Papers

We study the underdamped (kinetic) Langevin dynamics confined to a bounded domain $\Omega\subset\mathbb{R}^d$ by specular reflection of the velocity at the boundary. This process is the natural momentum-based analogue of the normally reflected overdamped Langevin diffusion, and it is used in practice for constrained sampling. Assuming only that the position marginal $\mu_x\propto e^{-U}$ satisfies a Poincaré inequality on $\Omega$ with constant $m>0$, $\nabla^2U\succeq-K\,\mathrm{Id}$ and $\Omega$ is a convex domain, we prove that the law converges to the Gibbs measure exponentially fast in $L^2$, with an explicit rate that scales like $\sqrt m$, which is optimal when $U$ is convex. Since the normally reflected overdamped dynamics converges exactly at rate $m$, this establishes a square-root acceleration for constrained sampling in the small-gap regime when $m$ is small, matching the acceleration known in the unconstrained case. The explicit rate is the same in the unconstrained setting of Fan–Li–Lu. The proof adapts the modified $L^2$ hypocoercivity method of Dolbeault–Mouhot–Schmeiser with the gap-shifted corrector of Fan–Li–Lu. The specular symmetry makes the transport operator antisymmetric, and that the corrector automatically selects the Neumann realization of the overdamped generator, which is precisely the boundary condition that keeps every auxiliary function inside the specular class. The Bochner identity used in the whole-space argument is replaced by a weighted Reilly formula, whose boundary contribution involves the second fundamental form of $\partial\Omega$ and is nonnegative for convex $\Omega$. Finally, we extend our results to the setting where the domain $\Omega$ is non-convex. We obtain an explicit contraction rate that depends on the domain.

Let $X_H$ denote the number of copies of a fixed graph $H$ in $G_{n, p}$. Gilmer and Kopparty conjectured that $X_H$ satisfies a local central limit theorem (LCLT) provided that $H$ is connected, $p \gg n^{-1/m(H)}$, and $n^2 (1-p) \gg 1$, where $m(H)$ is the maximum density. Following the work of Berkowitz, Sah and Sawhney confirmed this conjecture for every constant $p$, leaving the regime where $p=o(1)$ open. In this regime, the only case addressed in the literature is when $H=K_3$, where, in a recent paper, Araújo and Mattos confirmed the conjecture for $p \in (4n^{-1/2}, 1/2)$. This, together with a general result of Röllin and Ross, essentially settles the conjecture for the triangle. We generalise these results by showing that an LCLT holds for $H = K_r$ (for any fixed $r \ge 3$) in the regime $n^{-1/m(H)}\ll p\leq 1/2$, essentially settling the conjecture for cliques.

For any convex body $\mathcal{K}\subset\mathbb{R}^{n}$ containing a unit ball, the spectral gap of Hit-and-Run is $\Omega(1/(n^2 C_{\mathsf{PI}}))$, where $C_{\mathsf{PI}}$ is the Poincaré constant of the uniform distribution $\pi$ over $\mathcal{K}$. This implies that Hit-and-Run converges to a distribution within $\chi^2$-divergence $\varepsilon$ of the uniform distribution $\pi$ in $O(n^2 C_{\mathsf{PI}}\log(M/\varepsilon))$ steps from any starting distribution $\pi_0$ with $M=\chi^2(\pi_{0}\,\|\,\pi)$, thus refining the known bound of $O(n^2 R^2 \log(M/\varepsilon))$ by Lovász and Vempala (2004) in terms of the outer radius $R$; for nearly isotropic bodies, together with progress on the KLS conjecture, the complexity is $O(n^2\log n\log(M/\varepsilon))$, improving the dimension dependence from cubic to nearly quadratic while maintaining logarithmic dependence on the initial distance. It was an open problem to connect the convergence of Hit-and-Run to Poincaré/KLS constants as was done for the Ball walk by Kannan, Lovász and Simonovits (1997). Unlike Hit-and-Run, the Ball walk has an unavoidable linear dependence on (a stronger notion) of the initial warmness. We directly bound the spectral gap of the Hit-and-Run Markov chain by connecting it to functional isoperimetric constants, inspired by the recent analysis of In-and-Out. Rewriting the spectral gap in terms of dual certificates leads to the Babuška–Aziz constant studied in the analysis of PDEs; it is asymptotically bounded by the improved Poincaré constant, which we show can be bounded in terms of the usual Poincaré constant. The proof is based on duality and calculus, unlike known proofs of convergence for Hit-and-Run which are based on bounding the conductance. The same technique can be applied to Coordinate Hit-and-Run, resulting in a much improved mixing time of $O(n^3C_{\mathsf{PI}}\log(M/\varepsilon))$.

