Papers

We investigate stability properties of distribution-dependent stochastic Volterra equations with respect to changes in the coefficients, the Volterra kernels, and the initial condition. Under Lipschitz continuity assumptions on the coefficients, we first derive quantitative stability estimates for strong solutions with explicit error bounds. We then prove a general convergence theorem for strong solutions under substantially weaker assumptions, replacing Lipschitz continuity by a continuity assumption together with uniform linear growth of the approximating sequence. Finally, we study the associated distribution-dependent Volterra local martingale problem and prove the stability of its solutions under convergence of the coefficients, kernels, and initial distributions.

For two Gaussian random elements on a separable Hilbert space with common covariance operator, the fourth-order moment tensor structure of their mixture admits an operator-valued representation obtained from the purely quadratic component of the class-conditional second moment. We identify the tensor mechanism generating the completely diagonal coefficients of this fourth-order representation after covariance standardization, and show that these coefficients are governed entirely by the coordinatewise Cameron-Martin energy of the mean shift. This yields a necessary and sufficient condition, together with an explicit eigenvalue, for the Fisher discriminant to be an eigenfunction of the induced coordinatewise operator. We further introduce aggregated fourth-order spectral functionals for two complementary constructions, one built from fourth-order moments and the other from a product-of-expectations counterpart, and prove that they are asymptotically equivalent precisely when the underlying Gaussian measures are mutually singular. These results provide a spectral realization of the classical Cameron-Martin criterion through higher-order moment operators, explaining the probabilistic origin of fourth-order spectral quantities previously proposed for Gaussian discrimination and functional data classification and relating them to the "near-perfect classification" regime.

Let $A=(\xi_{ij})$ be an $n\times n$ random matrix with independent, not necessarily identically distributed, real entries satisfying \[ \mathbb E\xi_{ij}=0,\qquad \mathbb E\xi_{ij}^{2}=1,\qquad \sup_{z\in\mathbb R}\mathbb P(|\xi_{ij}-z|<a)\le b \] for fixed $a>0$ and $b\in(0,1)$. We prove that, for every $\delta\in(0,1)$, there are constants $c,C>0$, depending only on $a,b,\delta$, such that \[ \mathbb P\left( s_{n+1-l}(A)>Ct\frac l{\sqrt n} \right) \le \exp\!\left(-c\min\{tl,n\}\right) \] for every $t\ge1$ and every $1\le l\le(1-\delta)n$. Thus, with no moment assumption beyond variance, all but a fixed proportion of the largest singular values satisfy the optimal upper bound of order $l/\sqrt n$ with an exponential upper-tail estimate. Combined with the lower bound of the rectangular least singular value bound, this gives $s_{n+1-l}(A)\asymp l/\sqrt n$ with failure probability exponentially small in $l$. The same argument gives the rectangular scale $\sqrt{N+1}-\sqrt{n-l+1}$ for $N\times n$ matrices whenever $N-n+l\le(1-\delta)N$.

We study the linear growth rate of the range of size-conditioned Branching Random Walks (BRW) when the offspring distribution $\mu$ is critical and attracted to an $\alpha$-stable law. This is done via the infinite invariant BRW introduced by Le Gall & Lin and a new criterion which relates this growth rate of the range to a notion of dimension of the underlying tree in a general way. Then, in the transient case (that is, when the range does grow linearly), we extend the notion of branching capacity to this $\alpha$-stable case. We show that it is still related to the asymptotic probability that a BRW (or its infinite version) reaches a distant set in $\mathbb Z^d$, and we estimate the $\alpha$-stable branching capacity of balls.

Let $\zeta(G)$ denote the minimum number of parts in a partition of $V(G)$ in which every part induces either a clique or an independent set. Erdős and Gimbel asked whether, for $G_n\sim G(n,1/2)$, the difference $\chi(G_n)-\zeta(G_n)$ tends to infinity with high probability. We resolve this problem along the full sequence $n\to\infty$ and prove that $\mathbb P(\chi(G_n)-\zeta(G_n)\ge ((\log 2)^2/4)\log(200/153)\,n/(\log n)^3)\to1$. This gives a lower bound at the conjectured scale $n/(\log n)^3$. We also obtain a phase-resolved refinement: if $\delta_n$ is the fractional part of the standard independence-number center, then the coefficient may be replaced by $(\log 2)^2A_4(\delta_n)/4-o(1)$, where $A_4$ is explicit, continuous, nonconstant, and satisfies $A_4(\delta)>\log(200/153)$ for every $\delta\in[0,1]$. The proof uses signed cocoloring profiles supported on four consecutive class sizes and remains uniform across jumps of the natural class-size cutoff. An exact signed-overlap identity separates local cell rewards from a binary cycle-space factor. A canonical decomposition into high cells and a capped residual matching, together with an endpoint-table comparison and an injective restriction of residual even edge sets, yields the required second-moment bound. A bounded-differences argument then amplifies the resulting rare signed witness to a high-probability cocoloring.

