Tvrtko Tadić

Graduate Student
University of Washington
Department of Mathematics
Box 354350
Seattle, WA 98195-4350

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Research

Image choice:
Brownian motion on the edges of a square Brownian motion on a 512 by 512 net Limit case 3d Brownian motion on a 64 by 64 net 1 3d Brownian motion on a 64 by 64 net 2

General interests

I am interested in analysis of stochastic models (properties of stochastic processes, aplications, simulations, statistical and optimization approaches). Some of the models that I have been working on include:

• Processes indexed by graphs
• Markov processes
• Gaussian processes
• Stochastic differential equations

More details about the projects I am working on can be found below.

Markov processes on time-like graphs

This is my current project.

We often model real world process in which we have uncertainty with Markov processes. These processes are a collection of random variables indexed over an set T with conditions on the dependence of these variables. T is usually time, and as such in most cases a subset of real numbers. However, in general T can be anything (even a graph with some order).

In probability there are a lot of random processes that involve graphs. But most of them are either processes on graphs (where the state space of the process is, for example, the subset of vertices) or processes that study random graphs. In statistics, graphical models are used to model relationships, information retrieval, and language processing. There we have a countable number of random variables indexed by the vertices of a graph where the graph structure induces a set of conditional independence constraints (called the graphical Markov property).

In their paper Krzysztof Burdzy and Soumik Pal introduced a similar continuous model, which they called Markov processes on time-like graphs.

Video choice:
Construction of the process on a 8 by 8 net Simple coupling and branching process

Their model had hard restrictions on the degree each vertex should have. I am working on developing and further improving this model. The following titles represent type of things that I have been working on.

Geometry of time-like graphs

Time-like graphs (TLG's) represent the index set, that models processes that couple and branch. But in order to have a 'good definiton' of such a process we need to restrict our attention to so called TLG* graphs. TLG*'s are defined inductively, and it is not clear which graphs are TLG*'s. For instance all planar TLG's are TLG*'s, TLG*'s have a strucure of a topological lattice, but this is not enough to classify them. I have developed a stingy algorithm that provides us with the answer is a TLG a TLG*.

Construction of processes on TLG's

Processes indexed by TLG*'s can be constructed under very general conditions on distributions of processes along full-time paths. It turns out that the distribution of the process in the end doesn't depend on way we constructed the process.

Properties of processes on TLG's

Depending on the assumptions that we made, the process will have properties that depend on the structure of the graph. So we will have things like the spine-Markov property, time-Markov property, ... under certain assumptions we will also have the optional sampling theorem.

Asymptotics of Brownian motion on certain graphs

It is a natural question what happens when the cell sizes on certain configurations of a planar graph go to 0. We answer that question for two configurations and the Brownian motion idexed by them. Under certain conditions we will get that all the processes become the same along full-time paths, under other configurations they will be modified and in the end we will have the stochastic heat equation.