### $(a,k)$-regularized $C$-resolvent families: regularity and local properties.

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We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can be used to compute normalized additive functionals “on-the-fly” as well as their limiting occupation measures with a given precision degree that does not depend on the final time horizon. We...

Let $\U0001d504$ be a ${C}^{*}$-algebra, $G$ a compact abelian group, $\tau $ an action of $G$ by $*$-automorphisms of $\U0001d504,{\U0001d504}^{\tau}$ the fixed point algebra of $\tau $ and ${\U0001d504}_{F}$ the dense sub-algebra of $G$-finite elements in $\U0001d504$. Further let $H$ be a linear operator from ${\U0001d504}_{F}$ into $\U0001d504$ which commutes with $\tau $ and vanishes on ${\U0001d504}^{\tau}$. We prove that $H$ is a complete dissipation if and only if $H$ is closable and its closure generates a ${C}_{0}$-semigroup of completely positive contractions. These complete dissipations are classified in terms of certain twisted negative definite...

A class of evolution operators is introduced according to the device of Kato. An evolution operator introduced here provides a classical solution of the linear equation u'(t) = A(t)u(t) for t ∈ [0,T], in a general Banach space. The paper presents a necessary and sufficient condition for the existence and uniqueness of such an evolution operator.

In this paper we characterize the class of polynomially Riesz strongly continuous semigroups on a Banach space $X$. Our main results assert, in particular, that the generators of such semigroups are either polynomially Riesz (then bounded) or there exist two closed infinite dimensional invariant subspaces ${X}_{0}$ and ${X}_{1}$ of $X$ with $X={X}_{0}\oplus {X}_{1}$ such that the part of the generator in ${X}_{0}$ is unbounded with resolvent of Riesz type while its part in ${X}_{1}$ is a polynomially Riesz operator.

We give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded ${H}^{\infty}$-calculus and is based on elementary analysis.