The $\Lambda$-distribution of a compact matrix group is an invariant in algebraic probability theory that was recently introduced to study zero distributions of function field $L$-functions. It is encoded by the $\sigma$-moment generating function, a generalization of the Molien series of classical invariant theory. In this work, we compute the $\sigma$-moment generating functions of finite matrix groups in many new cases. In particular, we compute the asymptotic $\Lambda$-distributions for the infinite families of Weyl reflection groups of types $B_n/C_n$ and $D_n$, complementing the previously known case of reflection groups of type $A_n$, i.e., symmetric groups. We also establish a general result relating the shapes of $\sigma$-moment generating functions to the distributions of associated classical random variables, explaining a previous ad hoc observation for independent Gaussians arising from traces of powers on compact classical groups.

We consider the probability that the convex hull of the first $n$ partial sums of a $d$-dimensional random walk contains the origin. Under symmetric exchangeability of the increments and a general-position assumption, this absorption probability is distribution-free and admits an explicit formula, previously obtained by Kabluchko, Vysotsky and Zaporozhets [Geom. Funct. Anal. 27 (2017)] using characteristic polynomials of hyperplane arrangements. We give a different proof, based on a wall-crossing method which we develop here. Starting from a deterministic configuration of increments, we count the signed permutations for which the convex hull of the corresponding partial sums contains the origin and show that this count remains unchanged under generic deformations of the increments, and hence is the same for all configurations outside a natural exceptional set of measure zero. Evaluating the invariant at a single well-chosen configuration reduces the remaining calculation to the enumeration of permutation records combined with Wendel's theorem. Our method also reproves Wendel's theorem on convex hulls of random points with a sign-flip-invariant joint distribution and, in dimension one, Sparre Andersen's theorem. Finally, we derive new probabilistic representations and recurrence relations for the absorption probabilities of random-walk convex hulls and their random-bridge analogues.

Componentwise weak convergence of signed synaptic kernels does not, by itself, determine the fast-synapse limit of a sparse threshold-reset network. Within a causal event protocol with clamped refractoriness and smooth positive-delay kernels, we construct two families whose excitatory and inhibitory measures converge weakly to $\delta_0$ while their microscopic arrival orders are reversed. A target fires in the excitatory-first family and not in the inhibitory-first family precisely when $x+a-b<\theta\le x+a$. Strict margins preserve this response under perturbations of the target state, aggregate E/I pulse masses, and bounded drift. The macroscopic effect persists on a moderately sparse Dale-compatible random block graph with $q_N\to\infty$ and $q_N/N\to0$. The two systems share their graph and initial data. Along every deterministic joint scale $\varepsilon_N\downarrow0$, their population-averaged firing counts differ by $1/2+o_{L^1}(1)$. A bounded-degree construction and a later probe show that the discrepancy is macroscopic and can persist through reset. Fixed positive-delay kernels with finitely many classes admit a stable regime. Before grazing, typewise-mixing sparse networks converge to a delayed class mean-field system. Directed Erdos-Renyi graphs yield the bound $O_P(\lambda_N^{-1/2}+\|\pi_N-\pi\|_1)$ when $\lambda_N\to\infty$ and $\lambda_N/N\to0$. This separates stable averaging at a fixed delay from singular collapse. In the latter, componentwise weak convergence discards signed arrival-order information needed by the threshold-reset response.

The Bethe-Hessian is a symmetric matrix for which the negative spectrum has been observed to encode the informative structure of sparse stochastic block models. We prove that, in the stochastic block model where all vertices have expected degree $d>1$, the number of negative eigenvalues of the Bethe-Hessian is exactly the number predicted by the eigenvalues of the planted model lying outside the bulk spectrum. The condition $d>1$ is optimal, and matches a regime in which existing spectral approaches based on larger non-Hermitian matrices apply. Our result extends a theorem of Stephan and Zhu, who established the same conclusion under the assumption $d\geq 2$. Our improvement relies on two main ideas. First, we construct test vectors on the $2$-core, where degree fluctuations are substantially smaller, and then extend them to the entire graph while controlling the quadratic form. Second, we construct the test vectors using an isotropic basis of the underlying Markov random field, with coefficients adapted to each relevant planted eigenvalue. This allows us to control the fluctuations of the test vectors throughout the sparse regime.

