In this thesis, we study large deviations and parameter estimations for small noise diffusion processes. In Chapter 1, we start with the classical limit theorems to intuitively introduce large deviations and parameter estimations, which provide for further developments in the thesis.

The first part, consisting of Chapters 2 - 4, is on large deviations. In Chapter 2, we begin with the simple stochastic differential equation to explain the idea behind the proof of the nonlinear semigroup method, which is used to prove large deviations in Chapters 3 and 4. In the process, viscosity solutions and the Hamilton-Jacobi-Bellman equations are introduced...@en

In this thesis we investigate discrete space Markov processes with multiple layers and how these can be applied to physical systems such as RNA transcription. We show that the Markov processes of single particles satisfy an invariance principle (i.e., the limiting behaviour is Brownian with a certain drift) for both homogeneous and random environments. Additionally, we perform an in-depth numerical analysis of multiple particle systems in the context of RNA transcription and reproduce some known phenomena such as RNA polymerase clustering and cooperation.

The focus of this thesis is the hydrodynamic limit of the Brownian Energy Process (BEP) and the
Asymmetric Brownian Energy Process (ABEP) in infinite volume. The thesis starts by introducing some general theory about Markov processes, after which the topic of Markov duality is introduced. A number of relevant interacting particle systems (IPS) will be introduced, and we will show how the BEP and the ABEP can be related to the simpler Symmetric Inclusion Process (SIP) through Markov duality. Using these tools, the first main result is proven, which states that the hydrodynamic limit of the BEP is a weak solution to the heat equation. As a consequence of this, in the second main result we use the relation between the BEP and the ABEP to prove that the hydrodynamic limit of the ABEP is a weak solution to the viscous Burgers’ equation. We then attempt to show a similar result for a newly developed IPS, the Dynamic ABEP. Finally, we prove propagation of chaos for the BEP and the ABEP, where for the latter we argue that this is only possible in a finite volume. As a part of the proof of this we argue that the SIP in infinite volume is very ’similar’ to Independent Random Walkers (IRW) when we look at a long enough time-scale, where we give a rough sketch of a proof that significantly improves upon a quantification of this ’similarity’ established in the literature.

In this thesis the hydrodynamic limit of the Freezing Model is studied. The model consists of an integer line on which particles can get frozen to different degrees, analogous to jumping to another integer line, with certain rates and can get unfrozen with certain rates per frozen layer. The main result of the thesis is a proof that the hydrodynamic limit for the Freezing model converges to a system of PDE’s describing the particle density for each layer, either the ground layer or a frozen one. Firstly, it is proven that the position of a random walker in the Freezing Model, appropriately scaled, converges to a so-called Switching Brownian Motion. This together with duality is used in the rest of the proof. Secondly, it is proven that the expectation of the empirical field densities of the layers converges towards the solutions of the aforementioned PDE’s. Lastly, it is proven that the variance of the empirical field density converges to 0. Under an additional diffusive scaling of the system of PDE’s for the particle densities, a condition for diffusive behaviour is set up involving the ratio of the freezing and unfreezing rates. It is shown that the PDE’s collapse into the heat equation if this condition is satisfied. Finally, the case where the condition is not satisfied is investigated. The model is then no longer memoryless like a Markov process and shows sub-diffusive behaviour. The rescaled position of a single particle then no longer converges to Brownian motion, but to Brownian motion on the time scale on which the particle occupies the ground layer. This time scale is t β−1 with 1 < β < 2. This grows slower than t for large enough t. Additionally, simulations in Python were built to show that the model exhibits diffusive or nondiffusive behaviour depending on the jump rates of the process. Before everything, some mathematical preliminaries about probability theory, Markov theory, random walks and duality are given.

