Let $L = (L(t))_{t \geq 0}$ be a multivariate Lévy process with Lévy measure $\nu(dy) = \exp(-f(|y|))dy$ for a smoothly regularly varying function $f$ of index $\alpha > 1$. The process $L$ is renormalized as $X^\varepsilon(t) = \varepsilon L(\tau_\varepsilon t),\ t \in [0,T]$, for a scaling parameter $\tau_\varepsilon = o(\varepsilon^{-1})$, as $\varepsilon \to 0$. We study the behavior of the bridge $Y^{\varepsilon,x}$ of the renormalized process $X^\varepsilon$ conditioned on the event $X^\varepsilon(T) = x$ for a given end point $x \neq 0$ and end time $T > 0$ in the regime of small $\varepsilon$. Our main result is a sample path large deviations principle (LDP) for $Y^{\varepsilon,x}$ with a specific speed function $S(\varepsilon)$ and an entropy-type rate function $I_x$ on the Skorokhod space in the limit $\varepsilon \to 0+$. We show that the asymptotic energy minimizing path of $Y^{\varepsilon,x}$ is the linear parametrization of the straight line between 0 and $x$, while all paths leaving this set are exponentially negligible. We also infer a LDP for the asymptotic number of jumps and establish asymptotic normality of the jump increments of $Y^{\varepsilon,x}$. Since on these short time scales ($\tau_\varepsilon = o(\varepsilon^{-1})$) direct LDP methods cannot be adapted we use an alternative direct approach based on convolution density estimates of the marginals $X^\varepsilon(t),\ t \in [0,T]$, for which we solve a specific nonlinear functional equation.

The gene regulatory network models all the biochemical reactions between the different species (proteins, mRNA, etc.) present in a cell. A stochastic approach to this network, using jump processes, has been studied during the '70s (Kurtz), and it was established that in large populations, the rescaled process (thought as a concentrations process) converges towards the solution of an EDO, which is entirely deterministic. In order to obtain a random limit, Crudu, Debussche and Radulescu proposed a multiscale model. In this talk, I will propose a relevant stochastic representation of the multiscale model, for which it is possible to obtain the uniform convergence towards a PDMP and a CLT for the fluctuations around this limit.

We first introduce a brief review of the history of Brownian Motion up to the modern experiments where isolated Brownian particles are observed.Later, we introduce a one-space-dimensional wavefunction model of a heavy particle and a collection of light particles that might generate "Brownian-Motion-Like" trajectories as well as diffusive motion (displacement proportional to the square-root of time).This model satisfies two conditions that grant, for the temporal motion of the heavy particle:

1. An oscillating series with properties similar to those of the Ornstein-Uhlenbeck process;
2. A best quadratic fit with an "average" non-positive curvature in a proper time interval.

We note that Planck's constant and the molecular mass enter into the diffusion coefficient, while they also recently appeared in experimental estimates;to our knowledge, this is the first microscopic derivation in which they contribute directly to the diffusion coefficient.Finally, we discuss whether cat states are present in the thermodynamic ensembles.

(Joint for with W.D. Wick)

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