Patterns occur naturally in many physical and biological systems. By pattern, we mean a structure which has a complicated spatial dependence, but retains its shape as time passes. Prototypical examples are water waves, traveling pulses in neurons, convection cells, and tropical cyclones. This dissertation is concerned with the mathematical analysis of such patterns when they are subjected to random fluctuations in the environment, which we refer to as noise. The key questions which we address are stability, noise-induced motion, and long-time behavior of patterns. ... Read more

Classically, population dynamics are described by the deterministic Lotka-Volterra equations. These equations do not accurately reflect reality, where stochastic influences have a big impact on populations of species, due to unpredictable environmental and biological factors. Using the Wiener process to include these influences allows for more realistic results, but means that solutions can deviate to unrealistic population numbers. This thesis provides a self-contained proof of the existence, uniqueness and boundedness of solutions of such systems.
In many cases, population data for all species is unavailable. To make accurate estimates for the unknown or hidden data, the extended Kalman filter can be applied. Which, through a combination of the data and the mathematical model, creates an estimate for the population. An exponential bound for the error of this estimation is derived in expectation ... Read more

Backward stochastic differential equations (BSDE) are a fundamental tool in the mathematical modelling of financial problems. Through the famous nonlinear extensions to the Feynman-Kac formula, they do not merely provide a stochastic representation of the solution to large classes of partial differential equations such as pricing- or Hamilton- Jacobi-Bellman equations, but also include sensitivities, corresponding to derivatives of the solution, which are crucial in many financial mathematical applications. Henceforth, they simultaneously represent option pricing and hedging problems, and form a natural framework for the numerical treatment of stochastic optimal control.

The main challenge in the numerical approximation of such equations is the computation of conditional expectations over potentially high-dimensional spaces. In classical settings, where the dimensionality of the underlying randomness is moderate, many approaches have been proposed in the literature. However, for high-dimensional problems, one has to resort toMonte Carlo methods. In recent years, a new class of regression Monte Carlo methods has arisen in the literature, so called deep BSDE methods, which practically approximate the solution of BSDEs in a neural network regression Monte Carlo framework, after forming a suitable loss function motivated either by stochastic optimal control or the martingale representation theorem. These classes of methods can roughly be divided into two main categories. Forward methods, where the solution of the associated backward SDE is simultaneously optimized in a global optimization, minimizing a loss function stemming from a stochastic target problem reformulation. Alternatively, backward methods have been investigated, where the numerical resolution of the equation is decomposed into smaller sub-optimizations corresponding to a discrete set of points in a suitable time discretization. These methods enabled the numerical treatment of longstanding open challenges, such as the pricing and deltahedging of multi-asset financial options up to d = 100 risk factors and beyond.

The goal of this thesis is to analyze such modern machine learning based numerical methods, and apply them in the financial mathematical context. We propose numerical extensions of these methods in high-dimensional frameworks, analyze their convergence properties in discrete time, and investigate their robustness and accuracy in practical applications such as hedging and stochastic optimal control. Ourmain contributions in each chapter can be summarized as follows…
... Read more

In this dissertation, we aim to apply Fourier multiplier theory as a unifying method to advance the study of semigroup theory and further develop the Fourier multiplier theory itself. ... Read more

In this thesis, we study deterministic and stochastic abstract evolution equations which are of fractional order, either spatially or spatiotemporally.

The preliminary concepts necessary to interpret such equations are collected in the first chapter.

The second chapter is devoted to analyzing a class of equations which generalizes the spatial Whittle–Matérn stochastic partial differential equations (SPDEs) to space–time. We define solution concepts, establish their well-posedness and equivalence, and study their spatiotemporal regularity and covariance structure. The abstract spatial operator in these equations is assumed to be the negative generator of a strongly continuous semigroup on a separable Hilbert space; for certain spatial regularity results, it is moreover assumed to be analytic.

In the third chapter, we define various higher-order Markov properties for Hilbert-space-valued stochastic processes and investigate the relations between them. For solutions to linear abstract SPDEs, we identify two sets of additional conditions under which locality of the precision operator is either necessary or sufficient to satisfy the weakest Markov property. We show that the mild solutions from Chapter 2 satisfy a higher-order Markov property if the orders of the equations are non-fractional (i.e., integer) and that, conversely, a necessary condition for the weakest Markov property is, in general, not satisfied if this parameter is fractional. We moreover establish that an infinite-dimensional analog of the fractional Brownian motion can be obtained as a limiting case of this class of equations.

