arXiv subjects

Dario Trevisan

Publications and source records attributed to Dario Trevisan.

63 records · Page 4Linked to original sources

arXiv2015

Functional Cramer-Rao bounds and Stein estimators in Sobolev spaces, for Brownian motion and Cox processes

We investigate the problems of drift estimation for a shifted Brownian motion and intensity estimation for a Cox process on a finite interval $[0,T]$, when the risk is given by the energy functional associated to some fractional Sobolev space $H^1_0\subset W^{α,2}\subset L^2$. In both situations, Cramer-Rao lower bounds are obtained, entailing in particular that no unbiased estimators with finite risk in $H^1_0$ exist. By Malliavin calculus techniques, we also study super-efficient Stein type estimators (in the Gaussian case).

math.ST↗

arXiv2014

Well posedness of Lagrangian flows and continuity equations in metric measure spaces

We establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE's associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into $\mathbb{R}^\infty$. When specialized to the setting of Euclidean or infinite dimensional (e.g. Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of ${\sf RCD}(K,\infty)$ metric measure spaces object of extensive recent research fits into our framework. Therefore we provide, for the first time, well-posedness results for ODE's under low regularity assumptions on the velocity and in a nonsmooth context.

math.FA↗

arXiv2014

A short proof of Stein's universal multiplier theorem

We give a short proof of Stein's universal multiplier theorem, purely by probabilistic methods, thus avoiding any use of harmonic analysis techniques (complex interpolation or transference methods).

math.FA↗

arXiv2014

Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities

We prove a family of $L^p$ uncertainty inequalities on fairly general groups and homogeneous spaces, both in the smooth and in the discrete setting. The crucial point is the proof of the $L^1$ endpoint, which is derived from a general weak isoperimetric inequality.

math.CA↗

arXiv2013

Lagrangian flows driven by $BV$ fields in Wiener spaces

We establish the renormalization property for essentially bounded solutions of the continuity equation associated to $BV$ fields in Wiener spaces, with values in the associated Cameron-Martin space; thus obtaining, by standard arguments, new uniqueness and stability results for correspondent Lagrangian $L^\infty$-flows. An example related to Neumann elliptic problems is also discussed.

math.AP↗

arXiv2013

Zero noise limits using local times

We consider a well-known family of SDEs with irregular drifts and the correspondent zero noise limits. Using (mollified) local times, we show which trajectories are selected. The approach is completely probabilistic and relies on elementary stochastic calculus only.

math.PR↗

arXiv2013

BV-capacities on Wiener Spaces and Regularity of the Maximum of the Wiener Process

We define a capacity C on abstract Wiener spaces and prove that, for any u with bounded variation, the total variation measure |Du| is absolutely continuous with respect to C: this enables us to extend the usual rules of calculus in many cases dealing with BV functions. As an application, we show that, on the classical Wiener space, the random variable sup_{0\leqt\leqT} W_t admits a measure as second derivative, whose total variation measure is singular w.r.t. the Wiener measure.

math.PR↗

arXiv2013

BV-regularity for the Malliavin Derivative of the Maximum of the Wiener Process

We prove that, on the classical Wiener space, the random variable $\sup_{0\le t \le T} W_t$ admits a measure as second Malliavin derivative, whose total variation measure is finite and singular w.r.t.\ the Wiener measure.

math.PR↗

arXiv2011

Functions of Bounded Variation on the Classical Wiener Space and an Extended Ocone-Karatzas Formula

We prove an extension of the Ocone-Karatzas integral representation, valid for all $BV$ functions on the classical Wiener space. We establish also an elementary chain rule formula and combine the two results to compute explicit integral representations for some classes of $BV$ composite random variables.

math.PR↗