PhD Position in Probabilty Theory and Mathematical Analysis

Do you want to work on the interplay between probability theory, mathematical analysis and dynamic optimization theory? Join our team as one of our two new PhD candidates!

Job description

Dynamic stochastic optimization problems are ubiquitous in modern applications. Examples range from calculating the fastest flight route from Amsterdam to San Francisco to determine optimal investment strategies in financial markets or understanding how stochastic deviations in large-scale systems lead to phase transitions.

From a mathematical perspective, these problems can be formulated as a class of controlled Markov processes. Analyzing their infinitesimal behaviour leads naturally to the study of the Hamilton--Jacobi equation, a nonlinear partial differential equation that plays a fundamental role in optimal control theory, stochastic processes, and related fields.

In this project, you will explore the well-posedness of the Hamilton--Jacobi equation in novel, unexplored settings, either in curved finite-dimensional spaces, or in infinite-dimensional optimal transport frameworks. Key aspects include establising uniqueness of viscosity solutions and the representation of these solutions in terms of controlled stochastic processes. Depending on your interests, motivating applications may include filtering of partially observed processes, differential games, or multi-scale problems in statistical physics and chemistry.

This vacancy comprises two PhD positions, each emphasizing a different mathematical direction:

Position 1: Finite-Dimensional Curved Context

This project will focus on the theoretical development of the Hamilton–Jacobi equation in curved spaces, drawing upon techniques from differential geometry. The candidate will explore the role of curvature in dynamic optimization problems and analyze how stochastic control theory extends to non-Euclidean settings. Methods from (sub)-Riemannian geometry, geometric analysis, and PDE theory will be essential.

Position 2: Infinite-Dimensional Context in Optimal Transport

This project will extend the theory to infinite-dimensional spaces within the framework of optimal transport, the theory of how to best move “mass”. The candidate will explore how recent developments in this field can be applied to the study of Hamilton–Jacobi equations. This direction is highly relevant for applications in statistical physics, stochastic processes, and large-scale systems where transport and diffusion play a key role.

Both PhD candidates will be embedded in the Applied Probability group within the Department of Applied Mathematics, which has a strong research focus on controlled stochastic processes in contexts arising from statistical physics, finance, and beyond. The group also maintains close ties with the Analysis group, which specializes in the mathematical theory of partial differential equations, providing a rich and collaborative research environment.

Job requirements

You need to have:

• MSc degree in (applied/fundamental) mathematics or related field.
• Passion for fundamental research driven by scientific curiosity.
• Demonstrated proficiency in (controlled) stochastic processes or analysis of (non-linear) partial differential equations.
• Excellent communication skills in written and spoken English. Strong problem-solving skills and the ability to work both independently and collaboratively in an international research environment.

You will stand out if you have coursework or experience in the fields of differential geometry, optimal transport theory, differential games or filtering theory.

TU Delft (Delft University of Technology)

Delft University of Technology is built on strong foundations. As creators of the world-famous Dutch waterworks and pioneers in biotech, TU Delft is a top international university combining science, engineering and design. It delivers world class results in education, research and innovation to address challenges in the areas of energy, climate, mobility, health and digital society. For generations, our engineers have proven to be entrepreneurial problem-solvers, both in business and in a social context.

At TU Delft we embrace diversity as one of our core values and we actively engage to be a university where you feel at home and can flourish. We value different perspectives and qualities. We believe this makes our work more innovative, the TU Delft community more vibrant and the world more just. Together, we imagine, invent and create solutions using technology to have a positive impact on a global scale. That is why we invite you to apply. Your application will receive fair consideration.

Challenge. Change. Impact!

Faculty Electrical Engineering, Mathematics and Computer Science

The Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) brings together three scientific disciplines. Combined, they reinforce each other and are the driving force behind the technology we all use in our daily lives. Technology such as the electricity grid, which our faculty is helping to make completely sustainable and future-proof. At the same time, we are developing the chips and sensors of the future, whilst also setting the foundations for the software technologies to run on this new generation of equipment – which of course includes AI. Meanwhile we are pushing the limits of applied mathematics, for example mapping out disease processes using single cell data, and using mathematics to simulate gigantic ash plumes after a volcanic eruption. In other words: there is plenty of room at the faculty for ground-breaking research. We educate innovative engineers and have excellent labs and facilities that underline our strong international position. In total, more than 1000 employees and 4,000 students work and study in this innovative environment.

Click here to go to the website of the Faculty of Electrical Engineering, Mathematics and Computer Science.

Conditions of employment 
Doctoral candidates will be offered a 4-year period of employment in principle, but in the form of 2 employment contracts. An initial 1,5 year contract with an official go/no go progress assessment within 15 months. Followed by an additional contract for the remaining 2,5 years assuming everything goes well and performance requirements are met. 

Salary and benefits are in accordance with the Collective Labour Agreement for Dutch Universities, increasing from € 2901 per month in the first year to € 3707 in the fourth year. As a PhD candidate you will be enrolled in the TU Delft Graduate School. The TU Delft Graduate School provides an inspiring research environment with an excellent team of supervisors, academic staff and a mentor. The Doctoral Education Programme is aimed at developing your transferable, discipline-related and research skills. 

The TU Delft offers a customisable compensation package, discounts on health insurance, and a monthly work costs contribution. Flexible work schedules can be arranged.  

Will you need to relocate to the Netherlands for this job? TU Delft is committed to make your move as smooth as possible! The HR unit, Coming to Delft Service, offers information on their website to help you prepare your relocation. In addition, Coming to Delft Service organises events to help you settle in the Netherlands, and expand your (social) network in Delft. A Dual Career Programme is available, to support your accompanying partner with their job search in the Netherlands.  

Additional information
If you would like more information about this vacancy or the selection procedure, please contact dr. Richard Kraaij, via r.c.kraaij@tudelft.nl.

Application procedure
Are you interested in this vacancy? Please apply no later than 27 Apr 2025 via the application button and upload the following documents:

• CV.
• Motivational letter.
• Copy of your master thesis.

You can address your application to dr. Richard Kraaij.

Doing a PhD at TU Delft requires English proficiency at a certain level to ensure that the candidate is able to communicate and interact well, participate in English-taught Doctoral Education courses, and write scientific articles and a final thesis. For more details please check the Graduate Schools Admission Requirements. 

Please note:

• You can apply online. We will not process applications sent by email and/or post. 
• A pre-employment screening can be part of the selection procedure.
• For the final candidates, a knowledge security check will be part of the application procedure. For more information on this check, please consult Chapter 8 of the National Knowledge Security Guidelines. We carry out this check on the basis of legitimate interest.
• Please do not contact us for unsolicited services.