arXiv subjects

Leonardo Tolomeo

Publications and source records attributed to Leonardo Tolomeo.

At least 19 recordsLinked to original sources

arXiv2026

Cancellations for dispersive PDEs with random initial data

In this work, we provide a combinatorial formalism for dealing with the cancellations that have appeared recently in the context of dispersive PDEs with random initial data. The main idea is to transform iterated integrals encoded by decorated trees into words via an arborification map. This provides a formalism alternative to the one of molecules introduced by Deng and Hani (2023). It allows us to compute the cancellations coming from Wave turbulence and the proof of the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation.

math.AP↗

arXiv2026

Statistical mechanics of the radial focusing nonlinear Schrödinger equation in general traps

In this paper, we investigate the Gibbs measures associated with the focusing nonlinear Schrödinger equation with an anharmonic potential. We establish a dichotomy for normalizability and non-normalizability of the Gibbs measures in one dimension and higher dimensions with radial data. This extends a recent result of the third and fourth authors with Robert and Seong (2022), where the focusing Gibbs measures with a harmonic potential were addressed. Notably, in the case of a subharmonic potential, we identify a novel critical nonlinearity (below the usual mass-critical exponent) for which the Gibbs measures exhibit a phase transition. The primary challenge emerges from the limited understanding of eigenvalues and eigenfunctions of the Schrödinger operator with an anharmonic potential. We overcome the difficulty by employing techniques related to a recent work of the first two authors (2022).

math.AP↗

arXiv2026

Phase transition for invariant measures of the focusing Schrödinger equation

We consider the Gibbs measure for the focusing nonlinear Schrödinger equation on the one-dimensional torus $\mathbb T$, that was introduced in a seminal paper by Lebowitz, Rose and Speer (1988). We show that in the large torus limit, the measure exhibits a phase transition, depending on the size of the nonlinearity. This phase transition was originally conjectured on the basis of numerical simulation by Lebowitz, Rose and Speer (1988). Its existence is however striking in view of a series of negative results by McKean (1995) and Rider (2002).

math.AP↗

arXiv2025

Quasi invariant Gaussian measures for the nonlinear Schrödinger equation on $\mathbb T^2$

We study the transport of Gaussian measures under the flow of the 2-dimensional defocusing Schrödinger equation $i \partial_t u + Δu = |u|^{2k} u$ posed on $\mathbb T^2$. In particular, we show that the Gaussian measures with inverse covariance $\|u\|_{H^s}^2$, are quasi-invariant under the flow for $s>2$. Moreover, we show that the Radon-Nykodim density belongs to every $L^p$ space, locally in space. The proof relies on the physical-space energies introduced in [52], as well as a new abstract quasi-invariance argument that allows us to combine space-time estimates, along the flow with probabilistic bounds on the support of the measure.

math.AP↗

arXiv2025

A remark on the log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation

We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation built by Lebowitz-Rose-Speer (1988), formally given by $$ dρ\propto \exp\big(\frac 1 p\int_{\mathbb T} |u|^p d x - \frac 12\int_{\mathbb T} |\nabla u|^2 d x - \frac 12\int_{\mathbb T} |u|^2 d x\big) \mathbf 1_{\| u \|_{L^2(\mathbb T)}^2 \le K}dud\overline{u}. $$ When $2 \le p \le 4$, we show that these measures indeed satisfy a log-Sobolev inequality. When $p> 4$, we show a lower bound for the Hessian of the potential, which implies that the known techniques to show these inequalities cannot apply to the measure $ρ$.