We study the Random Quadratic Form (RQF) on a sphere in the presence of random Brownian forcing. We show that the forcing does not effectively change the law of the process but affects the synchronization properties of the system. While the RQF without forcing exhibits partial synchronization due to the intrinsic symmetries, the introduction of an arbitrarily small forcing results in long-term symmetry breaking and leads to full synchronization. In this work we focus on the small forcing regime and recover the multiscale behavior of the two-point process. We show that in the first stage the model converges to an anti-polar configuration due to the symmetries of the RQF and in the second stage the two clusters meet due to the symmetry breaking phenomenon. The model is motivated by continuous-time machine learning models such as Neural ODEs and continuous-time formulations of transformers. In particular, the results of this work explain the role of the bias and the scale of its initialization.

We establish quantitative mean-field convergence and propagation of chaos for repulsive Coulomb gradient flows at the bounded-density regularity of the limiting equation. The argument couples the dissipative modulated-energy identity with the normalized quadratic transport cost of the full $N$-particle law. The remaining negative mean-square force-error term is used through an exact completion of squares after mollification: the non-Lipschitz remainder is absorbed by this negative term, while a sharp first-order commutator estimate is applied to the mollified Lipschitz field. For the Coulomb equation, the sharp $L^\infty$ decay gives the density envelope $m(t)=\|\rho_0\|_{L^\infty}/(1+t\|\rho_0\|_{L^\infty})$. A density-adapted transport weight and mollification scale $m(t)^{-1/d}$ yield an Osgood comparison. Thus, for every $d\ge2$ and $\rho_0\in\mathcal P_2(\mathbb R^d)\cap L^\infty(\mathbb R^d)$, we obtain quantitative comparison with the global bounded-density Coulomb solution on every prescribed finite interval. For tensorized initial data, the normalized squared Wasserstein distance of the full $N$-particle law, the expected modulated energy, and the time-integrated mean-square force error are bounded by $N^{-2\gamma_{T,d}/d}$ for $d\ge3$ and $((1+\log N)/N)^{\gamma_{T,2}}$ for $d=2$, where $\gamma_{T,d}=(1+T\|\rho_0\|_{L^\infty})^{-c_d}$. For $d-2<s<d$, we also prove Riesz weak–strong stability for prescribed reference solutions in $L^\infty(0,T;B^{s-d+2}_{\infty,q})$, with Gronwall, Bihari, and Osgood comparisons according to $q$, together with uniqueness in the stated Besov class. Finally, an outlier construction separates modulated-energy convergence and Kac chaos from normalized Wasserstein convergence of the full $N$-particle law.

This cumulative habilitation thesis studies probabilistic circuits (PCs) as a powerful and tractable framework for reasoning and learning under uncertainty in artificial intelligence (AI). It first advocates for probability as a core language for AI, emphasizing its connections to logic and information theory; the conceptual simplicity of probabilistic reasoning—based primarily on the sum and product rules; the parallels between probabilistic inference and human cognition; and the role of probability in optimal decision making. However, probability also faces significant computational challenges, as probabilistic inference is NP-hard in almost all probabilistic models. PCs address these challenges through structural constraints that ensure exact computation of a wide range of inference queries in polynomial time, such as marginals, conditionals, most probable explanations, expectations, and more advanced inference tasks. This thesis synthesizes a decade of research across foundations, algorithmic developments, and empirical validation of PCs. Key contributions highlighted in this work are foundational theory of PCs, Bayesian approaches for learning PCs, scalable implementations and integration with deep learning, hybrid models that combine PCs with intractable models, and connections with symbolic machine learning paradigms. This is the first part of my Habilitation Thesis. The second part is omitted, as it comprises the cumulative part of the thesis and has been published at various venues (see Chapter 5).

