arXiv subjects

Sergio Polidoro

Publications and source records attributed to Sergio Polidoro.

21 records · Page 2Linked to original sources

arXiv2017

Harnack inequalities and Bounds for Densities of Stochastic Processes

We consider possibly degenerate parabolic operators in the form $$ \sum_{k=1}^{m}X_{k}^{2}+X_{0}-\partial_{t}, $$ that are naturally associated to a suitable family of stochastic differential equations, and satisfying the Hörmander condition. Note that, under this assumption, the operators in the form $Ł$ has a smooth fundamental solution that agrees with the density of the corresponding stochastic process. We describe a method based on Harnack inequalities and on the construction of Harnack chains to prove lower bounds for the fundamental solution. We also briefly discuss PDE and SDE methods to prove analogous upper bounds. We eventually give a list of meaningful examples of operators to which the method applies.

math.AP↗

arXiv2015

Harnack inequality for hypoelliptic second order partial differential operators

We consider nonnegative solutions $u:Ω\longrightarrow \mathbb{R}$ of second order hypoelliptic equations \begin{equation*} \mathscr{L} u(x) =\sum_{i,j=1}^n \partial_{x_i} \left(a_{ij}(x)\partial_{x_j} u(x) \right) + \sum_{i=1}^n b_i(x) \partial_{x_i} u(x) =0, \end{equation*} where $Ω$ is a bounded open subset of $\mathbb{R}^{n}$ and $x$ denotes the point of $Ω$. For any fixed $x_0 \in Ω$, we prove a Harnack inequality of this type $$\sup_K u \le C_K u(x_0)\qquad \forall \ u \ \mbox{ s.t. } \ \mathscr{L} u=0, u\geq 0,$$ where $K$ is any compact subset of the interior of the $\mathscr{L}$-propagation set of $x_0$ and the constant $C_K$ does not depend on $u$.

math.AP↗

arXiv2012

Two-sided bounds for degenerate processes with densities supported in subsets of R^N

We obtain two-sided bounds for the density of stochastic processes satisfying a weak Hörmander condition. In particular we consider the cases when the support of the density is not the whole space and when the density has various asymptotic regimes depending on the starting/final points considered (which are as well related to the number of brackets needed to span the space in Hörmander's theorem). The proofs of our lower bounds are based on Harnack inequalities for positive solutions of PDEs whereas the upper bounds derive from the probabilistic representation of the density given by the Malliavin calculus.

math.PR↗