We prove that for a GNS-symmetric quantum Markov semigroup, the complete modified logarithmic Sobolev constant is bounded by the inverse of its complete positivity mixing time. For classical Markov semigroups, this gives a short proof that every sub-Laplacian of a Hörmander system on a compact manifold satisfies a modified log-Sobolev inequality uniformly for scalar and matrix-valued functions. For quantum Markov semigroups, we show that the complete modified logarithmic Sobolev constant is comparable to the spectral gap up to the logarithm of the dimension. Such estimates are asymptotically tight for a quantum birth-death process. Our results, along with the consequence of concentration inequalities, are applicable to GNS-symmetric semigroups on general von Neumann algebras.

Let X be a complex Banach space and B be a closed linear operator with domain $\mathcal{D}(B) \subset X,\,\, a,b,c,d\in\mathbb{R},$ and $0 \lt \beta \lt \alpha.$ We prove that the problem

\begin{equation*}u(t) -(aB+bI)(g_{\alpha-\beta}\ast u)(t) -(cB+dI)(g_{\alpha}\ast u)(t) = h(t), \quad t\geq 0,\end{equation*}

where $g_{\alpha}(t)=t^{\alpha-1}/\Gamma(\alpha)$ and $h:\mathbb{R}_+\to X$ is given, has a unique solution for any initial condition on $\mathcal{D}(B)\times X$ as long as the operator B generates an ad-hoc Laplace transformable and strongly continuous solution family $\{R_{\alpha,\beta}(t)\}_{t\geq 0} \subset \mathcal{L}(X).$ It is shown that such a solution family exists whenever the pair $(\alpha,\beta)$ belongs to a subset of the set $(1,2]\times(0,1]$ and B is the generator of a cosine family or a C0-semigroup in $X.$ In any case, it also depends on certain compatibility conditions on the real parameters $a,b,c,d$ that must be satisfied.

In this paper, we study the embedding problem of an operator into a strongly continous semigroup. We obtain characterizations for some classes of operators, namely composition operators and analytic Toeplitz operators on the Hardy space $H^2$. In particular, we focus on the isometric ones using the necessary and sufficient condition observed by T. Eisner.

We study the quasi-ergodicity of compact strong Feller semigroups $U_t$, $t> 0$, on $L^2(M,\mu )$; we assume that M is a locally compact Polish space equipped with a locally finite Borel measue $\mu $. The operators $U_t$ are ultracontractive and positivity preserving, but not necessarily self-adjoint or normal. We are mainly interested in those cases where the measure $\mu $ is infinite and the semigroup is not intrinsically ultracontractive. We relate quasi-ergodicity on $L^p(M,\mu )$ and uniqueness of the quasi-stationary measure with the finiteness of the heat content of the semigroup (for large values of t) and with the progressive uniform ground state domination property. The latter property is equivalent to a variant of quasi-ergodicity which progressively propagates in space as $t \uparrow \infty $; the propagation rate is determined by the decay of . We discuss several applications and illustrate our results with examples. This includes a complete description of quasi-ergodicity for a large class of semigroups corresponding to non-local Schrödinger operators with confining potentials.

We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ($P_4$): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$, where $A$, $B$, $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$, $G$ and $F$ are bounded linear operators from $L^p([-2\pi,\,0];X)$ (respectively $B_{p,q}^s([-2\pi,\,0];X)$) into $X$, $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(t+\cdot )$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$. We completely characterize the well-posedness of ($P_4$) in the above two function spaces by using known operator-valued Fourier multiplier theorems.

We establish new local and global estimates for evolutionary partial differential equations in classical Banach and quasi-Banach spaces that appear most frequently in the theory of partial differential equations. More specifically, we obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient evolutionary partial differential equations. The estimates are achieved by introducing the notions of Schrödinger and general oscillatory integral operators with inhomogeneous phase functions and prove sharp local and global regularity results for these in Besov–Lipschitz and Triebel–Lizorkin spaces.