math.AP↗

arXiv2025

Hyperbolic $P(Φ)_2$-model on the plane

We study the hyperbolic $Φ^{k+1}_2$-model on the plane. By establishing coming down from infinity for the associated stochastic nonlinear heat equation (SNLH) on the plane, we first construct a $Φ^{k+1}_2$-measure on the plane as a limit of the $Φ^{k+1}_2$-measures on large tori. We then study the canonical stochastic quantization of the $Φ^{k+1}_2$-measure on the plane thus constructed, namely, we study the defocusing stochastic damped nonlinear wave equation forced by an additive space-time white noise (= the hyperbolic $Φ^{k+1}_2$-model) on the plane. In particular, by taking a limit of the invariant Gibbs dynamics on large tori constructed by the first two authors with Gubinelli and Koch (2021), we construct invariant Gibbs dynamics for the hyperbolic $Φ^{k+1}_2$-model on the plane. Our main strategy is to develop further the ideas from a recent work on the hyperbolic $Φ^3_3$-model on the three-dimensional torus by the first two authors and Okamoto (2021), and to study convergence of the so-called enhanced Gibbs measures, for which coming down from infinity for the associated SNLH with positive regularity plays a crucial role. By combining wave and heat analysis together with ideas from optimal transport theory, we then conclude global well-posedness of the hyperbolic $Φ^{k+1}_2$-model on the plane and invariance of the associated Gibbs measure. As a byproduct of our argument, we also obtain invariance of the limiting $Φ^{k+1}_2$-measure on the plane under the dynamics of the parabolic $Φ^{k+1}_2$-model.

math.AP↗

arXiv2025

Quasi-Gaussianity of the 2D stochastic Navier-Stokes equations

We study the qualitative properties of solutions to the 2D stochastic Navier-Stokes equations with forcing that is white in time and coloured in space. Our main result shows that the unique invariant measure of this system is equivalent to that of the corresponding Ornstein-Uhlenbeck process. Our method relies on a generalization of the "time-shifted Girsanov method" of [MS05, MRS22] to compare the laws of time marginals for dissipative SPDEs. This generalisation allows to not only compare solutions to a nonlinear equation to those of the corresponding linear equation, but also to directly compare two nonlinear equations. We use this to establish equivalence of the Navier-Stokes system to a "twisted" nonlinear system that leaves the Gaussian measure invariant. We further apply this method to establish similar equivalence statements for a family of hypoviscous Navier-Stokes equations.

math.PR↗

arXiv2025

Critical threshold for weakly interacting log-correlated focusing Gibbs measures

We study log-correlated Gibbs measures on the $d$-dimensional torus with weakly interacting focusing quartic potentials whose coupling constants tend to $0$ as we remove regularization. In particular, we exhibit a phase transition for this model by identifying a critical threshold, separating the weakly and strongly coupling regimes; in the weakly coupling regime, we show that the frequency-truncated measures converge to the base Gaussian measure (possibly with a renormalized $L^2$-cutoff), whereas, in the strongly coupling regime, we prove non-convergence of the frequency-truncated measures, even up to a subsequence. Our result answers an open question posed by Brydges and Slade (1996).

math.PR↗

arXiv2024

Stochastic quantization of the $Φ^3_3$-model

(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study the construction of the $Φ^3_3$-measure and complete the program on the (non-)construction of the focusing Gibbs measures, initiated by Lebowitz, Rose, and Speer (1988). This problem turns out to be critical, exhibiting the following phase transition. In the weakly nonlinear regime, we prove normalizability of the $Φ^3_3$-measure and show that it is singular with respect to the massive Gaussian free field. Moreover, we show that there exists a shifted measure with respect to which the $Φ^3_3$-measure is absolutely continuous. In the strongly nonlinear regime, by further developing the machinery introduced by the authors (2020), we establish non-normalizability of the $Φ^3_3$-measure. Due to the singularity of the $Φ^3_3$-measure with respect to the massive Gaussian free field, this non-normalizability part poses a particular challenge as compared to our previous works. In order to overcome this issue, we first construct a $σ$-finite version of the $Φ^3_3$-measure and show that this measure is not normalizable. Furthermore, we prove that the truncated $Φ^3_3$-measures have no weak limit in a natural space, even up to a subsequence. We also study the dynamical problem. By adapting the paracontrolled approach, in particular from the works by Gubinelli, Koch, and the first author (2018) and by the authors (2020), we prove almost sure global well-posedness of the hyperbolic $Φ^3_3$-model and invariance of the Gibbs measure in the weakly nonlinear regime. In the globalization part, we introduce a new, conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the variational formula and ideas from theory of optimal transport.

