We study evolutions of measures under flows of solutions to stochastic differential equations with interaction. Under suitable regularity assumptions on the coefficients, we prove moment bounds for the compact-set Jacobian norm. These bounds yield pathwise transfer of logarithmic Sobolev, concentration, and Talagrand transport inequalities from the initial measure to the random pushforward measure at time t. Finally, for a kernel model satisfying a pairwise dissipativity condition, we prove exponential decay of the expected dispersion around the random center of mass.
In this article we consider a system of n interacting points in R 2 with dynamics defined by a linear term and bounded C 1 perturbations with bounded Jacobian matrix. We study the asymptotic behavior of the configuration after recentering and rescaling by the linear flow. Next, we present the results about the convergence of the rescaled system and discuss the long-term behavior of some geometric properties, e.g. the convexity of the polygon formed by the interacting points. We prove that for almost all initial conditions the time average of the convexity indicator of the evolving polygon equals the convexity indicator of the limiting configuration of the rescaled system.
In this paper we present the conditions for the existence of intersections for functions of several independent Brownian motions on Carnot groups. These conditions are geometrical and local in nature, in the sense that they give a simple relation between the local geometry of a manifold, related to intersections, at a given point, with the structure of Lie algebra of left-invariant vector fields in a neighbourhood of the same point. We prove that these conditions are necessary and sufficient for the existence of intersections, if we exlcude from the consideration the points of a nowhere dense set on the manifold, related to intersections.
We study a Monte Carlo approximation scheme for stochastic differential equations with interaction. The initial measure is replaced by its empirical approximation constructed from independent samples, and the corresponding equation is discretized in time by the Euler-Maruyama scheme. For initial measures with finite moments of order q>2, we derive a quantitative estimate for the empirical approximation in the Wasserstein distance of order two. Combining this estimate with stability estimates for stochastic flows with interaction, we obtain a convergence bound for the total Euler-Maruyama-Monte Carlo approximation. We also compare the particle-based approximation with the deterministic tensor-grid construction used in earlier work.
In this work, we investigate a stochastic HLIV model incorporating viral dynamics, in which the mean-reverting inhomogeneous geometric Brownian motion process (IGBM for short) perturb the conversion rate from eclipse phase cells to infected cells. We first establish the existence and uniqueness of a positive solution and show that the model admits a unique global solution for any given initial condition. We then employ stochastic Lyapunov functions together with Fatou's lemma to derive sufficient conditions for the existence of a stationary distribution of positive solution, thereby indicating strong persistence of the disease. Finally, we conclude by summarizing the main results and implications of our study.