We prove a measurable realization theorem for hyperuniform Delone point processes. In dimensions \(d\geq2\), for every prescribed \(q\geq1\), every essentially free ergodic p.m.p.\ action of \(\mathbb R^d\) admits, at every sufficiently large prescribed intensity, a generating Delone realization whose return-time point process \(\eta\) is measurably isomorphic to the original action and whose Bartlett spectrum \(\sigma_\eta\) satisfies \[ \sigma_\eta(B_\varepsilon)=o(\varepsilon^{2q}) \qquad(\varepsilon\downarrow0). \] Thus arbitrarily high finite-order low-frequency suppression can be imposed without changing the prescribed measurable dynamics. The same realizations can be chosen with surface-order ball variance and linear rigidity to any prescribed finite order, while also being maximally rigid and almost surely bounded-displacement equivalent to a lattice. For essentially free Euclidean-motion actions whose translation subaction is ergodic, the construction can be made isotropic and \(V\)-ergodic, and hence \(V\)-weakly mixing. In dimension one, every essentially free ergodic flow admits generating Delone realizations with logarithmic interval discrepancy, maximal rigidity, and near-quadratic decay of the Bartlett spectrum at the origin.

We prove a functional stable limit theorem for the discretization error process of a local linearization scheme for stochastic differential equations with additive Brownian noise. The scheme includes the conditional mean of the second-order term involving the Brownian increment in the Taylor expansion of the drift. The leading error is then formed by centered quadratic terms in the Brownian increments, and the sharp normalization is \(n\sqrt n\). Under \(C^3\)-regularity and a Lyapunov-type condition on the drift, the scaled error process converges stably in \(C([0,1],\mathbb R^d)\) to the solution of the limiting linear stochastic differential equation. The martingale part of the limit is driven by a Brownian motion independent of the original \(\sigma\)-field, and its coefficient is determined by the Hessian of the drift and the covariance matrix of the additive noise.

In this paper, we study the hitting times of Borel right processes on a metric measure space $(E,d,\mu)$ whose heat kernels satisfy sub-Gaussian bounds. It is well known that if $X=(X_t)_{t\geq 0}$ is diffusion process without killing whose heat kernel satisfies a sub-Gaussian upper bound, then, under the volume growth condition $\mu(B(x,r))\asymp r^\alpha$, it satisfies \[ \IP^x[\tau_{B(x,r)}\le t]\le C_1\exp\left\{-C_2(r^\beta/t)^{1/(\beta -1)}\right\}, \] where $B(x,r):=\{y\in E: d(y,x)<r\}$, $\tau_{B(x,r)}:=\inf\{t>0:X_t\notin B(x,r)\}$, and $\beta$ is the walk dimension appearing in the sub-Gaussian heat kernel estimate. We extend this result to general Borel right processes, showing that under an upper bound condition on the volume growth, \[ \IP^x[\sigma_B\le t]\le C_3\exp \left\{-C_4\left(\frac{\widetilde d(x, B)^\beta}{t}\right)^{1/(\beta -1)}\right\}, \] where $B$ is a nearly Borel set, $\sigma_B$ denotes the first hitting time of $B$, and $\widetilde d(x,B)$ represents the distance from $x$ to $B$ after removing the influence of polar subsets of $B$. Furthermore, we show that for a Borel right process with a sub-Gaussian heat kernel lower bound, the hitting time distribution satisfies the corresponding lower bound \[ \IP^x[\sigma_B\le t]\ge C_5 \exp\left\{-C_6\cdot \left(\frac{\widetilde d(x,B)^\beta}{t}\right)^{1 /(\beta -1)}\right\}. \] We also characterize the relationship between the constants $C_i$, $3\leq i\leq 6$, and the constants appearing in the exponents of the corresponding heat kernel bounds. As an application of these hitting time estimates, we further study the small-time asymptotic behavior of $\IP^x[\sigma_B\leq t]$ as $t\downarrow 0$.

We establish a central limit theorem in Rényi divergence for independent and identically distributed lattice random variables $X_1, \cdots, X_n$ with zero mean, unit variance, and maximal span $h>0$. Let $S_n=(X_1+\cdots+X_n)/\sqrt n$. Let $Z_n$ denote the standard Gaussian distribution quantized on the support lattice of $S_n$. For every $\alpha>1$, with $\beta=\alpha/(\alpha-1)$, we prove that the Rényi divergence $D_\alpha(S_n\|Z_n)\to 0$ if and only if the divergence is finite at some convolution level and the strict sub-Gaussian condition $$ \mathbb E e^{tX}<e^{\beta t^2/2},\quad t\in\mathbb R,~ t\ne0 $$ holds. Under these conditions, we further derive an Edgeworth-type asymptotic expansion of the divergence to arbitrary order. These results provide a lattice counterpart of the Rényi entropic central limit theorem for continuous random variables due to Bobkov, Chisyakov and Götze (Ann. Probab. 47 (2019), 270–323).