In this thesis the theory of Markov processes and creation and annihilation operators will be used to derive the time evolution of a discrete reaction-diffusion system. More specifically, we make use of transition rates to construct the generator of a process. We then transform this generator through suitable quantum mechan- ical operators to arrive at our result, namely that the Poisson dis- tribution of a reaction-diffusion system stays Poisson distributed in time with varying rate. This applies for the univariate case, with only a simple birth and death process, as well as the multi- variate case. Furthermore, simulations support this result while also showing that an interacting particle system with pairwise an- nihilation does not evolve according to the Poisson distribution.

Various problems related to Random Walk in Random Environments have been researched intensively over the past 40 years by both the mathematics and physics communities. The setup of these problems is quite simple: one defines a Random Walk of choice and allows it to traverse a Random Environment that influences the Random Walk in a predetermined way.

In this thesis, we study the survival probability of a Random Walk in various random environments of traps. These Random Environments of traps are divided into two distinct classes: static and dynamic. In the static environment, the trap configuration is randomly chosen at time zero and remains fixed thereafter. In the dynamic case, the trap configuration evolves over time according to some predefined dynamics.

The problem in the static setting has a long history, and the situation is fairly well understood. For the dynamic environments, however, the state of the art is less developed, as there is no unified approach to deriving survival probabilities for different dynamic environments. Additionally, almost nothing is known about the interpolation between the models of random walk in static and dynamic environments. The interpolation model should exhibit exponential decay of survival probability in a fast environment, corresponding to the dynamic environment, and show subexponential decay of survival probability when the environment is slow, corresponding to the static environment.

One of the goals of this thesis is to compile various results of survival probabilities in different dynamic environments and to create a model that allows interpolation between exponential and subexponential decay of survival probability.

The Kalman filter is a recursive algorithm that estimates the state of a dynamic system subject to measurement and model noise. If all noise terms affecting the system are white Gaussian noise with known mean and variance, and all noise terms are independent of each other, then the Kalman filter is the optimal estimator for the state variable. When measurements are collected from multiple sources, the covariance between these sources should be known or the sources should be independent to ensure that the estimate made by the Kalman filter is optimal. When the covariance between dependent measurement sources is not known, various methods exist which provide a solution to this problem. This thesis discusses two methods: the H∞ filter and covariance intersection...

In this thesis, we study stochastic duality under hydrodynamic scaling in the context of interacting particles on a grid. The approach is inspired and motivated by the relation between duality and local equilibria. We identify duality relations in terms of the expectation of the density field for which the hydrodynamic limit is recovered. This is initially done both for symmetric inclusion and exclusion processes as well as for independent random walkers. We continue with the independent case and generalize to particles that also possess a, possibly scale-dependent, internal energy state. The results in this context assume generator convergence under scaling and are illustrated using run-and-tumble systems. This work also includes examples concerning instances of run-and-tumble processes that do not have convergence on a generator level. Apart from run-and-tumble processes, we examine the effect of reservoirs on the relevant duality relations and macroscopic profiles. The reservoirs are found to correspond with boundary conditions for the macroscopic profile.

We consider the problem of random walks moving around on a lattice Zd with an initial Poisson distribution of traps. We consider both static and moving traps. In the static case, we prove that the survival time has a decay of e−c t d /d +2 based on a heuristic argument. In the moving case we aim to prove the sub-exponential decay of the survival time in dimensions 1 and 2, as well as the exponential de- cay of survival in dimensions 3 and higher. We achieve the former by first expressing the survival probability in the range of a random walk and by showing that the asymptotic behavior of said range behaves in a sub-exponential and exponential way for dimensions 1/2 and ≥ 3 respectively. Further- more, we also show an upper bound for the survival time of the form lim supt →∞ 1 t log P(T ≥ t ) < 0. Following this we look at the situation where traps decay as ||x|| → ∞. Meaning less traps will be distributed further away from the origin. We show that in the static case, if the decay rate satisfies the condition px ≤ p(||x||) where p(r ) is non-increasing and r p(r ) is integrable and convergent, that the random walk will be transient, meaning that there will be a strictly positive chance of survival. Lastly, we then show that for the dynamically moving traps case, if the decay rate is "fast enough", meaning that if the Poisson parameter of the distribution of the traps ρ(x) is of the form 1/||x||2+α where α > d − 2, that there will also be a strictly positive probability of survival.