The fourth chapter concerns the deterministic natural Dirichlet problem for nonlocal abstract space–time operators posed on an arbitrary Banach space. We impose that the whole "past" of the solution equals a given function, i.e., its values are prescribed at all times preceding some (non-trivial) initial time. If the semigroup associated to the spatial operator is exponentially stable, we show that the problem is well-posed in an L^p-sense with p in [1, infinity]. Whenever such solutions are continuous, they satisfy a mild solution formula which expresses them in terms of the initial data and the semigroup, thus generalizing the well-known variation of constants formula for the first-order abstract Cauchy problem. A comparison to analogous solution concepts arising from Riemann–Liouville and Caputo type initial value problems is included.

Finally, in the fifth chapter, we study the convergence of a sequence of semilinear parabolic stochastic evolution equations posed on a sequence of Banach spaces approximating a limiting space. The abstract "discrete-to-continuum approximation" setting is encoded using projection and lifting operators. These allow us to define and compare the discretized equations, and to formulate conditions under which their solutions are well posed and convergent when lifted to a common state space. Our framework is applied to the case where the limiting problem is an SPDE whose linear part is a generalized Whittle–Matérn operator on a manifold, discretized by a sequence of geometric graphs constructed from a (random) point cloud. ... Read more

The topic of this dissertation lies in the field of operator algebras and non-commutative functional analysis. The dissertation studies structural properties of C*-algebras and von Neumann algebras, with a focus on the latter. New rigidity results are obtain for von Neumann algebras coming from Coxeter groups and/or graph products. Furthermore, new results are obtained on approximation properties of von Neumann algebras and C*-algebras coming from graph products. Last, this dissertation obtains sharp estimates on the norms of commutators in factors. ... Read more

We tackle the well-posedness of certain dynamical systems that result in non-autonomous quasi-linear problems in a critical setting, where the coefficients defining the flux and the Neumann boundary conditions depend on the solution itself. We want to show the existence and uniqueness of these solutions on a very short timescale.

The local well-posedness of quasi-linear problems in a critical setting by the maximal $L^p$-regularity theory from Chapter 18 of Hytönen, van Neerven, Veraar, and Weis (2024) are investigated, where we use the non-autonomous setting with non-constant domains from Di Giorgio, Lunardi, and Schnaubelt (2005). Dominant examples in the literature of such problems are problems with multidimensional, non-constant Neumann boundary conditions influencing the domain of the operator. In this thesis, we look for ways to ensure the short-timescale existence and uniqueness of solutions to these problems and research the possibility of applying them to the model problem with Neumann boundary conditions. By applying non-autonomous linear theory from Chapter 3 Part II of Yagi (2010), we find a result that allows us to determine the existence and uniqueness of short timescale mild solutions. When applied, however, we see that, unlike the work of Yagi (2010), we can only guarantee the local well-posedness of the model Neumann problem by using averaging functions because of our more strict critical setting. In the future, results can be based on different regularity types, or the given result can be applied on spaces with negative smoothness. ... Read more

In this thesis the question of existence and uniqueness of solutions to stochastic thin-film equations is investigated. The latter refers to a class of fourth-order, quasilinear, degenerate parabolic stochastic partial differential equations with (possibly nonlinear) gradient noise, which describe the evolution of a thin liquid film driven by surface tension and thermal fluctuations. Difficulties in their analysis arise due to the subtle interplay between the gradient noise term and the degenerate parabolic operator as well as the absence of a comparison principle for fourth-order equations.

Using stochastic compactness arguments the existence of weak martingale solutions is established for linear noise in effective dimension two (the physical dimension) and for nonlinear noise in dimension one. A key ingredient is the derivation of a-priori estimates on solutions to the equation as well as finding consistent approximations converging to a non-negative (and hence physically meaningful) limit. In the nonlinear noise case, the situation of an almost everywhere positive and non-negative initial value are treated separately. While in the former case the energy of the system can be estimated uniformly in time, the latter case allows only for a control of lower order functionals called α-entropies along the dynamics. While this forces us to rely on a weaker notion of solutions for non-negative initial values, it applies to a larger class of noises driving the equation.

Subsequently, based on stochastic maximal regularity techniques, stochastic thin-film equations are shown to be well-posed for strictly positive initial values in any spatial dimension until the profile touches down or blows up in suitable function spaces. In dimension one, the latter possibility is excluded by establishing a-priori estimates on the solution under the additional consideration of repulsive interaction forces between the molecules of the fluid and the substrate. Consequently, the equation admits unique solutions globally in time in this case for linear and nonlinear gradient noise terms. We also show that these solutions become as smooth as the noise permits. ... Read more