We study weighted $L^2$ Hessian estimates for parabolic terminal equations under the marginals of a prescribed diffusion, allowing singular initial laws and different principal coefficients in the equation and the diffusion. Under coefficient regularity, uniform ellipticity and bounded reference drift, we characterize the limiting optimal constant uniformly over shrinking subintervals with zero terminal data. Its finiteness is equivalent to linear well-posedness in the weighted parabolic Sobolev space on the full interval. For initial laws satisfying weighted doubling and coercivity conditions, this constant is the maximum of contributions from heat evolutions of rescaled initial measures, constant weights and exponential mixtures. The phase contributions are bounded above in terms of the coefficients at time zero outside the initial support and at spatial infinity. Under radial reference structure, we obtain exact coefficient formulas invariant under bounded changes of reference drift. In the matching case, with reference covariance $a$, principal matrix $a/2$ and normalized Hessian $a^{1/2}D^2u\,a^{1/2}$, the constant is $2$ for every initial law with time weight $t^\alpha$, $\alpha\ge1/2$; point starts attain $2\sqrt2$ at $\alpha=0$. Whenever the limiting constant is finite, increasing damping makes the full-interval Hessian constant converge to it and the value and gradient gains tend to zero. As applications, we obtain nonlinear solvability and residual stability for small Hessian perturbations. For solutions and candidates with a common regularity bound, secant estimates give linear stability under a relative spectral gap, while covariance interpolation extends residual stability beyond it.

We study systems of coupled Ornstein-Uhlenbeck diffusions with an absorbing failure state on finite graphs that converge locally weakly to a unimodular Galton-Watson tree. A single failure can trigger a cascade of threshold crossings, and the central question is when the cascade is subcritical or supercritical and what the cluster looks like near criticality. Under an explicitly stated front-decoupling hypothesis (the saturated-coupling regime), verified exactly on the Bethe lattice, the failed cluster of the root converges in law to the genealogy of a finite-type Galton-Watson process whose mean matrix M is computed from a single matrix-valued many-to-one formula. Conditional on this reduction we prove that criticality occurs at rho(M) = 1, that the quenched and annealed growth rates are the tangent at 0 and the value at 1 of a single convex real-analytic pressure phi(beta) = log rho(M(beta)), making the quenched/annealed gap a chord-tangent inequality, that on survival the end boundary of the failed cluster has Hausdorff dimension log rho(M), that at criticality P(|S| = n) ~ C n^{-3/2} and the conditioned cluster rescaled by n^{-1/2} converges to Aldous' Continuum Random Tree under a moment hypothesis, and that the subcritical domain is open and contractible but not convex, with the spectral gap 1 - rho(M) giving quantitative stability. Several natural claims in this setting are shown to be false, including the Hilbert-Schmidt/Krein-Rutman operator programme, the scalar many-to-one identity, and the spectral-radius/dimension identity. The reduction itself is proved unconditionally in the dissipative regime kappa < 1, where a pathwise front-tracking argument shows that back-reaction echoes decay geometrically in graph distance and beat the volume growth of the failed cluster; the strong-coupling regime remains open and is stated as a conjecture.

We prove a universality theorem for a broad class of two-dimensional gauge theories on compact surfaces. Each admissible conjugation-invariant Lévy process on a compact connected Lie group determines a universality class of lattice gauge theories whose continuum limit is the associated Markovian holonomy process. Our result can be seen as a gauge-theoretic analogue of invariance principles for random walks and Lévy processes. This framework includes the Yang–Mills holonomy process and the standard heat-kernel (Villain), Wilson, and Manton lattice actions. The proof uses state-sum formulas of independent interest, separating action-dependent spectral coefficients from action-independent topological coefficients determined by the surface and the marked ribbon type of the loop configuration.

