arXiv subjects

Rodrigo Bazaes

Publications and source records attributed to Rodrigo Bazaes.

8 recordsLinked to original sources

arXiv2026

Degenerate Diffusions on Continuum Percolation and Hamilton-Jacobi-Bellman Equations

We study degenerate controlled diffusions and Hamilton--Jacobi--Bellman equations posed on genuine continuum percolation clusters. The diffusion is constrained to evolve inside the infinite cluster and degenerates according to the distance to the irregular random boundary. Under suitable regularity and degeneracy assumptions on the diffusion matrix, we prove that the corresponding controlled diffusion never reaches the boundary and therefore admits a unique global strong solution. Using this result, we establish a stochastic control representation formula for viscosity solutions of degenerate Hamilton--Jacobi--Bellman equations posed on the random cluster, without imposing boundary conditions. We further verify that the structural assumptions introduced in this work, including quantitative integrability properties of the distance-to-the-boundary function and the associated degeneracy, hold for concrete continuum percolation models such as the Boolean model. In particular, although the diffusion never reaches the boundary, the boundary geometry remains a fundamental ingredient of the theory through the admissible degeneracy regime. The results establish a quenched framework for stochastic control and degenerate partial differential equations on genuine continuum percolation geometries, and identify a link between the analytic structure of the diffusion and the stochastic geometry of the underlying cluster. They also provide the foundational framework for the homogenization theory developed in the companion article [BMM26].

math.PR↗

arXiv2023

Effective Mass of the Fr\"ohlich Polaron and the Landau-Pekar-Spohn Conjecture

We prove that there is a constant $\overline C\in (0,\infty)$ such that the effective mass $m(\alpha)$ of the Fr\"ohlich Polaron satisfies $m(\alpha) \geq \overline C \alpha^4$, which is sharp according to a long-standing prediction of Landau-Pekar [19] from 1948 and of Spohn [36] from 1987. The method of proof, which demonstrates how the sharp quartic divergence rate of $m(\alpha)$ appears in a natural way, is based on analyzing the Gaussian representation of the Polaron measure and that of the associated tilted Poisson point process developed in [26]. Additionally, our technique here leads to accompanying results including, 1) an explicit identification of local interval process from [26] in the strong coupling limit in terms of functionals of the Pekar process [27], 2) strict monotonicity of the effective mass $m(\alpha)$ for all $\alpha>0$ and 3) the quartic divergence of $m(\alpha)$ for a generalized class of Polaron type interactions.

math-ph↗

arXiv2022

Positive and negative moments for directed polymers in random environment in weak disorder

Very recently, Junk [11] showed that for directed polymers in bounded random environments, the weak disorder (uniform integrable) phase implies that the polymer martingale is bounded in $L^p$ for some $p>1$ and also in $L^q$ for some $q<0$. Here, we establish this characterization of the weak disorder phase without requiring the boundedness assumption on the environments.

math.PR↗

arXiv2022

Subcritical Gaussian Multiplicative Chaos in the Wiener Space: Construction, Moments and Volume Decay

We construct and study properties of an infinite dimensional analog of Kahane's theory of Gaussian multiplicative chaos \cite{K85}. Namely, if $H_T(\omega)$ is a random field defined w.r.t. space-time white noise $\dot B$ and integrated w.r.t. Brownian paths in $d\geq 3$, we consider the renormalized exponential, weighted w.r.t. the Wiener measure $\mathbb P_0$. We construct the almost sure limit $\mu_\gamma$ in the {\it entire weak disorder (subcritical)} regime and call it {\it subcritical GMC} on the Wiener space. We show that $$ \mu_\gamma\Big\{\omega: \lim_{T\to\infty} \frac{H_T(\omega)}{T(\phi\star\phi)(0)} \ne \gamma\Big\}=0 \qquad \mbox{almost surely,} $$ meaning, $\mu_\gamma$ is supported only on $\gamma$-{\it thick paths}, and consequently, the normalized version is singular w.r.t. the Wiener measure. We characterize uniquely the limit $\mu_\gamma$ w.r.t. the mollification scheme $\phi$ in the sense of Shamov \cite{S14} and the random {\it rooted} measure $\mathbb Q_{\mu_\gamma}(d\dot B d\omega)= \mu_\gamma(d\omega,\dot B)P(d\dot B)$. We then determine the fractal properties of the measure around $\gamma$-thick paths: $-C_2 \leq \liminf_{r\to 0} r^2 \log \widehat\mu_\gamma(\|\omega\| < r) \leq \limsup_{r\to 0}\sup_\eta r^2 \log \widehat\mu_\gamma(\|\omega-\eta \| < r) \leq -C_1$ w.r.t a weighted norm $\|\cdot\|$. Here $C_1>0$ and $C_2<\infty$ are the uniform upper (resp. pointwise lower) H\"older exponents which are {\it explicit} in the entire weak disorder regime. Moreover, they converge to the scaling exponent of the Wiener measure as the disorder approaches zero. Finally, we establish negative and $L^p$ ($p>1$) moments for the total mass of $\mu_\gamma$ in the weak disorder regime.