One of the main challenges in molecular dynamics is overcoming the ‘timescale barrier’: in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behaviour on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory, as well as the algorithmic development, from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in molecular dynamics. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject.

We study the local and global existence and uniqueness of mild solution for a general class of abstract differential equations with state-dependent argument. In the last section, some examples on partial differential equations with state-dependent argument are presented.

Using probabilistic tools, we prove that any weak* continuous semigroup $(T_t)_{t \geqslant 0}$ of self-adjoint unital completely positive measurable Schur multipliers acting on the space $\mathrm {B}({\mathrm {L}}^2(X))$ of bounded operators on the Hilbert space ${\mathrm {L}}^2(X)$, where X is a suitable measure space, can be dilated by a weak* continuous group of Markov $*$-automorphisms on a bigger von Neumann algebra. We also construct a Markov dilation of these semigroups. Our results imply the boundedness of the McIntosh’s ${\mathrm {H}}^\infty $ functional calculus of the generators of these semigroups on the associated Schatten spaces and some interpolation results connected to ${\mathrm {BMO}}$-spaces. We also give an answer to a question of Steen, Todorov, and Turowska on completely positive continuous Schur multipliers.

This work studies the asymptotic behavior of a waves coupled system with a boundary dissipation of the fractional derivative type. We prove well-posedness and polynomial stability based on the semigroup approach, the energy method, and the result of stability.

Matryoshka dolls, the traditional Russian nesting figurines, are known worldwide for each doll’s encapsulation of a sequence of smaller dolls. In this paper, we exploit the structure of a new sequence of nested matrices we call matryoshkan matrices in order to compute the moments of the one-dimensional polynomial processes, a large class of Markov processes. We characterize the salient properties of matryoshkan matrices that allow us to compute these moments in closed form at a specific time without computing the entire path of the process. This simplifies the computation of the polynomial process moments significantly. Through our method, we derive explicit expressions for both transient and steady-state moments of this class of Markov processes. We demonstrate the applicability of this method through explicit examples such as shot noise processes, growth–collapse processes, ephemerally self-exciting processes, and affine stochastic differential equations from the finance literature. We also show that we can derive explicit expressions for the self-exciting Hawkes process, for which finding closed-form moment expressions has been an open problem since their introduction in 1971. In general, our techniques can be used for any Markov process for which the infinitesimal generator of an arbitrary polynomial is itself a polynomial of equal or lower order.

We define positively expansive semigroups of linear operators on Banach spaces. We characterize these semigroups in terms of the point spectrum of the infinitesimal generator. In particular, we prove that a positively expansive semigroup is neither uniformly bounded nor equicontinuous. We apply our results to the Lasota equation.

We examine a semigroup analogue of the Kumjian–Renault representation of C*-algebras with Cartan subalgebras on twisted groupoids. Specifically, we represent semigroups with distinguished normal subsemigroups as ‘slice-sections’ of groupoid bundles.

The goal of the paper is to obtain analogs of the sampling theorems and of the Riesz–Boas interpolation formulas which are relevant to the discrete Hilbert and Kak–Hilbert transforms in $l^{2}$ .

The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in $(0,\infty)$ . The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator. A Miyadera perturbation argument is used to prove that it is the generator of a positive, analytic semigroup on a weighted $L_1$ -space. Moreover, if the overall fragmentation rate does not vanish at infinity, then there is a unique stationary solution with given mass. Assuming further that the overall fragmentation rate diverges to infinity for large sizes implies the immediate compactness of the semigroup and that it eventually stabilizes at an exponential rate to a one-dimensional projection carrying the information of the mass of the initial value.

We prove the existence and asymptotic behaviour of the transition density for a large class of subordinators whose Laplace exponents satisfy lower scaling condition at infinity. Furthermore, we present lower and upper bounds for the density. Sharp estimates are provided if an additional upper scaling condition on the Laplace exponent is imposed. In particular, we cover the case when the (minus) second derivative of the Laplace exponent is a function regularly varying at infinity with regularity index bigger than $-2$ .