Rate of convergence for Wong-Zakai-type approximations of It\^o stochastic differential equations

We consider a class of stochastic differential equations driven by a one dimensional Brownian motion and we investigate the rate of convergence for Wong-Zakai-type approximated solutions. We first consider the Stratonovich case, obtained through the point-wise multiplication between the diffusion coefficient and a smoothed version of the noise; then, we consider It\^o equations where the diffusion coefficient is Wick-multiplied by the regularized noise. We discover that in both cases the speed of convergence to the exact solution coincides with the speed of convergence of the smoothed noise towards the original Brownian motion. We also prove, in analogy with a well known property for exact solutions, that the solutions of approximated It\^o equations solve approximated Stratonovich equations with a certain correction term in the drift.Comment: To appear on Journal of Theoretical Probabilit

Complex dynamics and multistability in nonlinear resonant nanosystems beyond the duffing critical amplitude

International audienceAnalytical multi-physics models which include main sources of nonlinearities for nanoresonators electrostatically actuated are developed in order to assess complex dynamics in nanosystems beyond the Duffing critical amplitude. In particular, multistability is investigated for doubly clamped beams and cantilevers. The bifurcation topology of a particular multistable behavior (up to five amplitudes for a given frequency) is parametrically identified and experimentally validated

On stochastic differential equations driven by the renormalized square of the Gaussian white noise

We investigate the properties of the Wick square of Gaussian white noises through a new method to perform non linear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove on Ito-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, Lipschitz continuity and linear growth condition, and produces existence and uniqueness results in the space where the noise lives. The linear case is studied in details and positivity of the solution is proved.Comment: 23 page

On Permutation Binomials over Finite Fields

Let $\mathbb{F}_{q}$ be the finite field of characteristic $p$ containing $q = p^{r}$ elements and $f(x)=ax^{n} + x^{m}$ a binomial with coefficients in this field. If some conditions on the gcd of $n-m$ an $q-1$ are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if $f(x) = ax^{n} + x^{m}$ permutes $\mathbb{F}_{p}$, where $n>m>0$ and $a \in {\mathbb{F}_{p}}^{*}$, then $p -1 \leq (d -1)d$, where $d = {{gcd}}(n-m,p-1)$, and that this bound of $p$ in term of $d$ only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of $\mathbb{F}_{q}$ from a permutation binomial over $\mathbb{F}_{q}$