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Ismael Bailleul

Publications and source records attributed to Ismael Bailleul.

At least 19 recordsLinked to original sources
arXiv2023

Spectrally cut-off GFF, regularized $Φ^4$ measure, and reflection positivity

We argue that the spectrally cut-off Gaussian free field $Φ_Λ$ on a compact Riemannian manifold or on $\mathbb{R}^n$ cannot satisfy the spatial Markov property. Moreover, when the manifold is reflection positive, we show that $Φ_Λ$ fails to be reflection positive. We explain the difficulties one encounters when trying to deduce the reflection positivity property of the measure exp$(-\|ρΦ_Λ\|_{L^4}^4) μ_{\text{GFF}}(dΦ)$ from the reflection positivity property of the Gaussian free field measure $μ_{\text{GFF}}$ in a naive way. These issues are probably well-known to experts of constructive quantum field theory but to our knowledge, no detailed account can be found in the litterature. Our pedagogical note aims to fill this small gap.

math.PR↗
arXiv2023

Wilson-Itô diffusions

We introduce Wilson-Itô diffusions, a class of random fields on $\mathbb{R}^d$ that change continuously along a scale parameter via a Markovian dynamics with local coefficients. Described via forward-backward stochastic differential equations, their observables naturally form a pre-factorization algebra à la Costello-Gwilliam. We argue that this is a new non-perturbative quantization method applicable also to gauge theories and independent of a path-integral formulation. Whenever a path-integral is available, this approach reproduces the setting of Wilson-Polchinski flow equations.

math.PR↗
arXiv2022

Diffusion in small time in incomplete sub-Riemannian manifolds

For incomplete sub-Riemannian manifolds, and for an associated second-order hypoelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities, with optimal constant in the exponent. Under similar conditions, we obtain the small-time logarithmic asymptotics of the heat kernel, and show concentration of diffusion bridge measures near a path of minimal energy. The first condition requires that we consider points whose distance apart is no greater than the sum of their distances to infinity. The second condition requires only that the operator not be too asymmetric.

math.PR↗
arXiv2021

Renormalised singular stochastic PDEs

Extended decorations on naturally decorated trees were introduced in the work of Bruned, Hairer and Zambotti on algebraic renormalization of regularity structures to provide a convenient framework for the renormalization of systems of singular stochastic PDEs within that setting. This non-dynamical feature of the trees complicated the analysis of the dynamical counterpart of the renormalization process. We provide a new proof of the renormalized system by-passing the use of extended decorations and working for a large class of renormalization maps, with the BPHZ renormalization as a special case. The proof reveals important algebraic properties connected to preparation maps.

math.PR↗
arXiv2018

Small-time fluctuations for the bridge of a sub-Riemannian diffusion

We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-Riemannian cut locus, then the fluctuations of the conditioned diffusion from the minimal energy path, suitably rescaled, converge to a Gaussian limit. The Gaussian limit is characterized in terms of the bicharacteristic flow, and also in terms of a second variation of the energy functional at the minimal path, the formulation of which is new in this context.

math.PR↗
arXiv2016

Random dynamical systems, rough paths and rough flows

We analyze common lifts of stochastic processes to rough paths/rough drivers-valued processes and give sufficient conditions for the cocycle property to hold for these lifts. We show that random rough differential equations driven by such lifts induce random dynamical systems. In particular, our results imply that rough differential equations driven by the lift of fractional Brownian motion in the sense of Friz-Victoir induce random dynamical systems.

math.PR↗
arXiv2016

Quasilinear generalized parabolic Anderson model equation

We present in this note a local in time well-posedness result for the singular $2$-dimensional quasilinear generalized parabolic Anderson model equation $$ \partial_t u - a(u)Δu = g(u)ξ$$ The key idea of our approach is a simple transformation of the equation which allows to treat the problem as a semilinear problem. The analysis is done within the elementary setting of paracontrolled calculus.

math.AP↗
arXiv2014

Rough integrators on Banach manifolds

We introduce a notion of p-rough integrator on any Banach manifolds, for any $p\geq 1$, which plays the role of weak geometric Holder p-rough paths in the usual Banach space setting. The awaited results on rough differential equations driven by such objects are proved, and a canonical representation is given if the manifold is equipped with a connection.