For any $\gamma \in \left[ {0,\frac{1}{3}} \right)$, we construct $\gamma$-Hölder-continuous martingale solutions to the two-dimensional stochastic incompressible Euler equations. These martingale solutions exhibit dissipative behavior and intermittency, thereby establishing the flexible part of the intermittent Onsager theorem. The proof rests on a stochastic and intermittent variant of the Newton–Nash iteration combined with the Littlewood–Paley decomposition scheme. This composite iteration scheme incorporates new stochastic pressure, Reynolds stress and intermittent perturbations to formalize the stochastic and intermittent fluctuations.

We prove a finite-entropy criterion for the entropic conditional central limit theorem. Let $(\xi_i,\eta_i)_{i\geq 1}$ be independent copies of a pair $(\xi,\eta)$, and set $W_n=n^{-1/2}\sum_{i=1}^n \xi_i$ and $\boldsymbol{\eta}_n=(\eta_1,\ldots,\eta_n)$. Under the assumptions that $\mathbb{E}\operatorname{Var}(\xi\mid\eta)<\infty$ and that the conditional law of $\xi$ given $\eta$ is absolutely continuous almost surely, we show that $\mathbb{E}h(W_n\mid\boldsymbol{\eta}_n)$ converges to the Gaussian entropy $\frac12\log(2\pi e\sigma^2)$, where $\sigma^2=\mathbb{E}\operatorname{Var}(\xi\mid\eta)$, if and only if $\mathbb{E}h(W_{n_0}\mid\boldsymbol{\eta}_{n_0})>-\infty$ for some $n_0$. The main technical ingredient is a continuity theorem for Fisher information under Gaussian smoothing, which allows us to replace the finite expected conditional Fisher-information assumption by a necessary and sufficient finite-entropy condition.

We consider a discrete-time financial market model where, in addition to finitely many dynamically traded assets, there are also (possibly infinitely many) static options to choose from. We introduce the concept of small cones of random variables and present a sufficient condition for the attainable positions in the market to be closed in probability. A fundamental theorem of asset pricing is shown in the present context. Utility maximization will also be considered. We will provide economically meaningful examples of infinite dimensional small cones to demonstrate the pertinence of our approach.

Starting from localized energy estimates, we prove finite speed of propagation for kinetic solutions to stochastic porous media equations with nonlinear conservative noise, the existence and uniqueness of which has recently been established. In particular, we propose a novel iteration technique which allows us to obtain a Stampacchia-type inequality involving one single integral quantity, despite the possibly different scaling behaviors of the porous media and the noise term. This allows us to apply stochastic filtering arguments developed for the case of linear source-type noise. Using related ideas, we identify flatness conditions on initial data which guarantee locally the occurrence of a waiting time phenomenon, i.e., the onset of forward propagation of the solution's support is locally delayed. The condition for the latter matches the one for the deterministic porous media equation up to a logarithmic correction in the case of critical nonlinearity in the noise, but it requires more and more flatness of the initial data as the nonlinearity tends towards linear conservative noise. This is in line with the expected behavior: In the case of linear conservative noise, instantaneous forward motion is possible due to the effects of stochastic transport, no matter how flat the initial profile is.

These notes were prepared for a series of lectures delivered at the 43rd Finnish Summer School on Probability and Statistics, held in Lammi, Finland, from May 26 to 30, 2025. They provide an introduction to McKean-Vlasov stochastic differential equations and their connections with non-linear Fokker-Planck equations and mean-field interacting particle systems. Particular attention is devoted to density-dependent equations and to kinetic-type models with degenerate noise. For the latter, we introduce the underlying non-Euclidean geometry and the associated anisotropic and intrinsic Hölder spaces, and present well-posedness and regularity results based on Schauder estimates and superposition principles.

Popular adaptive stochastic gradient descent (SGD) methods to train artificial intelligence (AI) systems include the RMSprop, the Adam, and the AdamW optimizers, where the adaptivity parts in Adam and AdamW basically just coincide with RMSprop. Such adaptive methods involve several hyperparameters including the regularization parameter $\epsilon$ (which ensures that one does not divide by 0 and is often chosen to be very close to zero such as $10^{-8}$ in PyTorch by default) and the second moment decay parameter $\beta$ (which is often chosen to be very close to $1$ such as 0.99 (RMSprop) and 0.999 (Adam and AdamW) in PyTorch by default). Despite the high relevance of such methods, it remains an open research problem to provide error estimates for such methods with the error constants being not exploding but uniformly bounded with the respect to the hyperparameters, even in the situation of convex stochastic optimization problems. It is the key contribution of this work to essentially solve this problem for RMSprop. Specifically, we bound the expectation of the stopped evaluation of the objective function at the RMSprop process from above by the sum of an initialization term that decays exponentially in the training time, a stochastic approximation remainder of order $\gamma_n$, and a memory error of order $( 1 - \beta)^2$ with the error constants being uniformly controlled over all admissible choices of the step sizes, the second moment decay parameter $\beta$ and the regularization parameter $\epsilon\in[0,1]$ (also covering $\epsilon=0$). Our non-asymptotic error estimates hold not just for all sufficiently large n but hold for every gradient step $n=1,2,3,...$ with all error constants being explicitly specified. The key innovative new feature in the proof of our analysis are suitable inverse moment bounds for the second moment process in RMSprop.

This article studies mean convergence of Banach space-valued random elements indexed in a family of finite measure spaces. We derive $L^p$-convergence theorems under (compact) uniform integrability in two regimes: a decaying-index-mass regime and a bounded-index-mass regime, the latter requiring a new dependence structure which is called diagonal negative dependence for the random elements and expressed via the self-product of the index measure. We provide examples showing that the conditions to obtain the results are sharp and strictly weaker than related conditions in the literature. As a further illustration for the index measure space framework, a functional law of large numbers in $L^p$ on the space of continuous functions is derived, where the random elements are solutions of stochastic differential equations driven by Brownian motions extracted from a common Brownian sheet.

By slightly adapting Wang's recent important argument for the homoscedastic setting, we prove that every finite heteroscedastic Gaussian mixture has only finitely many critical points, and, consequently, only finitely many modes.

Loss-versus-Rebalancing (LVR) is the dominant adverse-selection cost borne by liquidity providers on automated market makers. Under geometric Brownian motion, arbitrage profit scales with the probability of a profitable block, which vanishes as the block time $\Delta t \to 0$; this is the standing argument for ever-shorter blocks. Modeling the reference price instead as a jump-diffusion, I show that the constant-product LVR rate splits into a diffusion channel carrying the known multiplier $F(\gamma/(\sigma\sqrt{\Delta t}))$ and a jump channel $\lambda V \cdot G(\gamma;m,\delta^2)$ carrying no $\Delta t$, the two interacting only through an explicitly bounded remainder. The block schedule therefore governs only one channel. For symmetric jump laws the jump channel is moreover an exact lower bound, $\ell(\Delta t) \ge \lambda V G > 0$, so the rate does not vanish as $\Delta t \to 0$, and descends slowly, as $\sqrt{\Delta t}$. At Ethereum's calibrated 12-second slot the rate is 471 bp/yr against a floor of 125, so only three quarters of LP loss is schedule-addressable. At Solana's 400 ms slot the jump channel already dominates. Netting the rate against per-block consensus cost, the LP-side optimal block time is invariant in pool size and in every jump parameter $(\lambda,m,\delta)$: jumps shift the level of LP loss but not the planner's marginal tradeoff. Volatility, the fee tier, and consensus cost set the optimum, near 8 s. However, LVR is only one input to block-time welfare, so this bounds the LP-side contribution rather than settling the design question.

The fundamental theorem of statistical learning states that, under suitable measurability assumptions, finite Vapnik–Chervonenkis (VC) dimension guarantees that every proper consistent learning rule is probably approximately correct (PAC). Blumer, Ehrenfeucht, Haussler, and Warmuth showed, assuming the Continuum Hypothesis, that the "well-behavedness" condition of the concept class cannot be omitted: they constructed a concept class of Borel sets of VC dimension one admitting a consistent learning rule that is not PAC. We show that the Continuum Hypothesis is unnecessary. Working in Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC) alone, we construct a concept class of Borel sets on $[0,1]$ of VC dimension one and a proper consistent learning rule that is not PAC. More precisely, for a suitable Borel probability measure and target concept, the rule has true risk one at every sample size on a set of samples of outer probability one. Consequently, finite VC dimension and Borel measurability of the individual concepts do not suffice to guarantee that every proper consistent learning rule is PAC. The result shows, with no need of extra set-theoretical assumptions, that the additional regularity assumption in the fundamental theorem cannot in general be omitted.

We study extinction and extinguishment in a two-type continuous-state nonlinear branching model driven by Brownian branching noise and spectrally positive stable jumps. The populations are subject to nonlinear self-regulation and a mixed-sign predator–prey interaction: the second promotes the first, whereas the first suppresses the second. Two complementary structures are developed. An exact power–logarithmic cancellation functional removes the interaction drifts and yields stochastic Lyapunov estimates, nonexplosion, and boundary criteria. In the multiplicative regimes, integrating-factor identities and geometric Lévy factorizations express extinction through weighted exposure clocks and reduce the long-time analysis to effective decay rates. These methods yield almost-sure extinction criteria and identify a regime in which finite-time extinction and nonextinction coexist. On nonextinction, both populations remain positive at all finite times and converge jointly to zero, exhibiting joint extinguishment rather than positive persistence.

The dimer model and random lattice permutations are two fundamental objects at the interface of probability, combinatorics, and mathematical physics. We study these models on finite periodic boxes in $\mathbb Z^d$ within a common framework. For the dimer model, edges connecting arbitrary vertices carry a weight which depends on their relative displacement and dimer configurations are weighted through their occupied edges. Superimposing two independent perfect matchings gives the double-dimer model, whose configurations are collections of disjoint loops. Permutations, instead, are weighted through the spatial displacement of their jumps. For broad classes of weights of finite or infinite range we prove long-range order and the occurrence of macroscopic loops. This extends nearest-neighbour results to arbitrary-range edges and jumps. In particular, long-range weights yield long-range order and macroscopic loops already in dimensions $d=1,2$. In dimension two, this behaviour is qualitatively different from that of the nearest-neighbour model. We complement these results with sharp absence criteria.

Heavy-tailed $p$-value combination tests are attractive under unknown dependence because their null tails can be first-order robust even when the exact dependent null distribution is unavailable. That robustness does not resolve calibration: $\Pr\{T>q(\alpha)\}=\alpha+o(\alpha)$ neither quantifies the remaining size error nor determines its sign. We develop a second-order calibration theory for positive Half-Cauchy and reciprocal, or harmonic-mean, aggregation. After exact marginal standardization, extremal dependence is represented by an index-one exponent measure on the coordinate-face lattice. Its support separates axial, full-interior, proper-face, and mixed geometries. A Möbius decomposition shows how the noncompact weighted half-space probes every face, while quantitative face limits determine whether the correction is integrable, critically amplified, or nonintegrable. With common-factor inversion and a tail-to-calibration map, this yields machinery applicable beyond individual parametric copula families. The results include an integrable hidden-face transfer theorem, a sharp fixed-dimensional local-corner theorem with explicit higher-face control, a common-heavy-factor theorem, and size and critical-value expansions relative to independence. Gaussian, standard multivariate-$t$, positive Clayton, and max-linear pair-shock models exhibit distinct power, logarithmic, radial–angular, and proper-face mechanisms; the Gaussian weighted-half-space transfer is conditional on explicit face-boundary hypotheses. Thus dependence geometry determines the rate, coefficient, and direction of the calibration error left unresolved by first-order validity. Asymptotics use $t\to\infty$, equivalently vanishing significance levels.

Let \(q\in\mathbb N\), let \(G=(0,\infty)^q\), and let $ S:G\times E\longrightarrow E, (r,x)\longmapsto S_rx $ be a jointly measurable left action on an arbitrary measurable space \((E,\mathcal E)\). For \(\alpha=(\alpha_1,\ldots,\alpha_q)\in(0,\infty)^q\) set $ \chi_\alpha(r)=\prod_{i=1}^q r_i^{\alpha_i}. $ We study nonzero \(\sigma\)-finite measures \(\nu\) satisfying $ \nu(S_rA)=\chi_\alpha(r)^{-1}\nu(A), r\in G, A\in\mathcal E.$ Motivated by the scalar case \(q=1\) studied in [1] we derive equivalent conditions for the existence of an \(E\)-valued random element \(Z\) such that $ \nu(A) = \mathbb{E}\{\int_G\mathbb I_A(S_rZ)\prod_{i=1}^q \alpha_i r_i^{-\alpha_i-1}dr_i\}, A\in\mathcal E. $ We also characterise when two random elements generate the same homogeneous measure, using multihomogeneous moments and, after fixing an admissible product gauge, weighted transverse measures. When the corresponding weighted transverse measure is finite, tilting and gauge normalisation produce a canonical representer, unique in law on the prescribed gauge shell. Finally, we characterise stationarity under an action commuting with \(S\) and construct positive semidefinite tail-overlap kernels directly from \(\nu\).

