Supported by the

The

Northwest Probability Seminars are mini-conferences held at the University of Washington and organized in collaboration with the Oregon State University, the University of British Columbia and the University of Oregon. There is no registration fee.

The Scientific Committee for the NW Probability Seminar 2023 consists of Omer Angel (U British Columbia), Chris Burdzy (U Washington), Zhenqing Chen (U Washington), David Levin (U Oregon) and Axel Saenz Rodríguez (Oregon State U).

HOTEL INFORMATION

**
Hotels
near the University of Washington**.

The talks will take place in Mary Gates Hall 241
(see the **map**).

Parking on UW campus is free on Saturdays only after noon.
See
**
parking information**.

- 10:00 - 11:00
**Coffee**Mary Gates Commons - 11:00 - 11:50
**Shiping Cao**, University of Washington**Boundary Harnack principle for non-local operators**

We prove a scale invariant boundary Harnack principle on any open sets for a large class of non-local operators, that includes operators satisfying a two-sided jump kernel bound and an exit time estimate from balls on metric measure spaces that have volume doubling and reverse volume doubling properties. I will talk about some examples that the theorem applies, including any non-local operator that admits a two-sided mixed stable-like heat kernel bounds, non-local operators of divergence form with measurable coefficients and non-local operators of non-divergence form with Dini-continuous coefficients. Finally, I will introduce some applications of the scale invariant BHP.

This is a joint work with Zhen-Qing Chen. - 12:00 - 12:50
**Matthew Dickson**, University of British Columbia**Random Connection Models and Spherical Transforms on Hyperbolic Spaces**

Random connection models (RCMs) are continuum analogues of Bernoulli percolation, in that the vertices are distributed as a Poisson point process on some space and the edges then exist independently with a probability that is a given function of the positions of the two vertices in question. RCMs on $\mathbb{R}^d$ bear many similarities with Bernoulli percolation on $\mathbb{Z}^d$, while RCMs on hyperbolic $d$-spaces, $\mathbb{H}^d$, bear similarities with Bernoulli percolation on nonamenable graphs. In particular, various techniques can be used to derive mean-field critical exponents and prove the existence of a phase in which there are infinitely many infinite connected components. We will look at how the so-called spherical transform on $\mathbb{H}^d$ can be used to do this. - 1:00 - 3:00
**Lunch**, catered, Mary Gates Commons - 3:00 - 3:50
**Ruth J. Williams**, University of California, San Diego**"Birnbaum Lecture": Stochastic Networks: Bottlenecks, Entrainment and Reflection**

Stochastic models of complex networks with limited resources arise in a wide variety of applications in science andengineering, e.g., in manufacturing, transportation, telecommunications, computer systems, customer service facilities, and systems biology. Bottlenecks in such networks cause congestion, leading to queueing and delay. Sharing of resources can lead to entrainment effects. Understanding the dynamic behavior of such modern stochastic networks present challenging mathematical problems.

This talk will describe some developments in this area. A key feature will be dimension reduction, resulting from entrainment due to resource sharing. An example of bandwidth sharing in a data network will be featured. - 4:00 - 4:30
**Coffee**Mary Gates Commons - 4:30 - 5:20
**Radu Dascaliuc**, Oregon State University**Transformations of Stochastic Recursions and analysis of differential equations**

In this talk, we will explore connections between stochastic processes and nonlinear differential equations (DE) in the spirit of Le Jan and Sznitman’s probabilistic approach to the Navier-Stokes equations (NSE), as well as the classical work of William Feller connecting stochastic explosion of Markov processes and non-uniqueness for the associated Kolmogorov equations. In contrast to DE considered in Ito and McKean’s theory, the models we consider typically require a re-formulation in a Fourier space due to more complex non-local nonlinearities that may contain derivatives.

Our approach falls into the general framework of “probability on trees” and involves consideration of stochastic cascade models of Yule type, but with random intensities – the so-called doubly-stochastic Yule processes (DSY) $Y$, as well as the Solution Processes $X$ represented by stochastic recursions, such that the mean flow of $X$ satisfies a mild-type re-formulation of the DE. In the non-explosive case, the solution process $X$ is uniquely defined by the state of DSY at time $t>0$. In the explosive case, multiple solution processes for the same initial data can be constructed via a stochastic Picard iteration scheme, leading to non-uniqueness of the Cauchy problem for the DE. Moreover, transformations between solution processes corresponding to the same DSY model provide an intriguing way to connect existence and uniqueness problems of distinct DE. We will illustrate these points on a simple model we refer to as the alpha-Riccati equation, first considered by David Aldous and Paul Shields in the context of aging problems and, from different considerations, by Krishna Athreya. We then describe how this probabilistic approach can be used to directly connect NSE to a much simpler model previously considered by Stephen Montgomery-Smith, providing a new pathway to study NSE regularity problems.

Based on the joint work with Tuan Pham, Enrique Thomann, and Edward Waymire. - 6:00 No-host (likely subsidized)
**dinner**.- Restaurant:
**Mamma Melina**. There will be set menu, the first menu on this**list**. The cost per person will be 38 dollars (this does not include tax and gratuity) although it is likely that the conference will have funds to partly subsidize dinner. Please bring cash. There will be a vegetarian option. Wine and coffee can be ordered and paid for individually (cash only; alcohol will not be subsidized). Address: 5101 25th Ave NE, Seattle, WA 98105. See Google**map**. The restaurant is a 20 minute**walk**from Mary Gates Hall.

- Restaurant: