^d$ be a fractional Brownian motion of Hurst index $\alpha$ and let $f\colon \mathbb{R}^n\to \mathbb{R}^d$ be a Borel map. For every Borel set $A\subset \mathbb{R}^n$ we determine the almost sure value of the packing dimension of the image and graph of $B+f$ restricted to $A$. In order to do so, we assign different notions of dimension to jointly measurable stochastic processes. This generalizes a result of Xiao, who calculated the packing dimension of $B(A)$ by using the concept of packing dimension profiles introduced by Falconer and Howroyd. (TCPL 201) |

11:35 - 12:20 |
Mathav Murugan: Mathav Murugan: Boundary Harnack principle using elliptic Harnack inequality ↓ We shall see some motivating applications behind studying the boundary Harnack principle. Then,
I will present a recent generalization of boundary Harnack principle, that provides new results
even in R^n. This talk is based on joint work with Martin Barlow. (TCPL 201) |

12:20 - 12:21 |
Checkout by Noon ↓ 2-day workshop participants are welcome to use BIRS facilities (Corbett Hall Lounge, TCPL, Reading Room) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 12 noon. There is no coffee break service on Sunday afternoon, but self-serve coffee and tea are always available in the 2nd floor lounge, Corbett Hall. (Front Desk – Professional Development Centre) |

12:21 - 13:20 | Lunch (TCPL 201) |

13:20 - 14:05 |
Miklos Racz: From trees to seeds: on the inference of the seed from large random trees ↓ I will discuss the influence of the seed in models of randomly growing
trees; in particular, I will focus on the preferential attachment and
uniform attachment models. In both of these models, different seeds
lead to different distributions of limiting trees from a total
variation point of view. I will discuss the differences and
similarities in proving this for the two models. This is based on joint
work with Sebastien Bubeck, Ronen Eldan, and Elchanan Mossel. (TCPL 201) |