The stochastic FitzHugh-Nagumo equations are a system of stochastic partial differential equations that describes the propagation of action potentials along nerve axons. In the present work we obtain well-posedness and regularisation results for the FitzHugh-Nagumo equations with domain R^d. We begin by considering the weak critical variational setting, where we prove global well-posedness for the case d=1. We subsequently consider the strong variational setting, which allows us to extend our well-posedness results to d <= 4. To prove well-posedness and regularisation for arbitrary d, we consider the FitzHugh-Nagumo equations in the L^p(L^q)-setting. Building on earlier results for reaction-diffusion equations, we first prove well-posedness on the d-dimensional flat torus and use bootstrapping techniques to prove instantaneous regularisation of the solution. We subsequently extend the theory for reaction-diffusion equations to the unbounded domain R^d to finally prove well-posedness and regularisation for the FitzHugh-Nagumo equations on R^d. ... Read more

In probability theory, Lp spaces for p > 0 together with the topology of conver-
gence in probability have been widely applied. However, in that case we restrict
ourselves to only a part of all the measurable functions and to an underlying prob-
ability space. One of the main aims of this thesis is to generalize this concept to
the set of all measurable functions with the usual a.e. equivalence classes (which
we call L0) and (possibly) non-finite measure spaces. The other main aim is to
establish an ordered structure on this L0 space ... Read more

In this thesis, the semilinear Cahn-Hilliard-Gurtin equation is studied using the method of Maximal Regularity. In 2012, Wilke developed a linear theory in $L^p$-spaces, and achieved a local and global well-posedness result for large $p$. In 2013, Denk and Kaip developed a linear theory in mixed integrability $L^pL^q$-spaces, using the method of Newton polygons. In this thesis, we connected the recent weighted anisotropic Mikhlin multiplier theorem of Lorist with the method of Newton polygons, leading to a linear theory in time-weighted $L^pw_\alpha L^q$-spaces, which is novel. By a postulation that the linear theory also holds in domains, we are able to treat the local well-posedness in the recently developed critical space setting of Prüss et al. This approach draws upon recent advances in interpolation theory in the setting of fractional Sobolev spaces with power weights in time, such as exhibited in the work of Agresti and Veraar.By adapting the global well-posedness result of Wilke, we are able to treat the semilinear equation in less regular spaces, i.e. smaller integrability parameters $p$ and $q$, and with rough initial data. ... Read more

This master's thesis introduces a new $p$-dependent coercivity condition through which $L^p(\Omega; L^2([0, T]; X))$ estimates can be obtained for a large class of SPDEs in the variational framework. Using these estimates, we obtain existence and uniqueness results by using a Galerkin approximation argument. The framework that is built is applied to many SPDEs such as stochastic heat equations with Dirichlet and Neumann boundary conditions, Burger's equation and Navier-Stokes in 2D. Furthermore, we obtain known results for systems of SPDEs and higher order SPDEs using our unifying coercivity condition. We also obtain first steps towards a theory of higher order regularity of stochastic heat equations. ... Read more

When transforming PDE problems using Fourier and Laplace transforms, we can find functions that represent the problem, and which can be used to determine properties of the problem. We define such functions as symbols $P(\lambda,z)$. In general, we define the class of symbols $S(L_t\times L_x)$ are all functions which are represented by a polynomial of the form $R_P(\lambda,z):=\sum_{\ell\in I_P}\tau_\ell(\lambda,z)\phi_\ell(\lambda)\psi_\ell(z)$, where $\tau_\ell(\lambda,z)$ are $\rho$-homogeneous functions of $(\lambda,z)$ on the cones $L_t\times L_x$, and $\phi_\ell(\lambda)$ and $\psi_\ell(z)$ homogeneous functions of $\lambda$ on the cone $L_t$ and $z$ on the cone $L_x$ respectively. These functions have a certain $\gamma$-order $d_\gamma(P)$ that shows the order of the function relative to a relative weight $\gamma$, and a certain $\gamma$-principal part $\pi_\gamma P(\lambda,z)$, which is the part of $P$ that causes this $\gamma$-order.

For such a symbol, we define its Newton polygon $N(P)$ as a certain convex hull of points on $[0,\infty)^2$, which serves as a geometric description of the order of $P$. We define the weight function $W_P$ to be a positive polynomial with the orders found on the vertices of the Newton polygon $N(P)$. We define a notion of order functions, and show the order function of $P$ is $d_\gamma(P)$.

We define a notion of parameter-ellipticity and parabolicity for symbols in $S(L_t\times L_x)$ based on the Newton polygon $N(P)$, namely N-parameter-ellipticity and N-parabolicity. Using various results from the work of R. Denk and M. Kaip \cite{Denk Kaip} and results I introduced myself, we then prove the equivalence between N-parameter-ellipticity and having non-vanishing $\gamma$-principal parts. ... Read more

