We are interested in the law of the first passage time of an Ornstein–Uhlenbeck process to time-varying thresholds. We show that this problem is connected to the laws of the first passage time of the process to members of a two-parameter family of functional transformations of a time-varying boundary. For specific values of the parameters, these transformations appear in a realisation of a standard Ornstein–Uhlenbeck bridge. We provide three different proofs of this connection. The first is based on a similar result for Brownian motion, the second uses a generalisation of the so-called Gauss–Markov processes, and the third relies on the Lie group symmetry method. We investigate the properties of these transformations and study the algebraic and analytical properties of an involution operator which is used in constructing them. We also show that these transformations map the space of solutions of Sturm–Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we interpret our results through the method of images and give new examples of curves with explicit first passage time densities.

We consider a class of weakly interacting particle systems of mean-field type. The interactions between the particles are encoded in a graph sequence, i.e. two particles are interacting if and only if they are connected in the underlying graph. We establish a law of large numbers for the empirical measure of the system that holds whenever the graph sequence is convergent to a graphon. The limit is the solution of a non-linear Fokker–Planck equation weighted by the (possibly random) graphon limit. In contrast with the existing literature, our analysis focuses on both deterministic and random graphons: no regularity assumptions are made on the graph limit and we are able to include general graph sequences such as exchangeable random graphs. Finally, we identify the sequences of graphs, both random and deterministic, for which the associated empirical measure converges to the classical McKean–Vlasov mean-field limit.

This article deals with kinetic Fokker–Planck equations with essentially bounded coefficients. A weak Harnack inequality for nonnegative super-solutions is derived by considering their log-transform and adapting an argument due to S. N. Kružkov (1963). Such a result rests on a new weak Poincaré inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.

Delayed production can substantially alter the qualitative behaviour of feedback systems. Motivated by stochastic mechanisms in gene expression, we consider a protein molecule which is produced in randomly timed bursts, requires an exponentially distributed time to activate and then partakes in positive regulation of its burst frequency. Asymptotically analysing the underlying master equation in the large-delay regime, we provide tractable approximations to time-dependent probability distributions of molecular copy numbers. Importantly, the presented analysis demonstrates that positive feedback systems with large production delays can constitute a stable toggle switch even if they operate with low copy numbers of active molecules.

Let X be an Ornstein–Uhlenbeck process driven by a Brownian motion. We propose an expression for the joint density / distribution function of the process and its running supremum. This law is expressed as an expansion involving parabolic cylinder functions. Numerically, we obtain this law faster with our expression than with a Monte Carlo method. Numerical applications illustrate the interest of this result.

Motivated by the classical De Bruijn's identity for the additive Gaussian noise channel, in this paper we consider a generalized setting where the channel is modelled via stochastic differential equations driven by fractional Brownian motion with Hurst parameter H ∈ (0, 1). We derive generalized De Bruijn's identity for Shannon entropy and Kullback–Leibler divergence by means of Itô's formula, and present two applications. In the first application we demonstrate its equivalence with Stein's identity for Gaussian distributions, while in the second application, we show that for H ∈ (0, 1/2], the entropy power is concave in time while for H ∈ (1/2, 1) it is convex in time when the initial distribution is Gaussian. Compared with the classical case of H = 1/2, the time parameter plays an interesting and significant role in the analysis of these quantities.

This study investigates the phenomenon of targeted energy transfer (TET) from a linear oscillator to a nonlinear attachment behaving as a nonlinear energy sink for both transient and stochastic excitations. First, the dynamics of the underlying Hamiltonian system under deterministic transient loading is studied. Assuming that the transient dynamics can be partitioned into slow and fast components, the governing equations of motion corresponding to the slow flow dynamics are derived and the behaviour of the system is analysed. Subsequently, the effect of noise on the slow flow dynamics of the system is investigated. The Itô stochastic differential equations for the noisy system are derived and the corresponding Fokker–Planck equations are numerically solved to gain insights into the behaviour of the system on TET. The effects of the system parameters as well as noise intensity on the optimal regime of TET are studied. The analysis reveals that the interaction of nonlinearities and noise enhances the optimal TET regime as predicted in deterministic analysis.

In this work, depending on the relation between the Deborah, the Reynolds and the aspectratio numbers, we formally derived shallow-water type systems starting from a micro-macrodescription for non-Newtonian fluids in a thin domain governed by an elastic dumbbell typemodel with a slip boundary condition at the bottom. The result has been announced by theauthors in [G. Narbona-Reina, D. Bresch, Numer. Math. and Advanced Appl.Springer Verlag (2010)] and in the present paper, we provide a self-containeddescription, complete formal derivations and various numerical computations. Inparticular, we extend to FENE type systems the derivation of shallow-water models forNewtonian fluids that we can find for instance in [J.-F. Gerbeau, B. Perthame,Discrete Contin. Dyn. Syst. (2001)] which assume an appropriaterelation between the Reynolds number and the aspect ratio with slip boundary condition atthe bottom. Under a radial hypothesis at the leading order, for small Deborah number, wefind an interesting formulation where polymeric effect changes the drag term in the secondorder shallow-water formulation (obtained by J.-F. Gerbeau, B. Perthame). We also discussintermediate Deborah number with a fixed Reynolds number where a strong coupling is foundthrough a nonlinear time-dependent Fokker–Planck equation. This generalizes, at a formallevel, the derivation in [L. Chupin, Meth. Appl. Anal. (2009)] includingnon-linear effects (shallow-water framework).