Stochastic Differential Systems with Memory: Theory, Examples and Applications

The purpose of this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic time-lag model with Gaussian noise , and the classical ``heat-bath model of R. Kubo , modeling the motion of a ``large molecule in a viscous fluid. These examples are embedded in a general class of stochastic functional differential equations (sfde\u27s). We then establish pathwise existence and uniqueness of solutions to these classes of sfde\u27s under local Lipschitz and linear growth hypotheses on the coefficients. It is interesting to note that the above class of sfde\u27s is not covered by classical results of Protter, Metivier and Pellaumail and Doleans-Dade. In Chapter II, we prove that the Markov (Feller) property holds for the trajectory random field of a sfde. The trajectory Markov semigroup is not strongly continuous for positive delays, and its domain of strong continuity does not contain tame (or cylinder) functions with evaluations away from zero. To overcome this difficulty, we introduce a class of quasitame functions. These belong to the domain of the weak infinitesimal generator, are weakly dense in the underlying space of continuous functions and generate the Borel $\sigma$-algebra of the state space. This chapter also contains a derivation of a formula for the weak infinitesimal generator of the semigroup for sufficiently regular functions, and for a large class of quasitame functions. In Chapter III, we study pathwise regularity of the trajectory random field in the time variable and in the initial path. Of note here is the non-existence of the stochastic flow for the singular sdde $dx(t)= x(t-r) dW(t)$ and a breakdown of linearity and local boundedness. This phenomenon is peculiar to stochastic delay equations. It leads naturally to a classification of sfde\u27s into regular and singular types. Necessary and sufficient conditions for regularity are not known. The rest of Chapter III is devoted to results on sufficient conditions for regularity of linear systems driven by white noise or semimartingales, and Sussman-Doss type nonlinear sfde\u27s. Building on the existence of a compacting stochastic flow, we develop a multiplicative ergodic theory for regular linear sfde\u27s driven by white noise, or general helix semimartingales (Chapter IV). In particular, we prove a Stable Manifold Theorem for such systems. In Chapter V, we seek asymptotic stability for various examples of one-dimensional linear sfde\u27s. Our approach is to obtain upper and lower estimates for the top Lyapunov exponent. Several topics are discussed in Chapter VI. These include the existence of smooth densities for solutions of sfde\u27s using the Malliavin calculus, an approximation technique for multidimensional diffusions using sdde\u27s with small delays, and affine sfde\u27s

Molnupiravir plus usual care versus usual care alone as early treatment for adults with COVID-19 at increased risk of adverse outcomes (PANORAMIC): an open-label, platform-adaptive randomised controlled trial

Background: The safety, effectiveness, and cost-effectiveness of molnupiravir, an oral antiviral medication for SARS-CoV-2, has not been established in vaccinated patients in the community at increased risk of morbidity and mortality from COVID-19. We aimed to establish whether the addition of molnupiravir to usual care reduced hospital admissions and deaths associated with COVID-19 in this population. Methods: PANORAMIC was a UK-based, national, multicentre, open-label, multigroup, prospective, platform adaptive randomised controlled trial. Eligible participants were aged 50 years or older—or aged 18 years or older with relevant comorbidities—and had been unwell with confirmed COVID-19 for 5 days or fewer in the community. Participants were randomly assigned (1:1) to receive 800 mg molnupiravir twice daily for 5 days plus usual care or usual care only. A secure, web-based system (Spinnaker) was used for randomisation, which was stratified by age (<50 years vs ≥50 years) and vaccination status (yes vs no). COVID-19 outcomes were tracked via a self-completed online daily diary for 28 days after randomisation. The primary outcome was all-cause hospitalisation or death within 28 days of randomisation, which was analysed using Bayesian models in all eligible participants who were randomly assigned. This trial is registered with ISRCTN, number 30448031. Findings: Between Dec 8, 2021, and April 27, 2022, 26 411 participants were randomly assigned, 12 821 to molnupiravir plus usual care, 12 962 to usual care alone, and 628 to other treatment groups (which will be reported separately). 12 529 participants from the molnupiravir plus usual care group, and 12 525 from the usual care group were included in the primary analysis population. The mean age of the population was 56·6 years (SD 12·6), and 24 290 (94%) of 25 708 participants had had at least three doses of a SARS-CoV-2 vaccine. Hospitalisations or deaths were recorded in 105 (1%) of 12 529 participants in the molnupiravir plus usual care group versus 98 (1%) of 12 525 in the usual care group (adjusted odds ratio 1·06 [95% Bayesian credible interval 0·81–1·41]; probability of superiority 0·33). There was no evidence of treatment interaction between subgroups. Serious adverse events were recorded for 50 (0·4%) of 12 774 participants in the molnupiravir plus usual care group and for 45 (0·3%) of 12 934 in the usual care group. None of these events were judged to be related to molnupiravir. Interpretation: Molnupiravir did not reduce the frequency of COVID-19-associated hospitalisations or death among high-risk vaccinated adults in the community

Molnupiravir plus usual care versus usual care alone as early treatment for adults with COVID-19 at increased risk of adverse outcomes (PANORAMIC): an open-label, platform-adaptive randomised controlled trial

BackgroundThe safety, effectiveness, and cost-effectiveness of molnupiravir, an oral antiviral medication for SARS-CoV-2, has not been established in vaccinated patients in the community at increased risk of morbidity and mortality from COVID-19. We aimed to establish whether the addition of molnupiravir to usual care reduced hospital admissions and deaths associated with COVID-19 in this population.MethodsPANORAMIC was a UK-based, national, multicentre, open-label, multigroup, prospective, platform adaptive randomised controlled trial. Eligible participants were aged 50 years or older—or aged 18 years or older with relevant comorbidities—and had been unwell with confirmed COVID-19 for 5 days or fewer in the community. Participants were randomly assigned (1:1) to receive 800 mg molnupiravir twice daily for 5 days plus usual care or usual care only. A secure, web-based system (Spinnaker) was used for randomisation, which was stratified by age (<50 years vs ≥50 years) and vaccination status (yes vs no). COVID-19 outcomes were tracked via a self-completed online daily diary for 28 days after randomisation. The primary outcome was all-cause hospitalisation or death within 28 days of randomisation, which was analysed using Bayesian models in all eligible participants who were randomly assigned. This trial is registered with ISRCTN, number 30448031.FindingsBetween Dec 8, 2021, and April 27, 2022, 26 411 participants were randomly assigned, 12 821 to molnupiravir plus usual care, 12 962 to usual care alone, and 628 to other treatment groups (which will be reported separately). 12 529 participants from the molnupiravir plus usual care group, and 12 525 from the usual care group were included in the primary analysis population. The mean age of the population was 56·6 years (SD 12·6), and 24 290 (94%) of 25 708 participants had had at least three doses of a SARS-CoV-2 vaccine. Hospitalisations or deaths were recorded in 105 (1%) of 12 529 participants in the molnupiravir plus usual care group versus 98 (1%) of 12 525 in the usual care group (adjusted odds ratio 1·06 [95% Bayesian credible interval 0·81–1·41]; probability of superiority 0·33). There was no evidence of treatment interaction between subgroups. Serious adverse events were recorded for 50 (0·4%) of 12 774 participants in the molnupiravir plus usual care group and for 45 (0·3%) of 12 934 in the usual care group. None of these events were judged to be related to molnupiravir.InterpretationMolnupiravir did not reduce the frequency of COVID-19-associated hospitalisations or death among high-risk vaccinated adults in the community

Quelques remarques sur les facteurs des systèmes dynamiques gaussiens

We study the factors of Gaussian dynamical systems which are generated by functions depending only on a finite number of coordinates. As an application, we show that for Gaussian automorphisms with simple spectrum, the partition ${(X_0 ≤ 0), (X_0 > 0)}$ is generating