This paper is concerned with a comparison principle for viscosity solutions to Hamilton–Jacobi (HJ), –Bellman (HJB), and –Isaacs (HJI) equations for general classes of partial integro-differential operators. Our approach contributes to the literature in three ways: (1) We cast the Crandall–Ishii Lemma into a test function framework to tackle a wide class of second-order integro-differential operators in the spirit of the classical doubling of variables method. (2) We provide a unified approach to estimate the difference of Hamiltonians by adapting the probabilistic notion of couplings to an analytic setting. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology. We apply our theory to a variety of examples, in particular, to second-order differential operators and, more generally, generators of spatially inhomogeneous Lévy processes.

This thesis focuses on two main topics: large deviations for Markov processes and the well-posedness of Hamilton–Jacobi equations. The first two chapters provide an introduction to both areas. Chapter 1 explores the mathematical foundations of Hamilton–Jacobi equations, highlighting their applications in control theory and emphasising the role of viscosity solutions in handling situations where classical solutions fail. Chapter 2 introduces large deviations theory, starting from basic examples and leading to rigorous definitions. A key theme is the connection between large deviations and Hamilton–Jacobi equations, introduced through the Feng–Kurtz method. The subsequent chapters present the main research contributions of this thesis. Chapter 3 studies two examples of two-scale Markov processes and applies the Feng–Kurtz method to establish a large deviations principle. Chapter 4 transitions to the second theme of this thesis: the well-posedness of Hamil- ton–Jacobi equations. Motivated by the previous examples, we analyse a general class of Hamilton–Jacobi equations. We establish a comparison principle for viscosity solutions, demonstrating its applicability in a broad setting. Chapter 5 extends these results by proving the existence of viscosity solutions for a general class of Hamilton–Jacobi equations using Lyapunov control techniques. In the final chapter, Chapter 6, we investigate second-order Hamilton–Jacobi equations, presenting a novel proof of the comparison principle for viscosity solutions.

In this note, our aim is to show that families of smooth hypersurfaces of Rⁿ⁺¹ which are all “C¹ –close” enough to a fixed compact, embedded one, have uniformly bounded constants in some relevant inequalities for mathematical analysis, like Sobolev, Gagliardo– Nirenberg and “geometric” Calderòn–Zygmund inequalities.

We consider the context of molecular motors modelled by a diffusion process driven by the gradient of a weakly periodic potential that depends on an internal degree of freedom. The switch of the internal state, that can freely be interpreted as a molecular switch, is modelled as a Markov jump process that depends on the location of the motor. Rescaling space and time, the limit of the trajectory of the diffusion process homogenises over the periodic potential as well as over the internal degree of freedom. Around the homogenised limit, we prove the large deviation principle of trajectories with a method developed by Feng and Kurtz based on the analysis of an associated Hamilton–Jacobi–Bellman equation with an Hamiltonian that here, as an innovative fact, depends on both position and momenta.

In this survey we present the state of the art about the asymptotic behavior and stability of the modified Mullins–Sekerka flow and the surface diffusion flow of smooth sets, mainly due to E. Acerbi, N. Fusco, V. Julin and M. Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the strict stability property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under W^2,p–perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently “close” to a smooth strictly stable critical set E, both flows exist for all positive times and asymptotically “converge” to a translate of E.