Recherches

2025

1. Springer

   

   Zeros of the Brownian Sheet

   Keming Chen & Guillaume Woessner

   Journal of Theoretical Probability, 2025

   Abs DOI

   In this work, we firstly answer to a question raised by Khoshnevisan by proving that almost surely there is no projection of big enough rank changing the Hausdorff dimension of the zeros of the Brownian sheet. Secondly, we prove that almost surely for every projection whose rank is not matching the aforementioned condition, the projection of the zero set is the entirety of the projective space.

2024

1. Arxiv

   

   A martingale-type of characterisation of the Gaussian free field and fractional Gaussian free fields

   Juhan Aru & Guillaume Woessner

   2024

   Abs DOI

   We establish a martingale-type characterisations for the continuum Gaussian free field (GFF) and for fractional Gaussian free fields (FGFs), using their connection to the stochastic heat equation and to fractional stochastic heat equations. The main theorem on the GFF generalizes previous results of similar flavour and the characterisation theorems on the FGFs are new. The proof strategy is to link the resampling dynamics coming from a martingale-type of decomposition property to the stationary dynamics of the desired field, i.e. to the (fractional) stochastic heat equation.

2. Taylor and Francis

   

   On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model

   Milica Tomašević & Guillaume Woessner

   Stochastic Analysis and Applications, 2024

   Abs DOI

   In this work, we first prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in dimension d≥3. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Second, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in ℝd. We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model.