We consider the Langevin diffusion $dX_t = - \beta \nabla V(X_t) dt + \sqrt{2} dB_t$ for a general nonnegative real-analytic potential $V$ and a large parameter $\beta$. In the large-$\beta$ limit the process is confined to the zero set of $V$, assuming that it starts there. We derive a candidate limiting evolution on the zero set. To do so, the zero set is partitioned into strata according to a measure of local codimension known as the local learning coefficient and its multiplicity. It is then shown that the Dirichlet form associated with $X$ converges in a certain sense to a hierarchy of Dirichlet forms corresponding to a stochastic evolution on the strata. This evolution is strongly biased toward higher-dimensional, or"more singular", strata. This result is motivated by a question from Watanabe's singular learning theory regarding the learning dynamics of overparameterized statistical models and the generalization puzzle in deep learning. The result suggests a mechanism for the observation that stochastic gradient methods tend to be biased toward singular solutions that generalize well.
We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of the spectrum. Under the assumption $a\in C^1(\mathbb{T})$, we prove square-root asymptotics for the integrated density of states and obtain a small-energy expansion for the Lyapunov exponent as $E\to0^+$. The leading term of the Lyapunov exponent is linear and positive precisely under a nondegeneracy condition on the sampling function. In this case, we prove a large-deviation estimate for the transfer matrices with an explicit rate depending on $E$. As consequences, we obtain local H\"older continuity of the Lyapunov exponent and the integrated density of states near the bottom of the spectrum, as well as the Anderson localization. Some of these results extend the small-coupling results of Chulaevsky--Spencer and Bourgain--Schlag for Schr\"odinger operators generated by the doubling map to divergence-gradient type operators. The arguments adapt methods from those works, but require uniform control in the small-energy parameter and refined correlation estimates reflecting the divergence-gradient structure.
We prove absolute continuity of the running maximum $X^{\ast}_T=\sup_{0\leq s\leq T}X_s$ of one-dimensional time-inhomogeneous L\'evy--It\^o diffusions driven by a Brownian motion and an independent non-truncated pure-jump L\'evy process. Using Bismut's directional Malliavin calculus on the Wiener--Poisson space together with the running-maximum criteria of Song--Xie and Nakagawa--Suzuki, we reduce the problem to constructing an admissible direction $\Theta$. The key is to ensure that the directional derivative $D_\Theta X_t$ is strictly positive for all $t\in(0,T]$. We give explicit directions in two regimes. In the uniformly elliptic case with time-inhomogeneous coefficients, a purely Brownian perturbation yields an explicit positive integral representation for $D_\Theta X_t$, and hence $X^{\ast}_T$ admits a density without truncating the jump component. In a Brownian-degenerate pure-jump model with a bounded deterministic time-dependent jump weight $\kappa(t)$ that may vanish on subintervals, we prove absolute continuity under the minimal nondegeneracy-in-time condition $\int_0^t \kappa(s)^2\,ds>0$ for every $t>0$ and infinite activity of the L\'evy measure. A weighted Poisson positivity lemma is the key new input. Finally, we show that silent initial intervals can create atoms and we derive an explicit atom--density decomposition.
Given a strongly convex function $u$, equip $R^d$ with a Riemannian metric given by the Hessian $\nabla^2 u$. This is a so-called Hessian manifold. Given a probability density $\mu$ one may run a Langevin diffusion intrinsic to the manifold with stationary distribution $\mu$. Such (Hessian) manifold-valued Langevin diffusions are called Mirror Langevin diffusions (MLD) which have recently become popular. One of the questions we explore is whether, given $\mu$, one can choose $u$ to get an exponential convergence to equilibrium for the MLD, especially if $\mu$ is not strongly log-concave. Our results are based on Lyapunov function methods and give sufficient conditions for a Poincar\'e or a log-Sobolev inequality to hold for the MLD. These, in turn, imply exponential convergence. We also introduce a Markov chain approximation to the MLD given by a two step Gibbs sampler with stationary distribution $\mu$. This Markov chain is a variant of the Sinkhorn Markov chain introduced in arXiv:2307.16421 that is conjectured to converge to a time-inhomogeneous generalization of the MLD. Under suitable assumptions, we prove that the Markov chain has a guaranteed convergence rate in $\chi^2$ that is consistent with the diffusion time scale. Our proofs are based on ideas from entropic optimal transport and strong data processing inequalities.
Benjamin Capdeville, Young-Heon Kim, Soumik Pal· 0 citations
We study a penalisation problem for two-dimensional Brownian motion. Starting from the Wiener measure, we consider a family of probability measures obtained by weighting paths by a nonnegative functional $F_t$ depending on $t \geq 0$, $F_t$ being measurable with respect to the $\sigma$-algebra generated by the path up to time $t$. Under suitable assumptions on the penalisation process, we establish the weak convergence of these measures when $t \rightarrow \infty$. The limiting law is identified explicitly in terms of a $\sigma-$finite measure $\mathbf{W}^{(2)}$, which admits a path decomposition involving the last hitting time of a circle. This decomposition plays a central role in the analysis and yields a martingale representation of the limiting measure. where ordering and local time techniques are no longer available. The proofs rely on Laplace transform methods and Tauberian theorems, which replace excursion-theoretic tools and allow a precise identification of the limiting measure and its structural properties.
Joseph Najnudel, Thammadol Tansrivorarat· 0 citations
We study the one-dimensional stochastic partial differential equation \[ d_t X_t(x)=\frac{1}{2}\Delta X_t(x) +b_1\unicode{x1D7D9}_{\{X_t(x)>0\}} +\sqrt{X_t(x)}\dot W(t,x), \] where $b_1>0$, $\dot W$ is space-time white noise, and the initial condition is a nonnegative, compactly supported continuous function. We prove that its unique weak solution has the compact support property. The drift term lies outside the usual regularity assumptions for the Dawson--Girsanov theorem. Instead, the proof is based on a layer decomposition in which the solution is constructed as the monotone limit of sums of super-Brownian motions with random immigration rates determined recursively by the positivity sets of the preceding layers. By comparison, the compact support property extends to a broader class of bounded nonnegative drifts vanishing at the origin. Finally, support-radius estimates yield a comparison of the total mass of $X$ with a squared Bessel process, which allows us to show that $X$ has a positive extinction probability.
We consider a fractional Brownian motion $B$ with Hurst index $0<H<1/2$, and its maximiser $\tau$ on $[0,1]$. We show that the rescaled process $a^H(B_{\tau+\,\cdot\,/a}-B_\tau)$ converges in $C_{\mathrm{loc}}(\mathbb R)$ to a limiting tangent law that is $H$-self-similar, supported on nonpositive paths pinned at zero, and rerooting-rescaling invariant: rerooting the limit process at its maximum on any fixed compact interval separated from zero and rescaling again asymptotically reproduces the same law. We also identify the tangent law as the limit of two-sided finite-grid hard-wall laws as the mesh vanishes and both horizons diverge, which can informally be interpreted as conditioning fractional Brownian motion on a nonpositive path. As an application, we consider persistence probabilities for fractional Brownian motion: a tilted variant of $B$ yields a different tangent law with a finite left horizon and an infinite right horizon and we show that \[ \mathbb P(B_t\leq1\text{ for all }0\leq t\leq T) = \big(C+o(1)\big)\,T^{-(1-H)},\quad \text{as }T\to\infty, \] where the leading order coefficient $C\in(0,\infty)$ has an explicit representation in terms of the expected maximum and the tilted tangent law.