In this thesis, the hydrodynamic limit (HDL) for two trapping models is studied, the Random Waiting Time Model (RWTM) and the fractional kinetics process (FKP), on a discrete lattice Zd . The RWTM is studied for dimension d ≥ 1 and E[wi ],∞where wi denotes the waiting time at position i . On the other hand, the FKP is studied for d ≥ 3 and a nonexisting first finite moment. Instead, it is assumed that the waiting times wi follow a power law distribution. Two main results are presented in this thesis. Firstly, it is proven that the HDL for the RWTMconverges to the solution of the heat equation. The solution is deterministic. The proof consists of showing that the expectation value of the empirical density fields converge to the aforementioned solution by using the duality property and Doob’s theorem, and that the variance is finite and decays to 0. Additionally, it is proven that a rescaled random walk converges to a Brownian motion by using Lévy’s characterisation of Brownianmotion. Secondly, it is proven that the HDL for the FKP converges to a random measure of the solution of the fractional heat equation, defined in the Caputo sense. Hence, the solution is random. The proof consists of using similar techniques of the proof of the RWTM and of using that the limit of a sequence of random speed measures is again random. Before all of this, an introduction toMarkov processes, their semigroups, martingales, and generators is presented. Additionally, an introduction to random walks, Brownianmotion, and duality is included.

In this thesis, research was done in the area of interacting particle systems. Especially, the symmetric exclusion process with local perturbations was investigated. These perturbations, were in the form of sinks and sources, which add or take away particles at certain rates. Moreover, simulations were done for the asymmetric exclusion process. This process took place on a ring, with the addition of a source. For the symmetric exclusion process with sinks and sources, certain expressions were proven for the expected occupancy of a site. For the simulations, the main goal was to find out how the jumping rates and starting density, influenced the time to get to the fully occupied state, at different source rates. The first proof was for a source at an arbitrary site. From this expression, one could see that if the rates were recurrent, the system converged to the fully occupied state. If, however, the rates were transient, the system had a limiting density. Thereafter, it was shown that if a sink and source are placed in the same arbitrary site, the system always converged to a density, which under the Bernoulli measure, was not equal to the fully occupied state. The fact that the sink and source were in the same site, was an indispensable condition. Subsequently, the case of countably many sources was investigated. For which it was also shown that recurrent rates always yield a fully occupied state, as time tends to infinity. Whereas transient rates, once again, caused a limiting density. Moreover, the special case of a simple random walk in three dimensions or higher was investigated. If, for distances far away from the origin, the source rates could be bounded above by a certain function, then the system would not converge to the fully occupied state. Also, another proof showed that a recurrent set of source sites would always let the process converge to a fully occupied state. Lastly, similar conditions for one time dependent source at the origin were proven. Namely, it was shown for recurrent rates, that if the source dies out quick enough, as t tends to infinity, the system did not converge to the fully occupied state. For the simulations, it was found that the expected time to fill up was influenced by the starting density, both at small and large source rates, in a linear way. This conclusion could not be drawn for the rate difference: p-q. Due to the large error of the fit, a linear relation was not clearly seen.  Finally, some concluding remarks were made on both aspects of the thesis, along with some recommendations for future projects related to this topic.

In this thesis, wealth distribution in a closed economic system is examined by studying the simple inclusion process (SIP). The simple inclusion process is a model coming from statistical physics that models the jumping of particles in a graph. In the model particles have attraction among each other and each site also has a characteristic attraction parameter, denoted by the variable α. We regard the simple inclusion process as an agent based model for the economy for which sites represent agents and the particles represent wealth transferring from agent to agent. For the model, invariant measures are found that represent possible long term distributions of wealth in the system. We extend the model by looking at α as a parameter drawn from its own underlying probability distribution, ψ(α). In the view of wealth distribution, α can be seen as the wealth attraction an agent has. We set out to look for conditions on the distribution of α for which we obtain asymptotic power law behaviour of the resulting wealth distribution. It is shown that if a higher-order moment of ψ(α) diverges, the resulting wealth distribution will have a weak asymptotic power law lower bound. By setting stricter conditions on the distribution of α, we obtain that the wealth distribution will be asymptotically equal to a power law.