math.CA↗
arXiv2014

A flow-based approach to rough differential equations

These are lecture notes for a Master 2 course on rough differential equations driven by weak geometric Holder p-rough paths, for any p>2. They provide a short, self-contained and pedagogical account of the theory, with an emphasis on flows. The theory is illustrated by some now classical applications to stochastic analysis, such as the basics of Freidlin-Wentzel theory of large deviations for diffusions, or Stroock and Varadhan support theorem.

math.CA↗
arXiv2013

Flows driven by rough paths

We devise in this work a simple mechanism for constructing flows on a Banach space from approximate flows, and show how it can be used in a simple way to reprove from scratch and extend the main existence and well-posedness results for rough differential equations, in the context of dynamics on a Banach space driven by a finite dimensional Holder weak geometric p-rough path, for any 2<p; the explosion question under linear growth conditions on the vector fields, Taylor expansion and convergence rates for Euler estimates are also dealt with. We illustrate our approach by proving an existence and well-posedness result for some mean field stochastic rough differential equation.

math.PR↗
arXiv2013

Flows driven by Banach space-valued rough paths

We show in this note how the machinery of C^1-approximate flows devised in the work "Flows driven by rough paths", and applied there to reprove and extend most of the results on Banach space-valued rough differential equations driven by a finite dimensional rough path, can be used to deal with rough differential equations driven by an infinite dimensional Banach space-valued weak geometric Holder p-rough paths, for any p>2, giving back Lyons' theory in its full force in a simple way.

math.PR↗
arXiv2013

Path-dependent rough differential equations

We show in this work how the machinery of C^1-approximate flows introduced in our previous work "Flows driven by rough paths", provides a very efficient tool for proving well-posedness results for path-dependent rough differential equations on flows of the form dϕ= V h(dt) + F X(dt), for smooth enough path-dependent vector fields V,F = (V_1,...,V_\ell), any Holder weak geometric p-rough path X and any a-Holder path h, with a+1/p>1.

math.PR↗
arXiv2013

Large deviation principle for bridges of degenerate diffusions

We prove that bridges of subelliptic diffusions on a compact manifold, with distinct ends, satisfy a large deviation principle in a space of Holder continuous functions, with a good rate function, when the travel time tends to 0. This leads to the identification of the deterministic first order asymptotics of the distribution of the bridge under generic conditions on the endpoints of the bridge.

math.PR↗
arXiv2011

A probabilistic view on the general relativistic Boltzmann equation

A new probalistic approach to general relativistic kinetic theory is proposed. The general relativistic Boltzmann equation is linked to a new Markov process in a completely intrinsic way. This treatment is then used to prove the causal character of the relativistic Boltzmann model.

math-ph↗
arXiv2011

Sensitivity for Smoluchowski equation

This article investigates the question of sensitivity of the solutions of Smoluchowski equation on R_+^* with respect to parameters λin the interaction kernel K^lambda. It is proved that the solution is a C^1 function of (t,lambda) with values in a good space of measures under the hypotheses K^{lambda}(x,y) \leq phi(x)phi(y), for some sub-linear function phi, a (4+epsilon)-moment assumption on the initial condition, and that the derivative is a solution, in a suitable sense, of a linearized equation.

math.PR↗
arXiv2011

Probabilistic view on singularities and spacetime boundary

The aim of this article is to promote the use of probabilistic methods in the study of problems in mathematical general relativity. Two new and simple singularity theorems, whose features are different from the classical singularity theorems, are proved using probabilistic methods. Under some energy conditions, and without any causal or initial/boundary assumption, simple conditions on the energy flow imply probabilistic incompleteness. Also we introduce a probabilistic notion of spacetime boundary which has none of the pathological defects that the classical boundaries may have.

math.PR↗
arXiv2011

Spatial coagulation with bounded coagulation rate

We prove that the spatial coagulation equation with bounded coagulation rate is well-posed for all times in a given class of kernels if the convection term of the underlying particle dynamics has divergence bounded below by a positive constant. Multiple coagulations, fragmentation and scattering are also considered.

math.FA↗