Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class. The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach...

We study ergodic properties of stochastic dissipative systems with additive noise. We show that the system is uniformly exponentially ergodic provided the growth of nonlinearity at infinity is faster than linear. The abstract result is applied to the stochastic reaction diffusion equation in ${ℝ}^d$ with $d\le 3$.

We review some recent results in quantitative stochastic homogenization for divergence-form, quasilinear elliptic equations. In particular, we are interested in obtaining $L^{\infty }$-type bounds on the gradient of solutions and thus giving a demonstration of the principle that solutions of equations with random coefficients have much better regularity (with overwhelming probability) than a general equation with non-constant coefficients.

We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.

We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.

Consider an infinite dimensional diffusion process process on $T^{{𝐙}^d}$, where $T$ is the circle, defined by the action of its generator $L$ on $C^2\left(T^{{𝐙}^d}\right)$ local functions as $Lf\left(\eta \right)={\sum }_{i\in {𝐙}^d}\left(\frac{1}{2}{a}_i\frac{{\partial }^{2}f}{\partial {\eta }_i^2}+b_i\frac{\partial f}{\partial {\eta }_i}\right)$. Assume that the coefficients, $a_i$ and $b_i$ are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that $a_i$ is only a function of ${\eta }_i$ and that ${inf}_{i,\eta }{a}_i\left(\eta \right)\>0$. Suppose $\nu$ is an invariant product measure. Then, if $\nu$ is the Lebesgue measure or if $d=1,2$, it is the unique invariant measure. Furthermore, if $\nu$ is translation invariant, then...

Consider an infinite dimensional diffusion process process on TZd, where T is the circle, defined by the action of its generator L on C2(TZd) local functions as $Lf\left(\eta \right)={\sum }_{i\in {𝐙}^d}\left(\frac{1}{2}{a}_i\frac{{\partial }^{2}f}{\partial {\eta }_i^2}+b_i\frac{\partial f}{\partial {\eta }_i}\right)$. Assume that the coefficients, ai and bi are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that ai is only a function of ${\eta }_i$ and that ${inf}_{i,\eta }{a}_i\left(\eta \right)>0$. Suppose ν is an invariant product measure. Then, if ν is the Lebesgue measure or if d=1,2, it is the unique invariant measure. Furthermore, if ν is translation...

In this paper we study upper bounds for the density of solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $Hgt;1/3$. We show that under some geometric conditions, in the regular case $Hgt;1/2$, the density of the solution satisfies the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case $Hgt;1/3$ and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper...