arXiv subjects

Jules Pitcho

Publications and source records attributed to Jules Pitcho.

6 recordsLinked to original sources

arXiv2026

On the existence of Markovian measures on continuous paths

Let $\mathbf{\eta}$ be a positive Radon measure on the space of continuous paths from R into a locally compact Polish space $Y$, and assume that $\mathbf{\eta}$ admits an invariant measure. We explicit conditions on $\mathbf{\eta}$ such that successive Markovianisation of $\mathbf{\eta}$ over any dense and countable set of times have all limit points (in the weak-star topology on Radon measures) satisfying the strong Markov property. We show that if Y is a locally compact Polish group, and $\mathbf{\eta}$ is left or right translation invariant, then $\mathbf{\eta}$ satisfies such conditions. Our proof uses the Zermalo-Fraenkel and Axiom of Dependant Choice axiomatisation of set theory, in which countable products of compacts are compact.

math.AP↗

arXiv2025

On the zero-noise limit for SDE's singular at the initial time

We investigate the zero-noise limit for SDE's driven by Brownian motion with a divergence-free drift singular at the initial time and prove that a unique probability measure concentrated on the integral curves of the drift is selected. More precisely, we prove uniqueness of the zero-noise limit for divergence-free drifts in $L^1_{loc}((0,T];BV(\mathbb{T}^d;\mathbb{R}^d))\cap L^q((0,T);L^p(\mathbb{T}^d;\mathbb{R}^d))$ where $p$ and $q$ satisfy a Prodi-Serrin condition. The vector field constructed by Depauw [C. R. Acad. Sci. Paris, 2003] lies in this class and we show that for almost every intial datum, the zero-noise limit selects a probability measure concentrated on several distinct integral curves of this vector field.

math.PR↗

arXiv2024

On vanishing diffusivity selection for the advection equation

We study the advection equation along vector fields singular at the initial time. More precisely, we prove that for divergence-free vector fields in $L^1_{loc}((0, T ]; BV (\mathbb{T}^d;\mathbb{R}^d))\cap L^2((0, T ) \times\mathbb{T}^d;\mathbb{R}^d)$, there exists a unique vanishing diffusivity solution. This class includes the vector field constructed by Depauw, for which there are infinitely many distinct bounded solutions to the advection equation.

math.AP↗

arXiv2024

A remark on selection of solutions for the transport equation

We prove that for bounded, divergence-free vector fields in $L^1_{loc}((0,+\infty);BV_{loc}(R^d;R^d))$, regularisation by convolution of the vector field selects a single solution of the transport equation for any integrable initial datum. We recall the vector field constructed by Depauw in [10], which lies in the above class of vector fields. We show that the transport equation along this vector field has at least two bounded weak solutions for any bounded initial datum.

math.AP↗

arXiv2024

On the stochastic selection of integral curves of a rough vector field

We prove that for bounded, divergence-free vector fields b in L^1_{loc}((0,1];BV(\T^d;\R^d)), there exists a unique incompressible measure on integral curves of b. We recall the vector field constructed by Depauw in [Depauw, C. R. Math. Acad. Sci. Paris, 2003], which lies in the above class, and prove that for this vector field, the unique incompressible measure on integral curves exhibits stochasticity.

math.AP↗

arXiv2024

On the lack of selection for the transport equation over a dense set of vector fields

We construct a set of bounded vector fields dense in $L^p((0,2);W^{s,p}_{loc}(\mathbb{R}^2;\mathbb{R}^2))$ for $1\leq p <+\infty$ and $0\leq s<1$ with $p<1/s$ for which smooth regularisation of the vector field does not give a selection criterion for the continuity equation, thereby showing that the two examples constructed in [Calc. Var. Partial Differ. Equ. 59 (2019), Ciampa, Crippa and Spirito] and [Ann. Math. Qu\'e. 46 (2022), De Lellis and Giri] are generic.

math.AP↗