arXiv subjects

Franco Flandoli

Publications and source records attributed to Franco Flandoli.

112 records · Page 7Linked to original sources

arXiv2009

Well-posedness of the transport equation by stochastic perturbation

We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type.

math.AP↗

arXiv2007

On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model

We study the pathwise regularity of the map $$ ϕ\mapsto I(ϕ) = \int_0^T < ϕ(X_t), dX_t>$$ where $ϕ$ is a vector function on $\R^d$ belonging to some Banach space $V$, $X$ is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A \emph{stochastic current} is a continuous version of this map, seen as a random element of the topological dual of $V$. We give sufficient conditions for the current to live in some Sobolev space of distributions and we provide elements to conjecture that those are also necessary. Next we verify the sufficient conditions when the process $X$ is a $d$-dimensional fractional Brownian motion (fBm); we identify regularity in Sobolev spaces for fBm with Hurst index $H \in (1/4,1)$. Next we provide some results about general Sobolev regularity of Brownian currents. Finally we discuss applications to a model of random vortex filaments in turbulent fluids.

math.PR↗

arXiv2004

Statistics of a vortex filament model

A random field composed by Poisson distributed Brownian vortex filaments is constructed. The filament have a random thickness, length and intensity, governed by a measure $γ$. Under appropriate assumptions on $γ$ we compute the scaling law of the structure function and get the multifractal scaling as a particular case.

math.PR↗

arXiv2002

On a relation between stochastic integration and geometric measure theory

Two problems are addressed for the path of certain stochastic processes: a) do they define currents? b) are these currents of a classical type? A general answer to question a) is given for processes like semimartingales or with Lyons-Zheng structure. As to question b), it is shown that Hölder continuous paths with exponent $γ> 1/2$ define integral flat chains.

math.PR↗