Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations

We consider a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times\R^d$, where $I$ is a right-halfline. We prove logarithmic Sobolev and Poincar\'e inequalities with respect to an associated evolution system of measures $\{\mu_t: t \in I\}$, and we deduce hypercontractivity and asymptotic behaviour results for the evolution operator $G(t,s)$

On coupled systems of Kolmogorov equations with applications to stochastic differential games

We prove that a family of linear bounded evolution operators $({\bf G}(t,s))_{t\ge s\in I}$ can be associated, in the space of vector-valued bounded and continuous functions, to a class of systems of elliptic operators $\bm{\mathcal A}$ with unbounded coefficients defined in I\times \Rd (where $I$ is a right-halfline or $I=\R$) all having the same principal part. We establish some continuity and representation properties of $({\bf G}(t,s))_{t \ge s\in I}$ and a sufficient condition for the evolution operator to be compact in C_b(\Rd;\R^m). We prove also a uniform weighted gradient estimate and some of its more relevant consequence

Generalized Gaussian Estimates for Elliptic Operators with Unbounded Coefficients on Domains

We consider second-order elliptic operators A in divergence form with coefficients belonging to Lloc∞(Ω), when Ω ⊆ ℝd is a sufficiently smooth (unbounded) domain. We prove that the realization of A in L2(Ω), with Neumann-type boundary conditions, generates a contractive, strongly continuous and analytic semigroup (T(t)) which has a kernel k satisfying generalized Gaussian estimates, written in terms of a distance function induced by the diffusion matrix and the potential term. Examples of operators where such a distance function is equivalent to the Euclidean one are also provided

Functional inequalities for some generalised Mehler semigroups

We consider generalised Mehler semigroups and, assuming the existence of an associated invariant measure $\sigma$, we prove functional integral inequalities with respect to $\sigma$, such as logarithmic Sobolev and Poincar\'{e} type. Consequently, some integrability properties of exponential functions with respect to $\sigma$ are deduced

On vector-valued Schrödinger operators with unbounded diffusion in Lp spaces

We prove generation results of analytic strongly continuous semigroups on Lp(Rd, Rm) (1 < p< ∞) for a class of vector-valued Schrödinger operators with unbounded coefficients. We also prove Gaussian type estimates for such semigroups

Kernel estimates for nonautonomous Kolmogorov equations with potential term

Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients and a possibly unbounded potential term

Multimodal Dependent Type Theory

We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode theory allow us to use the same type theory to compute and reason in many modal situations, including guarded recursion, axiomatic cohesion, and parametric quantification. We reproduce examples from prior work in guarded recursion and axiomatic cohesion, thereby demonstrating that MTT constitutes a simple and usable syntax whose instantiations intuitively correspond to previous handcrafted modal type theories. In some cases, instantiating MTT to a particular situation unearths a previously unknown type theory that improves upon prior systems. Finally, we investigate the metatheory of MTT. We prove the consistency of MTT and establish canonicity through an extension of recent type-theoretic gluing techniques. These results hold irrespective of the choice of mode theory, and thus apply to a wide variety of modal situations

On coupled systems of PDEs with unbounded coefficients

We study the Cauchy problem associated with parabolic systems of the form Dtu = A(t)u in Cb (Rd; Rm), the space of continuous and bounded functions f: Rd → Rm. Here A(t) is a coupled nonautonomous elliptic operator acting on vector-valued functions, having diffusion and drift coefficients which change from equation to equation. We prove existence and uniqueness of the evolution operator G(t, s) which governs the problem in Cb (Rd; Rm) and its positivity. The compactness of G(t, s) inCb (Rd; Rm) and some of its consequences are also studied. Finally, we extend the evolution operator G(t, s) to the Lp-spaces related to the so called “evolution system of measures” and we provide conditions for the compactness of G(t, s) in this setting