math.PR↗

arXiv2022

Hamilton-Jacobi-Bellman Equations in Random Geometries: Homogenization on Continuum Percolation Clusters

We develop a quenched homogenization theory for optimal control problems related to Hamilton--Jacobi--Bellman equations on random geometries arising from continuum percolation. The underlying state space is the infinite connected component of a continuum percolation model conditioned to contain the origin. The relevant law of the environment is no longer translation invariant, and the geometry of the state space becomes part of the homogenization problem. The associated controlled diffusion is allowed to degenerate according to the distance to the random boundary of the cluster. The degeneracy regime is determined by a balance between a negative-moment threshold for the distance-to-boundary function of the cluster and the coercivity of the Hamiltonian. We prove that the rescaled value functions converge, locally in $L^p$ on the rescaled random domains, almost surely to a deterministic limit governed by an effective Hamiltonian. The effective Hamiltonian admits dual variational characterizations involving a class of curl-free gradients satisfying an induced mean-zero condition. The resulting effective theory retains information about the continuum percolation geometry, the degeneracy of the diffusion, and the nonstationarity induced by conditioning on the infinite component. The proof introduces a variational framework for homogenization nonstationary conditioned laws. Its main ingredients are random shifts adapted to the geometry of the cluster, a two-step min--max construction for admissible gradients, and a novel relative entropy structure intrinsic to the stochastic control representation. The latter reveals a new connection between relative entropy and effective theories for nonlinear stochastic control problems and applies equally well in the general setting of stationary ergodic random media on $\mathbb R^d$ and is therefore of independent interest.

math.AP↗

arXiv2021

Quenched and averaged large deviation rate functions for random walks in random environments: the impact of disorder

In 2003, Varadhan [V03] developed a robust method for proving quenched and averaged large deviations for random walks in a uniformly elliptic and i.i.d. environment (RWRE) on $\mathbb Z^d$. One fundamental question which remained open was to determine when the quenched and averaged large deviation rate functions agree, and when they do not. In this article we show that for RWRE in uniformly elliptic and i.i.d. environment in $d\geq 4$, the two rate functions agree on any compact set contained in the interior of their domain which does not contain the origin, provided that the disorder of the environment is sufficiently low. Our result provides a new formulation which encompasses a set of sufficient conditions under which these rate functions agree without assuming that the RWRE is ballistic (see [Y11]), satisfies a CLT or even a law of large numbers ([Zer02,Ber08]). Also, the equality of rate functions is not restricted to neighborhoods around given points, as long as the disorder of the environment is kept low. One of the novelties of our approach is the introduction of an auxiliary random walk in a deterministic environment which is itself ballistic (regardless of the actual RWRE behavior) and whose large deviation properties approximate those of the original RWRE in a robust manner, even if the original RWRE is not ballistic itself.

math.PR↗

arXiv2021

The effect of disorder on quenched and averaged large deviations for random walks in random environments: boundary behavior

For a random walk in a uniformly elliptic and i.i.d. environment on $\mathbb Z^d$ with $d \geq 4$, we show that the quenched and annealed large deviations rate functions agree on any compact set contained in the boundary $\partial \mathbb{D}:=\{ x \in \mathbb R^d : |x|_1 =1\}$ of their domain which does not intersect any of the $(d-2)$-dimensional facets of $\partial \mathbb{D}$, provided that the disorder of the environment is~low~enough. As a consequence, we obtain a simple explicit formula for both rate functions on $\partial \mathbb{D}$ at low disorder. In contrast to previous works, our results do not assume any ballistic behavior of the random walk and are not restricted to neighborhoods of any given point (on the boundary $\partial \mathbb{D}$). In addition, our~results complement those in [BMRS19], where, using different methods, we investigate the equality of the rate functions in the interior of their domain. Finally, for a general parametrized family of environments, we~show that the strength of disorder determines a phase transition in the equality of both rate functions, in the sense that for each $x \in \partial \mathbb{D}$ there exists $\varepsilon_x$ such that the two rate functions agree at $x$ when the disorder is smaller than $\varepsilon_x$ and disagree when its larger. This further reconfirms the idea, introduced in [BMRS19], that the disorder of the environment is in general intimately related with the equality of the rate functions.

math.PR↗

arXiv2020

Localization at the boundary for conditioned random walks in random environment in dimensions two and higher

We introduce the notion of \emph{localization at the boundary} for conditioned random walks in i.i.d. and uniformly elliptic random environment on $\mathbb{Z}^d$, in dimensions two and higher. Informally, this means that the walk spends a non-trivial amount of time at some point $x\in \mathbb{Z}^{d}$ with $|x|_{1}=n$ at time $n$, for $n$ large enough. In dimensions two and three, we prove localization for (almost) all walks. In contrast, for $d\geq 4$ there is a phase-transition for environments of the form $ω_{\varepsilon}(x,e)=α(e)(1+\varepsilonξ(x,e))$, where $\{ξ(x)\}_{x\in \mathbb{Z}^{d}}$ is an i.i.d. sequence of random variables, and $\varepsilon$ represents the amount of disorder with respect to a simple random walk. The proofs involve a criterion that connects localization with the equality or difference between the quenched and annealed rate functions at the boundary.

math.PR↗