math.PR↗

arXiv2024

Focusing $Φ^4_3$-model with a Hartree-type nonlinearity

(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) Lebowitz, Rose, and Speer (1988) initiated the study of focusing Gibbs measures, which was continued by Brydges and Slade (1996), Bourgain (1997, 1999), and Carlen, Fröhlich, and Lebowitz (2016) among others. In this paper, we complete the program on the (non-)construction of the focusing Hartree Gibbs measures in the three-dimensional setting. More precisely, we study a focusing $Φ^4_3$-model with a Hartree-type nonlinearity, where the potential for the Hartree nonlinearity is given by the Bessel potential of order $β$. We first construct the focusing Hartree $Φ^4_3$-measure for $β> 2$, while we prove its non-normalizability for $β< 2$. Furthermore, we establish the following phase transition at the critical value $β= 2$: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. We then study the canonical stochastic quantization of the focusing Hartree $Φ^4_3$-measure, namely, the three-dimensional stochastic damped nonlinear wave equation (SdNLW) with a cubic nonlinearity of Hartree-type, forced by an additive space-time white noise, and prove almost sure global well-posedness and invariance of the focusing Hartree $Φ^4_3$-measure for $β> 2$ (and $β= 2$ in the weakly nonlinear regime). In view of the non-normalizability result, our almost sure global well-posedness result is sharp. In Appendix, we also discuss the (parabolic) stochastic quantization for the focusing Hartree $Φ^4_3$-measure. We also construct the defocusing Hartree $Φ^4_3$-measure for $β> 0$.

math.PR↗

arXiv2024

Quasi-invariance of the Gaussian measure for the two-dimensional stochastic cubic nonlinear wave equation

We consider the stochastic damped nonlinear wave equation $\partial_t^{2}u+\partial_t u+u-Δu +u^{3} = \sqrt{2} {\langle{\nabla}\rangle^{-s}} ξ$ on the two-dimensional torus $\mathbb T^2$, where $ξ$ denotes a space-time white noise and $s>0$. We show that the measure $\vecμ_s$ corresponding to the unique invariant measure for the flow of the associated linear equation is quasi-invariant under the nonlinear stochastic flow.

math.PR↗

arXiv2024

A remark on Gibbs measures with log-correlated Gaussian fields

We study Gibbs measures with log-correlated base Gaussian fields on the $d$-dimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson's argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove non-normalizability of the Gibbs measure. When $d = 2$, our argument provides an alternative proof of the non-normalizability result for the focusing $Φ^4_2$-measure by Brydges and Slade (1996). Furthermore, we provide a precise rate of divergence, where the constant is characterized by the optimal constant for a certain Bernstein's inequality on $\mathbb{R}^d$. We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) non-normalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction.

math.PR↗

arXiv2024

Sharp quasi-invariance threshold for the cubic Szegő equation

We consider the 1-dimensional cubic Szegő equation with data distributed according to the Gaussian measure with inverse covariance operator $(1-\partial_x^2)^\frac s2$, where $s>\frac12$. We show that, for $s>1$, this measure is quasi-invariant under the flow of the equation, while for $s<1$, $s\neq \frac34$, the transported measure and the initial Gaussian measure are mutually singular for almost every time. This is the first observation of a transition from quasi-invariance to singularity in the context of the transport of Gaussian measures under the flow of Hamiltonian PDEs.

