Abstract. We investigate the asymptotic properties as t → ∞ of the following differential equation in the Hilbert space H:where the map a : R + → R + is nonincreasing and the potential G : H → R is of class C 1 . If the coefficient a(t) is constant and positive, we recover the so-called "Heavy Ball with Friction" system. On the other hand, when a(t) = 1/(t + 1) we obtain the trajectories associated to some averaged gradient system. Our analysis is mainly based on the existence of some suitable energy function. When the function G is convex, the condition ∞ 0 a(t) dt = ∞ guarantees that the energy function converges toward its minimum. The more stringent condition ∞ 0 e − t 0 a(s) ds dt < ∞ is necessary to obtain the convergence of the trajectories of (S) toward some minimum point of G. In the one-dimensional setting, a precise description of the convergence of solutions is given for a general nonconvex function G. We show that in this case the set of initial conditions for which solutions converge to a local minimum is open and dense.

This paper deals with a natural stochastic optimization procedure derived from the so-called Heavy-ball method differential equation, which was introduced by Polyak in the 1960s with his seminal contribution [Pol64]. The Heavy-ball method is a second-order dynamics that was investigated to minimize convex functions f . The family of second-order methods recently received a large amount of attention, until the famous contribution of Nesterov [Nes83], leading to the explosion of large-scale optimization problems. This work provides an in-depth description of the stochastic heavy-ball method, which is an adaptation of the deterministic one when only unbiased evalutions of the gradient are available and used throughout the iterations of the algorithm. We first describe some almost sure convergence results in the case of general non-convex coercive functions f . We then examine the situation of convex and strongly convex potentials and derive some non-asymptotic results about the stochastic heavy-ball method. We end our study with limit theorems on several rescaled algorithms.

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The killing of antigen-bearing cells by clonal populations of cytotoxic T lymphocytes (CTLs) is thought to be a rapid phenomenon executed uniformly by individual CTLs. We combined bulk and single-CTL killing assays over a prolonged time period to provide the killing statistics of clonal human CTLs against an excess of target cells. Our data reveal efficiency in sustained killing at the population level, which relied on a highly heterogeneous multiple killing performance at the individual level. Although intraclonal functional heterogeneity was a stable trait in clonal populations, it was reset in the progeny of individual CTLs. In-depth mathematical analysis of individual CTL killing data revealed a substantial proportion of high-rate killer CTLs with burst killing activity. Importantly, such activity was delayed and required activation with strong antigenic stimulation. Our study implies that functional heterogeneity allows CTL populations to calibrate prolonged cytotoxic activity to the size of target cell populations.

For any a ą 0, consider the hypocoercive generators yB x `aB 2 y ´yB y and yB x ´axB y `B2 y ´yB y , respectively for px, yq P R{p2πZq ˆR and px, yq P R ˆR. The goal of the paper is to obtain exactly the L 2 pµ a q-operator norms of the corresponding Markov semi-group at any time, where µ a is the associated invariant measure. The computations are based on the spectral decomposition of the generator and especially on the scalar products of the eigenvectors. The motivation comes from an attempt to find an alternative approach to classical ones developed to obtain hypocoercive bounds for kinetic models.