In this thesis the behaviour of a Bose Einstein condensate is explored that consists of bosons that annihilate. In order to do this a system where bose einstein condensation occurs is modeled as a Zero Range process which is a special case of a Markov process.
First we made a single slow site Zero range process with uniform rates and see how this would be influenced by annihilating particles. For this system it is shown that, in the large system limit, the number of particles in the slow site is given by a differential equation. A series of realization of different sizes of this system are done to support that the fraction of the particles in the ground state converges
to the differential equation.
In order to relate this mathematical approach to physical Bose Einstein condensates it is established that the equilibrium distributions of multiple bosons in one system is the same as the detailed balance of a zero range process where the transition rates depend on the occupation of a site, with an indication function, that is equal to 0 when there are no particles at that site and some constant 𝑐 if there is one
or more particles at that site.
In statistical physics we need to take the energy of the state of the system into account. If the particles do not have interaction with each other this is the sum over all particles of the energy of the particle site. As a zero range model only has rates with one particle hop we can see that the difference in energy of the system is equal to the difference in energy of the sites of hop. In order to stay in detailed
balance the rate for the specific hop must be exp(BΔ𝐸) times the opposite rate, where B = 1 / 𝑘_b 𝑇.
Now we can go from any system with a set of spin orbitals and energies of those spin orbitals and design a zero range model where the spin orbitals correspond to sites of the zero range model with rates between those sites, such that the systems detailed balance distribution is the same distribution as the equilibrium of the physical system. In this zero range model an annihilation term is introduced.
This can be any function of occupation for the site, but in this report a simple relation that decreases when Δ𝐸 increases is used.
In order to see the effect on such systems the system is numerically approached for a 100 particles system in a 3 dimensional harmonic potential. This system is too complicated to get an differential equation that can be easily solved. Therefore it is approximated. This numerical approximation appears to have similar behaviour to the numerical approximation done in [3]. However the results are not identical.
The advantage of the model in this thesis is that an annihilation therm can be implemented. This approximation of a 3 dimensional harmonic oscillator with annihilating particles is done. If annihilation is slow enough, the exited sites form a different equilibrium compared to a Bose Einstein condensate of non annihilating particles. This happens as long as there is a condensate in the ground state.

In this thesis we study an interacting particle system: the Symmetric Inclusion Process with slowly varying inhomogeneities (SIP(α)). In the SIP(α) particles display random walk like behaviour subjected to an attractive type of interaction whilst evolving in an inhomogeneous environment. We set out to prove its hydrodynamic limit. The main tool helping us for obtaining the hydrodynamic limit is the self-duality property of the process.

In several experiments, enzymes have shown an in increase in diffusivity in the presence of their substrate. The enhancement in diffusivity ranged from as low as 28% for urease to 80% in the case of alkaline phosphatase. There are two main competing theories. One asserts that catalytically driven boosts propel the enzyme forward in ‘leaps’, while the other argues that phoretic activity due to attractive or repulsive surface interactions on the enzyme are responsible for the enhanced diffusivity. At the moment of writing, no consensus has been reached on the mode of enhancement. A novel agent-based lattice model for the diffusion of enzymes was derived in this Bachelor’s Thesis in accordance to the phoretic theory of enhanced diffusion. The model that was developed is a stochastic model which describes the interactions at the particle level. The model showed that phoresis produces enhanced diffusion in the order that was expected for the enzyme urease. Furthermore, the model showed pattern formation in the case of repulsive interactions between enzyme and substrate. The instability of this phase transition was investigated using linear stability analysis on the continuum limit of the lattice model. This yielded a restriction on the strength of the interactions for which pattern formation could occur.