math.AP↗

arXiv2023

Ergodicity for the hyperbolic $P(Φ)_2$-model

We consider the problem of ergodicity for the $P(Φ)_2$ measure of quantum field theory under the flow of the singular stochastic (damped) wave equation $u_{tt} + u_t + (1-Δ) u + {:}\,p(u)\mspace{2mu}{:} = \sqrt 2 ξ$, posed on the two-dimensional torus $\mathbb T^2$. We show that the $P(Φ)_2$ measure is ergodic, and moreover that it is the unique invariant measure for (the Markov process associated to) this equation which belongs to a fairly large class of probability measures over distributions. The main technical novelty of this paper is the introduction of the new concepts of asymptotic strong Feller and asymptotic coupling {restricted to the action of a group}. We first develop a general theory that allows us to deduce a suitable support theorem under these hypotheses, and then show that the stochastic wave equation satisfies these properties when restricted the action of translations by shifts belonging to the Sobolev space $H^{1-\varepsilon} \times H^{-\varepsilon}$. We then exploit the newly developed theory in order to conclude ergodicity and (conditional) uniqueness for the $P(Φ)_2$ measure.

math.PR↗

arXiv2022

Focusing Gibbs measures with harmonic potential

In this paper, we study the Gibbs measures associated to the focusing nonlinear Schrödinger equation with harmonic potential on Euclidean spaces. We establish a dichotomy for normalizability vs non-normalizability in the one dimensional case, and under radial assumption in the higher dimensional cases. In particular, we complete the programs of constructing Gibbs measures in the presence of a harmonic potential initiated by Burq-Thomann-Tzvetkov (2005) in dimension one and Deng (2013) in dimension two with radial assumption.

math.PR↗

arXiv2022

Yet another notion of irregularity through small ball estimates

We introduce a new notion of irregularity of paths, in terms of control of growth of the size of small balls by means of the occupation measure of the path. This notion ensures Besov regularity of the occupation measure and thus extends the analysis of Catellier and Gubinelli (2016) to general Besov spaces. On stochastic processes this notion is granted by suitable properties of local non-determinism.

math.PR↗

arXiv2022

Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations

We consider the Cauchy problem for the fractional nonlinear Schrödinger equation (FNLS) on the one-dimensional torus with cubic nonlinearity and high dispersion parameter $α> 1$, subject to a Gaussian random initial data of negative Sobolev regularity $σ<s-\frac{1}{2}$, for $s \le \frac 12$. We show that for all $s_{*}(α) <s\leq \frac{1}{2}$, the equation is almost surely globally well-posed. Moreover, the associated Gaussian measure supported on $H^{s}(\mathbb T)$ is quasi-invariant under the flow of the equation. For $α< \frac{1}{20}(17 + 3\sqrt{21}) \approx 1.537$, the regularity of the initial data is lower than the one provided by the deterministic well-posedness theory. We obtain this result by following the approach of DiPerna-Lions (1989); first showing global-in-time bounds for the solution of the infinite-dimensional Liouville equation for the transport of the Gaussian measure, and then transferring these bounds to the solution of the equation by adapting Bourgain's invariant measure argument to the quasi-invariance setting. This allows us to bootstrap almost sure global bounds for the solution of (FNLS) from its probabilistic local well-posedness theory.

math.AP↗

arXiv2021

Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus

We study an optimal mass threshold for normalizability of the Gibbs measures associated with the focusing mass-critical nonlinear Schrödinger equation on the one-dimensional torus. In an influential paper, Lebowitz, Rose, and Speer (1988) proposed a critical mass threshold given by the mass of the ground state on the real line. We provide a proof for the optimality of this critical mass threshold. The proof also applies to the two-dimensional radial problem posed on the unit disc. In this case, we answer a question posed by Bourgain and Bulut (2014) on the optimal mass threshold. Furthermore, in the one-dimensional case, we show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz, Rose, and Speer (1988). This normalizability at the optimal mass threshold is rather striking in view of the minimal mass blowup solution for the focusing quintic nonlinear Schrödinger equation on the one-dimensional torus.

math.PR↗