We study Gaussian random dynamical systems with coordinatewise $\tanh$ nonlinearity, where finite precision is modeled by nearest-grid rounding after each step. Gaussian symmetry reduces the dynamics to an exact Markov chain for the normalized squared radius. Rounding makes the origin absorbing, and the total variation distance to the absorbing equilibrium equals the survival probability of the absorption time. At fixed width, we identify the critical gain and prove an absorption cutoff with Gaussian profile as the mesh tends to zero. At fixed precision, global contraction yields a large-dimension absorption cutoff, while positive drift produces metastability. In the supercritical regime, we prove a large-dimension cutoff to a nonzero invariant law and show that, at fixed dimension, its mass near the repelling origin has a power-law asymptotic. All six main results are formalized in Lean 4 on top of Mathlib and independently checked against restatements that import only Mathlib.

We fix an integer $d\ge1$ and a symmetric positive-definite matrix $Q\in\mathbb{R}^{d\times d}$. Let $V:\mathbb{R}^d\to\mathbb{R}$ be finite, set \[ Z_\mu:=\int_{\mathbb{R}^d}e^{-V(x)}\,dx\in(0,\infty), \qquad d\mu(x):=Z_\mu^{-1}e^{-V(x)}\,dx, \] and assume that $\mu$ has finite second moment and that \[ x\longmapsto \frac12\langle Qx,x\rangle-V(x) \] is convex. Let $\nu$ be a compactly supported log-concave probability measure with support $K$, and let $\nabla\Phi$ be the Brenier map from $\mu$ to $\nu$. For $v\in\mathbb{R}^d$, define \[ w_K(v):= \sup_{y\in K}\langle y,v\rangle - \inf_{y\in K}\langle y,v\rangle. \] We prove that \[ \partial_{vv}\Phi \le 0.587 \sqrt{\langle Qv,v\rangle}\,w_K(v) \qquad(v\in\mathbb{R}^d) \] in the sense of distributions. We show that $\nabla\Phi$ has an everywhere-defined globally Lipschitz representative such that \[ \operatorname{Lip}(\nabla\Phi) \le 0.587 \sqrt{\|Q\|_{\mathrm{op}}}\,\operatorname{diam}(K). \] The directional Hessian estimate is affinely covariant, whereas the global Lipschitz estimate is dimension-free. The result also applies to singular or lower-dimensional targets. In particular, it removes the $\sqrt d$ loss in Kolesnikov's estimate for the Brenier map from Gaussian measure to normalised Lebesgue measure on a convex body. We also prove new bounds that depend only on the support for compactly supported semi-log-concave targets, which includes targets with bounded negative curvature.

We prove time-uniform self-normalised Bennett inequalities for a martingale $(M_n)_{n\geq0}$ in a separable Hilbert space, with $M_0=0$ and increments bounded in norm by one. Writing $V_n$ for its predictable covariance process and $h(u)=(1+u)\log(1+u)-u$ for the Bennett rate function, we show that, for every regularisation parameter $\rho>0$, the process \[\exp\left\{\rho\,h\!\left(\frac{\left\lVert (V_n+\rho I)^{-1/2}M_n\right\rVert}{\sqrt{\rho}}\right)-\frac12\log\det\left(I+\rho^{-1}V_n\right)\right\},\qquad n\geq0,\] is a nonnegative supermartingale with initial value one, where $\det$ is the Fredholm determinant. Ville's inequality yields time-uniform Bennett and Bernstein bounds for $\lVert (V_n+\rho I)^{-1/2}M_n\rVert$. The result permits conditional covariance increments of infinite rank; in finite dimensions, the resulting boundaries sharpen existing martingale-transform and determinant-based variational bounds. The same construction extends to compensated marked point processes with bounded jumps. Mixing these supermartingales over $\rho$ gives simultaneous control over the regularisation parameter. Consequences include an upper law of the iterated logarithm for the regularised ellipsoidal radius in separable Hilbert spaces and, in finite dimensions, spectrum-sensitive finite-time bounds and an upper law of the iterated logarithm for the unregularised radius $\lVert V_n^{-1/2}M_n\rVert$, whose constant $1$ is sharp over the class. We also obtain a time-uniform Bernstein inequality for the martingale norm $\lVert M_n\rVert$ with dependence on $\operatorname{tr}(V_n)$ and $\lVert V_n\rVert_{\mathrm{op}}$.

