National Science Centre 2023/51/B/ST1/01270
Resetting: Lévy processes, random walks, branching random walks and their functionals
Within this project we plan to analyse a stochastic process X with resetting that evolves as a
Markov process Y between independent of Y renewal epochs at which a resetting mechanism
is applied. The most common choice is a Poisson process modelling the resetting epochs and the
resetting mechanism based on multiplying the position prior resetting by fixed constant. The special
case of zero constant corresponds to so-called total resetting that moves the position of the stochastic
process to zero. We are going to consider a random walks with resetting as well that behaves like
a regular random walk with additional resetting mechanism performed at the renewal epochs Tn.
In this case usually the time between consecutive resettings is modeled by a geometric distribution.
Similarly, we plan to consider a branching random walks with resettings and generalised Ornstein-Uhlenbeck process.
Stochastic resetting is prevalent in nature and in real systems. It appears in many applied models in
biology, physics, actuarial science, queueing theory, financial mathematics, engineering applications
etc. For example, this process is used in the AIMD algorithm to model the Transmission Control
Protocol, the dominant protocol for data transfer over the internet. The subject of resetting or restart
has been also in the limelight recently due to so-called search processes which corresponds to random
search strategies when prior information about the target is missing or when the searcher itself can
only move diffusively. It appears as the fluid limit scaling for some queueing models (with binomial
catastrophe rates) used in modeling population growth subject to mild catastrophes. Such processes
can be viewed as a particular example of the so-called shot noise model, which is used in models
of earthquakes and layers of sedimentary material accumulated in depositional environments, but not
subjected to subsequent erosion. It can model avalanches or neuron firings as well. Other applications
of stochastic resetting have been also discussed in the context of backtrack recovery by RNA poly-
merase, enzymatic velocity, pollination strategies, enzymatic inhibition, stochastic thermodynamics,
and quantum mechanics. Furthermore, realisation of stochastic resetting was confirmed in switch-
ing holographic optical tweezers as well. The risk process with resetting has been found to be an
indispensable part of modelling mortgage lending and so-called micro-insurance polices.
Within this project we plan to study the following, strongly dependent on each other, topics:
1. Lévy processes and random walks with resetting: nonequilibrium stationary states, time-space
asymptotics of the transition density and exit problems;
2. Branching random walk in random environment with resetting: population size and maximal
displacement;
3. Influence of the heavy-tailed environment on the branching process with resetting;
4. Persistence of the Lévy process and the random walk with resetting and persistence of the
sample means of these processes;
5. Yaglom limits for the processes with resetting;
6. Persistence of the integrated Brownian motion and the integrated random walk with resetting
for the positive half-line;
7. Generalised Ornstein-Uhlenbeck process: nonequilibrium stationary states, exponential functional, time-space asymptotics of the transition density and exit problems.
The analysis of distributions and all kinds of properties of Markov evolutions is an active field of the
present-day probability theory. The richness and timeliness of open problems, coming from theory-orientated
questions and the behaviour of Markov processes, stimulates development of the theory of
stochastic processes, potential theory, and various applications. There are also numerous deep links
with other areas of mathematics. To the best of our knowledge, this project will be a first attempt to
create a unified theory of the multivariate processes with additional proportional regulations appearing
at independent renewal epochs. Questions related to stationary behaviour, NESS, exit times, right-
most particle position, persistence, quasi-stationary laws, etc. posed in this project are in the centre of
the present interest of many researchers due to its importance for applications and other mathematical
problems.