In 2016 Hieber and Kashiwabara showed that the three dimensional primitive equations admit a unique, global, strong solution for all initial data in a closed subspace of the Bessel space $H^{2/p,p}(\Omega)$ provided $p\geq6/5$, being this the first result in the general $L^p$-setting. Their approach consisted in studying the properties of the hydrostatic Stokes operator $A_p$ defined on the solenoidal subspace $L^p_{\overline{\sigma}}(\Omega)$ of $L^p(\Omega)$. In 2017 Giga et. al. further proved that the hydrostatic Stokes operator $A_p$ admits a bounded $H^\infty$-calculus, obtaining maximal $L^q-L^p$ regularity estimates for the linearized primitive equations in a much simpler way. In this work we will study Giga et. al.'s and Hieber and Kashiwabara’s works particularized for the $L^2$-case as well as all the necessary literature to replicate the proofs. The goal of the thesis is to present an extended version of Giga et. al.’s proof to make it more accessible. Although the $L^p$-case is not studied for lack of time, we differentiate between the Sobolev-Slobodeckij, Bessel potential and Besov spaces to accentuate how we could extend the proofs to the $L^p$-setting. ... Read more

The subject of this thesis is the study of the multilinear Muckenhoupt weight classes and the quantitative boundedness of operators with respect to these weights in both the scalar-valued and the vector-valued setting. This includes the study of multisublinear Hardy-Littlewood maximal operators, sparse forms, and multilinear Rubio de Francia extrapolation methods. ... Read more

In dit verslag behandelen we de Extrapolatie Stelling van Yano: een stelling die voor operators van de vorm T : f(x) -> T(f)(x) in bepaalde omstandigheden een afschatting geeft van de absolute integraal over T(f)(x) in termen van f(x) zelf. We zullen zien dat deze afschatting, en meer afschattingen van soortgelijke vorm, onder de juiste voorwaarden continuïteit impliceert voor de operator. Om deze stelling te kunnen behandelen zullen we wat theorie over vectorruimtes opgebouwd uit functies, en enkele voorbeelden relevant voor de stelling, introduceren. Ook behandelen we wat theorie omtrent operators, en passen we de Extrapolatie Stelling van Yano toe op enkele operators. ... Read more

In [8] Kwapien proved that every mean zero function f ∈ L∞[0, 1] we can
write as f = g ◦ T − g for some g ∈ L∞[0, 1] and some measure preserving
transformation T of [0, 1]. However, as was discovered in [4] there is a gap
in the proof for the case that f is not continuous. The aim of this bachelor
thesis is filling in that gap in the proof. We first extend Kwapien’s proof for continuous functions to certain other measure spaces. Thereafter, we use the method of proof suggested by Kwapien, to proof the theorem for mean zero function f ∈ L∞[0, 1] for which λ(f⁻¹({x})) = 0 for all x ∈ R. Using this result we then proof that every mean zero function f ∈ L∞[0, 1] can be written as a sum f =(g₁ ◦ T₁ − g₁) + (g₂ ◦ T₂ − g₂) where g₁, g₂ ∈ L∞[0, 1] and where T₁, T₂ are
measure preserving transformations of [0, 1]. We finish this thesis with an
application of Kwapien’s theorem in the study to singular traces ... Read more

Het plankprobleem gaat over het overdekken van convexe vormen met hypervlakken. De gegeven stellingen en bewijzen in het artikel \The plank problem for symmetric bodies" van Keith Ball zijn lastig te begrijpen voor bacholor wiskunde studenten. In dit verslag wordt verduidelijking gegeven van dit artikel zodat deze begrijpelijk en toegankelijk wordt voor anderen. Hierbij speelt de trace-class een belangrijke rol en komt er veel lineaire algebra over symmetrische matrices voorbij. ... Read more

Recently, by Z. Shen, resolvent estimates for the Stokes operator were established in L^p(Ω) when Ω is a Lipschitz domain in R^d, with d≥3 and |1/p-1/2|<1/(2d)+ε. This result implies that the Stokes operator generates a bounded analytic semigroup in L^p(Ω) in the case that Ω is a three-dimensional Lipschitz domain and 3/2-ε<p<3+ε. To fully understand the work of Z. Shen, a lot of background information is needed. In this thesis the resolvent estimates are studied in detail in the case d=3. In the end the results of Shen are extended to resolvent estimates in L^p(w,Ω), where Ω is a three-dimensional Lipschitz domain, |1/p-1/2|<1/6, and w∈A_2p/3∩RH_3/(3-p) is a weight function that belongs to an intersection of a Muckenhoupt weight class and satisfies a reverse Hölder inequality. ... Read more

The normal distribution is a very important distribution in probability theory and statisticsand has a lot of unique properties and characterizations. In this report we look at the proof of two of these characterizations and create counterparts of a normal distribution on abstract spaces, such as vector spaces and groups, which we shall call Gaussians. When we look at R^d, all these Gaussians coincide, along with a Gaussian vector in the normal sense, called multivariate normal. Furthermore, for one Gaussian we prove that it has exponential integrability properties. ... Read more