Lower bounds for the density of locally elliptic It\^{o} processes

We give lower bounds for the density $p_T(x,y)$ of the law of $X_t$, the solution of $dX_t=\sigma (X_t) dB_t+b(X_t) dt,X_0=x,$ under the following local ellipticity hypothesis: there exists a deterministic differentiable curve $x_t, 0\leq t\leq T$, such that $x_0=x, x_T=y$ and $\sigma \sigma ^*(x_t)>0,$ for all $t\in \lbrack 0,T].$ The lower bound is expressed in terms of a distance related to the skeleton of the diffusion process. This distance appears when we optimize over all the curves which verify the above ellipticity assumption. The arguments which lead to the above result work in a general context which includes a large class of Wiener functionals, for example, It\^{o} processes. Our starting point is work of Kohatsu-Higa which presents a general framework including stochastic PDE's.Comment: Published at http://dx.doi.org/10.1214/009117906000000458 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Filamentary structure in the Orion molecular cloud

A large scale 13CO map (containing 33,000 spectra) of the giant molecular cloud located in the southern part of Orion is presented which contains the Orion Nebula, NGC1977, and the LI641 dark cloud complex. The overall structure of the cloud is filamentary, with individual features having a length up to 40 times their width. This morphology may result from the effects of star formation in the region or embedded magnetic fields in the cloud. We suggest a simple picture for the evolution of the Orion-A cloud and the formation of the major filament. A rotating proto-cloud (counter rotating with respect to the galaxy) contians a b-field aligned with the galaxtic plane. The northern protion of this cloud collapsed first, perhaps triggered by the pressure of the Ori I OB association. The magnetic field combined with the anisotropic pressure produced by the OB-association breaks the symmetry of the pancake instability, a filament rather than a disc is produced. The growth of instabilities in the filament formed sub-condensations which are recent sites of star formation

A generic construction for high order approximation schemes of semigroups using random grids

Our aim is to construct high order approximation schemes for general semigroups of linear operators $P_{t},t\geq 0$. In order to do it, we fix a time horizon $T$ and the discretization steps $h_{l}=\frac{T}{n^{l}},l\in \mathbb{N}$ and we suppose that we have at hand some short time approximation operators $Q_{l}$ such that $P_{h_{l}}=Q_{l}+O(h_{l}^{1+\alpha })$ for some $\alpha >0$. Then, we consider random time grids $\Pi (\omega )=\{t_0(\omega )=0<t_{1}(\omega )<...<t_{m}(\omega )=T\}$ such that for all $1\le k\le m$, $t_{k}(\omega )-t_{k-1}(\omega )=h_{l_{k}}$ for some $l_{k}\in \mathbb{N}$, and we associate the approximation discrete semigroup $P_{T}^{\Pi (\omega )}=Q_{l_{n}}...Q_{l_{1}}.$ Our main result is the following: for any approximation order $\nu$, we can construct random grids $\Pi_{i}(\omega )$ and coefficients $c_{i}$, with $i=1,...,r$ such that $$P_{t}f=\sum_{i=1}^{r}c_{i}\mathbb{E}(P_{t}^{\Pi _{i}(\omega )}f(x))+O(n^{-\nu})$$% with the expectation concerning the random grids $\Pi _{i}(\omega ).$ Besides, $\text{Card}(\Pi _{i}(\omega ))=O(n)$ and the complexity of the algorithm is of order $n$, for any order of approximation $\nu$. The standard example concerns diffusion processes, using the Euler approximation for~$Q_l$. In this particular case and under suitable conditions, we are able to gather the terms in order to produce an estimator of $P_tf$ with finite variance. However, an important feature of our approach is its universality in the sense that it works for every general semigroup $P_{t}$ and approximations. Besides, approximation schemes sharing the same $\alpha$ lead to the same random grids $\Pi_{i}$ and coefficients $c_{i}$. Numerical illustrations are given for ordinary differential equations, piecewise deterministic Markov processes and diffusions

Riesz transform and integration by parts formulas for random variables

We use integration by parts formulas to give estimates for the $L^p$ norm of the Riesz transform. This is motivated by the representation formula for conditional expectations of functionals on the Wiener space already given in Malliavin and Thalmaier. As a consequence, we obtain regularity and estimates for the density of non degenerated functionals on the Wiener space. We also give a semi-distance which characterizes the convergence to the boundary of the set of the strict positivity points for the density

Regularity of probability laws by using an interpolation method

We study the problem of the existence and regularity of a probability density in an abstract framework based on a "balancing" with approximating absolutely continuous laws. Typically, the absolutely continuous property for the approximating laws can be proved by standard techniques from Malliavin calculus whereas for the law of interest no Malliavin integration by parts formulas are available. Our results are strongly based on the use of suitable Hermite polynomial series expansions and can be merged into the theory of interpolation spaces. We then apply the results to the solution to a stochastic differential equation with a local H\"ormander condition or to the solution to the stochastic heat equation, in both cases under weak conditions on the coefficients relaxing the standard Lipschitz or H\"older continuity requests