On the joint residence time of N independent two-dimensional Brownian motions

We study the behavior of several joint residence times of N independent Brownian particles in a disc of radius $R$ in two dimensions. We consider: (i) the time T_N(t) spent by all N particles simultaneously in the disc within the time interval [0,t]; (ii) the time T_N^{(m)}(t) which at least m out of N particles spend together in the disc within the time interval [0,t]; and (iii) the time {\tilde T}_N^{(m)}(t) which exactly m out of N particles spend together in the disc within the time interval [0,t]. We obtain very simple exact expressions for the expectations of these three residence times in the limit t\to\infty.Comment: 8 page

A Delayed Black and Scholes Formula I

In this article we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic differential delay equation (sdde). We believe that the proposed model is sufficiently flexible to fit real market data, and is yet simple enough to allow for a closed-form representation of the option price. Furthermore, the model maintains the no-arbitrage property and the completeness of the market. The derivation of the option-pricing formula is based on an equivalent martingale measure

Windings of the 2D free Rouse chain

We study long time dynamical properties of a chain of harmonically bound Brownian particles. This chain is allowed to wander everywhere in the plane. We show that the scaling variables for the occupation times T_j, areas A_j and winding angles \theta_j (j=1,...,n labels the particles) take the same general form as in the usual Brownian motion. We also compute the asymptotic joint laws P({T_j}), P({A_j}), P({\theta_j}) and discuss the correlations occuring in those distributions.Comment: Latex, 17 pages, submitted to J. Phys.

The Enskog Process

The existence of a weak solution to a McKean-Vlasov type stochastic differential system corresponding to the Enskog equation of the kinetic theory of gases is established under natural conditions. The distribution of any solution to the system at each fixed time is shown to be unique. The existence of a probability density for the time-marginals of the velocity is verified in the case where the initial condition is Gaussian, and is shown to be the density of an invariant measure.Comment: 38 page

Explicit formulae in probability and in statistical physics

We consider two aspects of Marc Yor's work that have had an impact in statistical physics: firstly, his results on the windings of planar Brownian motion and their implications for the study of polymers; secondly, his theory of exponential functionals of Levy processes and its connections with disordered systems. Particular emphasis is placed on techniques leading to explicit calculations.Comment: 14 pages, 2 figures. To appear in Seminaire de Probabilites, Special Issue Marc Yo

Neutral Higgs-Boson Pair Production at Hadron Colliders: QCD Corrections

Neutral Higgs-boson pair production provides the possibility of studying the trilinear Higgs couplings at future high-energy colliders. We present the QCD corrections to the gluon-initiated processes in the limit of a heavy top quark in the loops and the Drell-Yan-like pair production of scalar and pseudoscalar Higgs particles. The pp cross sections are discussed for LHC energies within the Standard Model and its minimal supersymmetric extension. The QCD corrections are large, enhancing the total cross sections significantly.Comment: 26 pages, latex, 9 ps figure

Network medicine links SARS-CoV-2/COVID-19 infection to brain microvascular injury and neuroinflammation in dementia-like cognitive impairment

Background Dementia-like cognitive impairment is an increasingly reported complication of SARS-CoV-2 infection. However, the underlying mechanisms responsible for this complication remain unclear. A better understanding of causative processes by which COVID-19 may lead to cognitive impairment is essential for developing preventive and therapeutic interventions. Methods In this study, we conducted a network-based, multimodal omics comparison of COVID-19 and neurologic complications. We constructed the SARS-CoV-2 virus-host interactome from protein-protein interaction assay and CRISPR-Cas9-based genetic assay results and compared network-based relationships therein with those of known neurological manifestations using network proximity measures. We also investigated the transcriptomic profiles (including single-cell/nuclei RNA-sequencing) of Alzheimer’s disease (AD) marker genes from patients infected with COVID-19, as well as the prevalence of SARS-CoV-2 entry factors in the brains of AD patients not infected with SARS-CoV-2. Results We found significant network-based relationships between COVID-19 and neuroinflammation and brain microvascular injury pathways and processes which are implicated in AD. We also detected aberrant expression of AD biomarkers in the cerebrospinal fluid and blood of patients with COVID-19. While transcriptomic analyses showed relatively low expression of SARS-CoV-2 entry factors in human brain, neuroinflammatory changes were pronounced. In addition, single-nucleus transcriptomic analyses showed that expression of SARS-CoV-2 host factors (BSG and FURIN) and antiviral defense genes (LY6E, IFITM2, IFITM3, and IFNAR1) was elevated in brain endothelial cells of AD patients and healthy controls relative to neurons and other cell types, suggesting a possible role for brain microvascular injury in COVID-19-mediated cognitive impairment. Overall, individuals with the AD risk allele APOE E4/E4 displayed reduced expression of antiviral defense genes compared to APOE E3/E3 individuals. Conclusion Our results suggest significant mechanistic overlap between AD and COVID-19, centered on neuroinflammation and microvascular injury. These results help improve our understanding of COVID-19-associated neurological manifestations and provide guidance for future development of preventive or treatment interventions, although causal relationship and mechanistic pathways between COVID-19 and AD need future investigations

Semi-Groups of Isometries and the Representation and Multiplicity of Weakly Stationary Stochastic Processes

1 online resource (PDF, 27 pages