## LMS PiNE Lectures

#### 10, 11, 14 September 2020

#### Semimartingale methods for Markov chains, interacting particle systems and random growth models

Speakers: Conrado da Costa (Durham), Frankie Higgs (Lancaster), George Liddle (Lancaster), and Chak Hei Lo (Edinburgh).

##### Aims

Sponsored by the LMS and organized by the PiNE research network, this short lecture series is aimed primarily at postgraduate students and early-career researchers in probability theory or neighbouring fields, but all are welcome.

The lectures are in three blocks, with several linking themes running through, but each block should also be essentially self-contained and can be viewed in isolation.

##### Course outline

**Foster–Lyapunov methods for Markov chains** Chak Hei Lo (3 lectures on **10–11 September**).

We will start the course by presenting various results using the semimartingale approach for Markov chains.
These results include Foster–Lyapunov criteria by which a suitable Lyapunov function can determine whether a process is transient or recurrent.
We will then move on to some applications on these methods, including to some random walks on strips and some interacting particles systems, such as voter models.

**Interacting particle systems and martingales** Conrado da Costa (3 lectures on **10–11 September**).

The purpose of these lectures is to explore martingale methods for the characterization, construction, and analysis of interacting particle systems,
including proofs of scaling limit behaviour.
The first lecture will focus on the construction of a class of interacting particle systems: a family of coupled birth and death chains.
The second lecture will study martingale problems for this model. The third lecture will explore martingale methods for the study of a mean-field voter model.

**Planar random growth and scaling limits** George Liddle and Frankie Higgs (2 lectures on **14 September**).

Conformal growth models are motivated
by some real-world growth processes, and are constructed using conformal maps.
We will introduce the one-parameter Hastings–Levitov model, which is used to describe Laplacian growth and allows us to vary between off-lattice versions of
many well studied models.
Then we investigate the "small particle" scaling limit, which often entails finding a martingale and relating its behaviour to
its analogue for the proposed continuum limit.

##### How to watch the talks

The talks were delivered live via Zoom and streamed on Youtube.

At the moment the live stream videos are still available via the PiNE **Youtube** channel.

The pdf slides from the lectures are also available, linked from the schedule below.

Subsequently recordings of the lectures will be made available for download from the LMS YouTube channel.

## Programme

#### Public access to watch any of the talks is via the PiNE Youtube channel.

**Note:** all times are BST (UTC +1).