We study the geometry that spectral duality induces on the simplex of reset distributions for absorbed Markov processes with geometric resetting. In the Fisher–Rao metric, selected by Čencov's theorem, the separatrix $\Sigma$ of reset-neutral distributions is rigid: under the square-root embedding $\pi\mapsto2\sqrt\pi$ its interior is a totally geodesic subsphere of the positive octant, itself a lower-dimensional Fisher–Rao simplex. The reset response reduces to a finite structure: the response functionals span a subspace whose dimension $r$ equals the number of active orbits of the duality involution, and the local separatrix is its annihilator. At the vertices of the simplex we prove a sign theorem valid for every $r$, which recovers the two-zone phenomenon of the first paper of this series. The invariant $r$ also settles the question of global orientation: for $r=1$, under a scalar sign condition satisfied by the canonical realisation, the sign of the response is fixed by the side of $\Sigma$; for $r\ge2$ the orientation cannot stay constant, and we give the criterion under which no global sign law holds, with counterexamples due to T. Newton showing that it is met in the abstract class. The biased random walk with multi-site geometric resetting realises the whole picture explicitly.

Motivated by Fisher's and Hill's \cite{Fisher,Hill} ideas of ranking and pivotal quantities, we introduce a planar Poisson process (PPP) model in which the negative component \(\Pi_-\), acted upon by a choice operator C, supplies a set of past benchmarks that divide the future points of the positive component \(\Pi_+\) into ranked categories. The benchmarks act as separators in an ordered paintbox while simultaneously acquiring the dual role of past standards established by the choice operator. The exceptional homogeneity properties of the PPP offer an infinitude of possibilities which absorb many existing combinatorial structures and enrich the toolbox of Bayesian distribution-free inference. A particular benchmark generating mechanism considered here (the exponential race) amounts to the device of splitting into spacings of nonhomogeneous order statistics. On the methodological side, the paper aims to highlight the role of order as important characteristic of an exchangeable structure, complementary to the description in terms of the components size. Moreover, we advocate the viewpoint that the order induced by a latent strength parameter is intrinsically inherited from the distant-past temporal sampling order, hence the decoupled orders may coexist within the framework of population duality without disturbing size-biasedness of components and the full exchangeability within the sample. This implies that the indistinguishability of components in the nonlinear CRP, sometimes regarded as nonexchangeability, does not in fact destroy exchangeability and should be reconciled with the arrival ordering within the paradigm of ordered structures.

We develop two Brownian stopping methods: one uses hitting probabilities and capacity, the other moments of weighted exit distributions. On a closed $m$-dimensional Riemannian manifold, $m\geq 2$, the first gives low-energy spectral inequalities for equilibrium measures supported on sets of zero volume. Combining scales gives heat observability and null controllability from a full-support measure carried by a dense set of Hausdorff dimension $m-2$, minimal under a bounded $H^1$-trace condition. For every nondecreasing unbounded $\Phi$ with $\Phi(4E)\leq D\Phi(E)$, the measure can be chosen so that the optimal spectral constant is comparable to $\Phi(E)$ and the small-time control cost to $T^{-1/2}\Phi(T^{-1})^{1/2}$. Squaring the Feynman-Kac martingale doubles the potential: for radial $V$, the boundary second moment is log-convex in $\log r$ if $r^2V(r)$ is nondecreasing, wherever the regular solution of $(\tfrac12\Delta+2V)g=0$ stays positive; the factor $2$ is sharp when $V(0)>0$. On small geodesic balls, weighted second moments give an almost-monotone frequency and a weighted three-radius inequality; bounds for its logarithmic derivative give doubling estimates, which, after a ground-state transform, feed the Remez theorem of Logunov-Malinnikova. The normalised first-moment exit law gives an averaged boundary-variance identity for Laplace eigenfunctions. At $r=c\lambda^{-1/2}$, outside a set carrying $O(c^{2/m})$ of the $\varphi^2$-mass, the component of $\{\varphi(x)\varphi>0\}\cap B(x,r)$ through $x$ contains a concentric ball of nearly full radius, fills nearly all of $B(x,r)$, and nearly matches its first Dirichlet eigenvalue. First-moment stopping also controls positive superlevel components. Reflected Brownian motion and local time give boundary-to-collar $L^\infty$ estimates for Steklov eigenfunctions on bounded Lipschitz domains, and an $L^2$ estimate for $C^2$ domains.

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 prove various statistical properties (i.e., decay of correlations, central limit theorems with convergence rates, maximal large deviations and almost sure invariance principles) for non-smooth observables in the form of indicators functions in polynomially mixing hyperbolic dynamical systems. Such results were not known (or even expected to hold) for slowly mixing hyperbolic systems, although they are important for applications to physics and other sciences.

