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Henri Elad Altman

Publications and source records attributed to Henri Elad Altman.

8 recordsLinked to original sources
arXiv2026

A Rough Functional Breuer-Major Theorem

We extend the functional Breuer-Major theorem by Nourdin and Nualart (2020) to the space of rough paths. The proof of tightness combines the multiplication formula for iterated Malliavin divergences, due to Furlan and Gubinelli (2019), with Meyer's inequality and a Kolmogorov-type criterion for the r-variation of cadlag rough paths, due to Chevyrev et al. (2022). Since martingale techniques do not apply, we obtain the convergence of the finite-dimensional distributions through a bespoke version of Slutsky's lemma: First, we overcome the lack of hypercontractivity by an iterated integration-by-parts scheme which reduces the remaining analysis to finite Wiener chaos; crucially, this argument relies on Malliavin differentiability of the nonlinearity but not on chaos decay and, as a consequence, encompasses the centred absolute value function. Second, in the spirit of the law of large numbers, we show that the diagonal of the second-order process converges to an explicit symmetric correction term. Finally, we compute all the moments of the remaining process and, through a fine combinatorial analysis, show that they converge to those of the Stratonovich Brownian rough path perturbed by an antisymmetric area correction, as computed by a suitable amendment of Fawcett's theorem. All of these steps benefit from a major combinatorial reduction that is implied by the original argument of Breuer and Major (1983).

math.PR↗
arXiv2025

Scaling Limits of Line Models in Degenerate Environment

We consider a 2-dimensional model of random walk in random environment known as line model. The environment is described by two independent families of i.i.d. random variables dictating rates of jumps in vertical, respectively horizontal directions, and whose values are constant along vertical, respect. horizontal lines. When jump rates are heavy-tailed in one of the directions, we prove that the random walk becomes superdiffusive in that direction, with an explicit scaling limit written as a two-dimensional Brownian motion time-changed (in one of the components) by a process introduced by Kesten and Spitzer in 1979. In the case of a fully degenerate environment, a sufficient condition for non-explosion is provided, and conjectures on the associated scaling limit are formulated.

math.PR↗
arXiv2020

On the gradient dynamics associated with wetting models

We consider several critical wetting models. In the discrete case, these probability laws are known to converge, after an appropriate rescaling, to the law of a reflecting Brownian motion, or of the modulus of a Brownian bridge, according to the boundary conditions. In the continuous case, a corresponding convergence result is proven in this paper, which allows to approximate the law of a reflecting Brownian motion by the law of Brownian meander tilted by its local time near the origin. On the other hand, these laws can be seen as the reversible probability measure of some Markov processes, namely, the dynamics which are encoded by integration by parts formulae. After proving the tightness of the associated reversible dynamics in the discrete case, based on heuristic considerations on the integration by parts formulae, we provide a conjecture on the limiting process, which we believe to satisfy a Bessel SPDE as introduced in a recent work by Elad Altman and Zambotti.

math.PR↗
arXiv2019

Bessel SPDEs with general Dirichlet boundary conditions

We generalise the integration by parts formulae obtained in arXiv:1811.00518v5 [math.PR] to Bessel bridges on $[0,1]$ with arbitrary boundary values, as well as Bessel processes with arbitrary initial conditions. This allows us to write, formally, the corresponding dynamics using renormalised local times, thus extending the Bessel SPDEs of arXiv:1811.00518v5 [math.PR] to general Dirichlet boundary conditions. We also prove a dynamical result for the case of dimension $2$, by providing a weak construction of the gradient dynamics corresponding to a $2$-dimensional Bessel bridge.

math.PR↗
arXiv2019

Bessel SPDEs and renormalised local times

In this article, we prove integration by parts formulae (IbPFs) for the laws of Bessel bridges from 0 to 0 over the interval [0,1] of dimension smaller than 3. As an application, we construct a weak version of an SPDE having the law of a one-dimensional Bessel bridge (i.e. the law of a reflected Brownian bridge) as reversible measure, the dimension 1 being particularly relevant in view of applications to scaling limits of dynamical critical pinning models. We also exploit the IbPFs to conjecture the structure of the SPDEs associated with Bessel bridges of all dimensions smaller than 3.

math.PR↗
arXiv2017

Bismut-Elworthy-Li formulae for Bessel processes

In this article we are interested in the differentiability property of the Markovian semi-group corresponding to the Bessel processes of nonnegative dimension. More precisely, for all $δ\geq 0$ and $T>0$, we compute the derivative of the function $x \mapsto P^δ_{T} F (x) $, where $(P^δ_{t})_{t \geq 0}$ is the transition semi-group associated to the $δ$ - dimensional Bessel process, and $F$ is any bounded Borel function on $\mathbb{R}_{+}$. The obtained expression shows a nice interplay between the transition semi-groups of the $δ$ - and the $(δ+ 2)$-dimensional Bessel processes. As a consequence, we deduce that the Bessel processes satisfy the strong Feller property, with a continuity modulus which is independent of the dimension. Moreover, we provide a probabilistic interpretation of this expression as a Bismut-Elworthy-Li formula.

math.PR↗