Post-Doctoral Research Visit F/M Multi-type growth fragmentation equations with singular coefficients

Context

Every year Inria International Relations Department has a few postdoctoral positions in order to support Inria
international collaborations.
The postdoctoral contract will have a duration of 12 to 24 months. The default start date is November 1st, 2024
and not later than January, 1st 2025. The postdoctoral fellow will be recruited by one of the Inria Centres in
France but it is recommended that the time is shared between France and the partner’s country (please note
that the postdoctoral fellow has to start his/her contract being in France and that the visits have to respect Inria
rules for missions)

Candidates for postdoctoral positions are recruited after the end of their Ph.D. or after a first post-doctoral
period: for the candidates who obtained their PhD in the Northern hemisphere, the date of the Ph.D. defense
shall be later than September 1, 2022; in the Southern hemisphere, later than April 1, 2022.
In order to encourage mobility, the postdoctoral position must take place in a scientific environment that is truly
different from the one of the Ph.D. (and, if applicable, from the position held since the Ph.D.); particular attention
is thus paid to French or international candidates who obtained their doctorate abroad.

In a recent paper [7], Denis Villemonais and Alexander Watson gave a prob-
abilistic solution to prove the existence, uniqueness and existence of a spectral
gap for the growth fragmentation equation, using in particular quasi-stationary
distribution theory [2]. The goal of this work was, on the one hand, to determine general conditions
for the existence and uniqueness of a semigroup satisfying this equation, and on
the other hand, to describe its long-time behavior.

The aim of the post-doc project is three fold: it concerns the extension of
this result to physically or mathematically relevant situations, the extension of
abstract quasi-stationary results to non-exponential convergence situations, and
of the study of control problem for branching processes.

First, we aim at extending these methods and results to the situation where
k(·, (0, +∞)) is not locally bounded, to cases with accumulation of jumps toward
0, to models with several types of particles, and to prove a sufficient criterion
ensuring that the spectrum is discrete. This last property is of great importance
in order to derive fluctuation results for the branching process associated to the
growth fragmentation phenomenon.

Second, we aim to prove that the limiting behaviour of this equation is
exponential with a well defined limiting profile, under assumptions which only
ensure non-exponential convergence. To do this, we will consider the problem
of proving the existence and convergence toward a quasi-stationary distribution
in the situation where the recent results [2] do not hold, and which represent
an open challenge in the theory of quasi-stationary distribution, but in some
very specific cases such as some birth and death processes (see e.g. [4, 3]) and
diffusion processes [6].

Third, we will consider the problem of controlling branching processes for
the maximization of the eigenvalue (or asymptotic growth rate) under appro-
priate constraint involving e.g. the dispersion of the number of individuals at fragmentation time. We will start the investigation by considering models where
explicit computations are possible, such as in [1] where explicit formulas from
refracted Lévy processes are used.

These researches will be conducted in the perimeter of the associate Inria
team MAGO, with collaborators from UCL and Inria Nancy.

References
[1] Cavalli, B. On a family of critical growth-fragmentation semigroups and
refracted Lévy processes. Acta Appl. Math., 2020, 166, 161-186
[2] N. Champagnat and D. Villemonais. General criteria for the study of quasi-
stationarity. Electronic Journal of Probability, 28, 2023.
[3] E.A. Van Doorn and P.K. Pollett. Quasi-stationary distributions for discrete-
state models. European journal of operational research, 230(1), 1-14, 2013.
[4] M. Kijima, M.G. Nair, P.K. Pollett and E.A. Van Doorn. Limiting condi-
tional distributions for birthdeath processes. Advances in Applied Probabil-
ity, 29(1), 185-204, 1997.
[5] Maillard, P. and Paquette, E. Interval fragmentations with choice: equidis-
tribution and the evolution of tagged fragments. Ann. Appl. Probab., 2022,
32, 3537-3571
[6] W. Oçafrain. Polynomial rate of convergence to the Yaglom limit for Brow-
nian motion with drift. Electron. Commun. Probab., 25: 1-12, 2020.
[7] D. Villemonais and A. Watson. A quasi-stationary approach to the long-
term asymptotics of the growth-fragmentation equation. Arxiv preprint
2202.12553, 2024.

Required qualifications: PhD thesis in probability theory / stochastic pro-
cesses. The selected candidate will be based in Nancy and will make visits to
UCL, UK, when needed. It is also expected that the selected candidate will play
a role in the organization of the meetings of the associate Inria team MAGO
and will be interested by the other subjects of the team, including models with
coagulation and discussion with physicists.

• Subsidized meals
• Partial reimbursement of public transport costs
• Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
• Possibility of teleworking (after 6 months of employment) and flexible organization of working hours
• Professional equipment available (videoconferencing, loan of computer equipment, etc.)
• Social, cultural and sports events and activities
• Access to vocational training
• Social security coverage

2788€ gross/month