Let $G$ be the Boolean hypercube which carries uniform measure $\lambda$, and let $T_\mu$ denote convolution by a finite positive measure $\mu$ on $G$. For $\psi_\mu(u)=\sup\{u\lambda(\{T_\mu f\geq u\}):f\geq 0,\|f\|_1=1\},$ we prove Talagrand's convolution conjecture (Talagrand, 1989): if $\mu_a=((1+a)\delta_1/2+(1-a)\delta_{-1}/2)^{\otimes n}$ and $0<a<1$, then $\psi_{\mu_a}(u)\leq C_a/\sqrt{\log u}$ for every $u > 1$ and $n\geq1$, where $C_a$ depends only on $a$. The proof utilizes the reverse-heat and Boolean-bridge framework of Chen (2025) and the localized terminal-discrepancy method of Xiang and Zhang (2026). We introduce a new power coupling: each reverse edge ratio is split into two geometric powers. This choice produces a switched exponential weight which restores the exact reverse jump rate of the perturbed coordinate. The resulting endpoint comparison yields an anti-concentration profile estimate without the iterated-logarithmic factor. The proof was discovered by the Odin Automatic AI Research Agent.

We study lower tails of support functions of isotropic convex bodies. An endpoint negative moment estimate, combined with a spherical cap argument, gives polynomial deviation estimates for the inradius of random projections in every dimension; in the unconditional class this yields the optimal order of the median inradius and shows that the cube is extremal. The same argument gives estimates for convex hulls of independent rotations and, after an additional projection, a mixed rotation-projection theorem. We also prove weighted lower and upper estimates for Minkowski averages of rotated polars and their random projections, as well as an inradius dependent estimate for the geometric distance of global averages from the Euclidean ball. A family of isotropic product cylinders shows that the latter estimate is sharp, up to absolute constants, throughout the possible range of the inradius. For the cube, an explicit negative moment computation gives a quantitative projected $(m,k,n)$-profile and, in full dimension, recovers the classical order $n/\ln(en)$ for bounded geometric distance. Finally, we determine the sharp volume profile of intersections of independent rotations in the unconditional isotropic class and show that the cube is extremal.

In this article, we consider discontinuous martingales on tangent bundles over submanifolds of Euclidean space. First, we introduce a connection rule on tangent bundles and establish the Itô calculus for discontinuous semimartingales on tangent bundles. Then we focus on harmonic maps with respect to non-local Dirichlet forms and show that the derivative of harmonic maps along infinitesimal symmetries induces discontinuous martingales on tangent bundles. This process may be viewed as a stochastic Jacobi field along the image martingale. We also introduce the stochastic parallel transport of tangent vectors along càdlàg semimartingales on Riemannian submanifolds with projected jumps. Using the parallel transport, we obtain the mean-value property for the differential of harmonic maps involving a jump part expressed through the second fundamental form. We also obtain a derivative formula for harmonic maps for isotropic Lévy processes with a Brownian component on compact Riemannian manifolds.

While Harris recurrent Markov chains (in discrete time) automatically exhibit wide-sense regenerative structure, we construct a Harris recurrent Markov process (in continuous time) that is not wide-sense regenerative, thereby giving a negative answer to the open problem first raised in the 1990s and later posed in Glynn(2011). The counterexample exhibits the following rigidity property: every almost surely finite random time that is independent of the state observed at that time must be almost surely constant. A Cantor set linearly independent over the rationals plays a key role in the construction, turning calendar time into an algebraic record of the path already traversed.

We consider for an arbitrary fixed $\rho$ and for each positive integer $n$ a multilayer feedforward artificial neural network with $\rho$ layers, $n$ neurons in the first layer (the input layer) and only one neuron, the output neuron, in the last layer. Very roughly formulated, the main result is that if the distribution of weights of connections from a layer to the next are, for all large $n$, approximated well by a fixed continuous (but otherwise arbitrary) curve which does not depend on $n$, and if the values of the $n$ input neurons are independently and identically distributed with a continuous probability density function, then there is a number $\psi$ such that for all $\varepsilon > 0$ the probability that the value of the output neuron is in $[\psi - \varepsilon, \psi + \varepsilon]$ tends to 1 as $n$ tends to infinity.