37 research outputs found
Stochastic embedding of dynamical systems
Most physical systems are modelled by an ordinary or a partial differential
equation, like the n-body problem in celestial mechanics. In some cases, for
example when studying the long term behaviour of the solar system or for
complex systems, there exist elements which can influence the dynamics of the
system which are not well modelled or even known. One way to take these
problems into account consists of looking at the dynamics of the system on a
larger class of objects, that are eventually stochastic. In this paper, we
develop a theory for the stochastic embedding of ordinary differential
equations. We apply this method to Lagrangian systems. In this particular case,
we extend many results of classical mechanics namely, the least action
principle, the Euler-Lagrange equations, and Noether's theorem. We also obtain
a Hamiltonian formulation for our stochastic Lagrangian systems. Many
applications are discussed at the end of the paper.Comment: 112 page
Stochastic derivatives for fractional diffusions
In this paper, we introduce some fundamental notions related to the so-called
stochastic derivatives with respect to a given -field . In
our framework, we recall well-known results about Markov--Wiener diffusions. We
then focus mainly on the case where is a fractional diffusion and where
is the past, the future or the present of . We treat some
crucial examples and our main result is the existence of stochastic derivatives
with respect to the present of when solves a stochastic differential
equation driven by a fractional Brownian motion with Hurst index . We
give explicit formulas.Comment: Published in at http://dx.doi.org/10.1214/009117906000001169 the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Plongement stochastique des systèmes lagrangiens
4 pagesWe define an operator which extends classical differentiation from smooth deterministic functions to certain stochastic processes. Based on this operator, we define a procedure which associates a stochastic analog to standard differential operators and ordinary differential equations. We call this procedure stochastic embedding. By embedding lagrangian systems, we obtain a stochastic Euler-Lagrange equation which, in the case of natural lagrangian systems, is called the embedded Newton equation. This equation contains the stochastic Newton equation introduced by Nelson in his dynamical theory of brownian diffusions. Finally, we consider a diffusion with a gradient drift, a constant diffusion coefficient and having a probability density function. We prove that a necessary condition for this diffusion to solve the embedded Newton equation is that its density be the square of the modulus of a wave function solution of a linear Schrödinger equation
An exponentially averaged Vasyunin formula
We prove a Vasyunin-type formula for an autocorrelation function arising from
a Nyman-Beurling criterion generalized to a probabilistic framework. This
formula can also be seen as a reciprocity formula for cotangent sums, related
to the ones proven in [BC13], [ABB17].Comment: This paper has been written from results already stated in a previous
version of another paper in 2018, but has been now submitted separately.
arXiv admin note: text overlap with arXiv:1805.0673
Théorème de Noether stochastique
4 pagesThe stochastic embedding procedure associates a stochastic Euler-Lagrange equation (SEL) to the standard Euler-Lagrange equation (EL). Can we derive (SEL) from a generalized least action principle? To address this question, we develop a stochastic calculus of variation initiated by Yasue. We give a stochastic analog F of the lagrangian action functional. We introduce a notion of stationarity according to which the solutions of (SEL) are the stationary points of F. This notion of stationarity brings coherence to stochastic calculus of variation with respect to stochastic embedding. Finally, we prove a stochastic Noether theorem which introduces an original notion of stochastic first integral
Lemme de coherence et théorème de Noether stochastique
4 pagesThe stochastic embedding procedure associates a stochastic Euler-Lagrange equation (SEL) to the standard Euler-Lagrange equation (EL). Can we derive (SEL) from a generalized least action principle? To address this question, we develop a stochastic calculus of variation initiated by Yasue. We give a stochastic analog F of the lagrangian action functional. We introduce a notion of stationarity according to which the solutions of (SEL) are the stationary points of F. This notion of stationarity brings coherence to stochastic calculus of variation with respect to stochastic embedding. Finally, we prove a stochastic Noether theorem which introduces an original notion of stochastic first integral
Dynamical properties and characterization of gradient drift diffusions
We study the dynamical properties of the Brownian diffusions having as diffusion coefficient matrix and as drift vector. We
characterize this class through the equality , where
(resp. ) denotes the forward (resp. backward) stochastic derivative of
Nelson's type. Our proof is based on a remarkable identity for
and on the use of the martingale problem. We also give a new formulation of a
famous theorem of Kolmogorov concerning reversible diffusions. We finally
relate our characterization to some questions about the complex stochastic
embedding of the Newton equation which initially motivated of this work.Comment: 16 page