In this paper we started by explaining what a Markov chain is. After this we defined some key concepts such as stationarity, reversibility and ergodicity which were used throughout the rest of the paper. Next, the classical central limit theorem was stated in order to refresh the reader of this theorem and to show that, because of the dependence, this theorem is not applicable for the Markov chains we discussed earlier. In the next section we started working on the Kipnis-Varadhan central limit theorem. We defined the Martingale central limit theorem and used it to show that the additive functions can be approximated by a martingale. This had the result that the Martingale central limit theorem implies the central limit theorem for the additive functions. Furthermore, we also derived an equation for calculating the limiting variance. For this we used the key concepts discussed in section 2 and another concept called spectral measures. The section is finished by a simple example in order to illustrate what we have discussed previously. In the next section we used simulations in one-dimension and two-dimensions to show that, in a random environment, the distribution of the additive functions indeed converges to that of the normal distribution. Because the probability distributions defined on the random environment was a symmetric distribution the Kipnis-Varadhan central limit theorem was not yet needed since the Martingale central limit theorem already applied. After this we defined some key concepts for a Markov chain working in continuous time. The Kipnis-Varadhan central limit theorem was proven for these Markov chains as well. Furthermore, a different equation was derived to find the limiting variance for continuous time Markov chains. In section 7 the random conductance model was explained. In this chapter the random conductance model was used in order to make simulations to show that the Kipnis-Varadhan central limit theorem works. After simulating the Markov chain in continuous time we indeed saw that the histograms show convergence towards the normal distribution. This shows that for the simulations the Kipnis-Varadhan central limit theorem works.

Quantum communication has been shown to be vastly superior to classical communication in many problems. However no general statements exist which tells us how much better quantum communication is to its classical counterpart. In this thesis it was studied the minimum amount of classical bits required to exactly simulate a quantum communication process. The quantum communication process specifically studied was a quantum prepare and measurement communication problem. It has been shown that the calculation of the amount of classical bits of communication required for simulation reduces to a minimization-maximization optimization problem. Several results have been presented for for solving this optimization problem and in addition a link was made between classical simulations of quantum communication and a recent debate on the reality of the quantum state.

In this thesis we study criticality in the context of the dissipative Abelian sandpile model. The model is linked to a simple trapped random walk, giving a practical method to determine criticality for certain landscapes of dissipative sites. The main results concern the lifetime of the random walk, especially the divergence of its first moment for traps placed on spherical shells. For the one dimensional case the point of divergence is determined with reasonable precision. In higher dimensions the divergence is shown to be possible for an infite amount of shells. The connection between the sandpile model and a random walk is shown mathematically and further researched via simulation.

In this thesis we will study the ergodic measures and the hydrodynamic limit of independent run-and-tumble particle processes, i.e., an interacting particle system for particles with an internal energy source, which makes them move in a preferred direction that changes at random times. We start by providing some basic concepts and theory of Markov processes and interacting particle systems. Afterwards, we define our model on the particle state space $\mathbb{Z}^d \times S$, with $S$ a finite space of internal states, by giving its generator, and we prove a duality result with a similar process which we will use repeatedly throughout this thesis. Then we show that the product Poisson measures with constant parameter are ergodic, and are also the only ergodic probability measures for this process in the space of so-called tempered measures, i.e., measures with bounded factorial moments. Lastly we prove the hydrodynamic limit of this process on $\mathbb{Z} \times S$ by showing that the evolution of the macroscopic density is a weak solution to a PDE.

In this thesis, the diffusive limit of active particle motion in R^d is studied via a technique based on homogenisation. Thereafter, this study is extended to active particle motion on a Riemannian manifold.

Furthermore, as an application of active particle motion, a connection is made with the Dirac equation. On the basis of this connection, a Monte Carlo method is developed to find the ground state of a Dirac equation with static potential. The core idea of this method is based on the Diffusion Monte Carlo method for the Schrödinger equation.