Integrals of spectral densities are frequently used to summarize spectral characteristics of linear processes. This work studies self-normalization for estimators of such integrals based on sequential periodograms and establishes weak convergence of the corresponding processes. For linear functionals of the spectral density, self-normalization yields pivotal limiting distributions that are free of unknown spectral quantities. For non-linear functionals, however, additional components with distinct covariance structures may arise in the limiting process. We demonstrate this phenomenon for the integrated squared spectral density.

We investigate Sylvester's classical four-point problem for ball-convex bodies, where convex hulls are replaced by intersections of balls of a fixed radius $R$. We show that the Efron–Buchta identities persist in this framework, so that the distribution of the number of vertices of the $R$-ball-convex hull of $n$ uniform random points is determined by the moments of the volumes of the hulls of its subsamples, and vice versa. For three and four uniform random points in an arbitrary planar ball-convex body, we determine the Sylvester probabilities of the $R$-ball-convex hull. We also give an extension of Groemer's inequality for the expected area of the ball-convex hull of $n$ random points and show that it is minimised by the disc of the same area. In particular, this yields an isoperimetric inequality for the ball-convex analogue of the affine length of the boundary curve. For the unit disc, we can express the Sylvester probabilities using dilogarithmic functions of the radius $R$, and give explicit values at $R=1$. As $R\to\infty$, these recover the classical Sylvester probabilities. As another consequence, we determine the exact distribution function of the circumradius of three uniform random points in he disc on $[1,\infty)$.

We study the relationship between the Cramér transform and Tukey's half-space depth for log-concave probability measures. For the uniform probability measure $\mu_K$ on a convex body $K\subseteq\mathbb{R}^n$, we prove the sharp pointwise comparison $$\Lambda_K^*(x)\leq -\log q_K(x)\leq \Lambda_K^*(x)+\frac12\log n+C,\qquad x\in\operatorname{int}(K),$$ where $C$ is an absolute constant. The order $\log n$ is optimal, as shown by the Euclidean ball. The proof combines exponential tilting, one-dimensional log-concavity, and self-concordance of the Cramér transform. As consequences, we obtain sharp-order moment and tail estimates for $\Lambda_K^*$ and identify $\exp(\Lambda_K^*(x))$, up to polynomial factors in the dimension, with the number of independent samples needed for $x$ to be captured by their random convex hull. We also establish an $O(n^2)$ variance bound for the logarithmic half-space depth and use it to derive a general criterion for sharp thresholds of random convex hulls. In particular, this criterion applies to the uniform measures on $\ell_p$-balls for every $p>1$. These results establish a quantitative link between large-deviation cost, geometric depth, and sampling complexity in high-dimensional convex geometry.

We prove that the stationary density fluctuation field of the one-dimensional weakly asymmetric simple inclusion process converges to the energy solution of the stochastic Burgers equation. The key technical ingredient is a second-order Boltzmann-Gibbs principle tailored to the microscopic current. As a consequence, we also obtain Ornstein-Uhlenbeck fluctuations for the symmetric inclusion process in a weakly condensing regime, a result of independent interest.

Easo, Severo and Tassion [EST25] proved two theorems about minimal edge cutsets in infinite graphs: that non-triviality of the uniform critical percolation parameter is equivalent to exponential growth of the number of minimal edge cutsets of size n separating a vertex from infinity, and that uniform transience is an alternative sufficient assumption. We show that the naive extensions of (the original contribution to) these theorems to site percolation and vertex cutsets are false; our counterexample involves modifying a rooted binary tree and replacing edges with gadgets that allow many choices for where to block (in constructing a cutset) one of two local routes, while keeping every vertex uniformly close to a supercritical, uniformly transient shortcut skeleton.

Dickman and Tomé (1991) introduced models on $Z$ in which $k$ consecutive occupied sites give birth at rate $\lambda$, individual particles die at rate 1, and the configuration is subject to nearest neighbor stirring. They claimed that the pair creation model ($k=2$) has a continuous phase transition and was in the universality class of directed percolation, but the triplet creation model ($k=3$) has a discontinuous (or first-order) phase transition when the stirring rate is large. Hinrichsen (2000a) disputed the second claim and argued that first-order phase transitions in 1+1-dimensional nonequilibrium systems with fluctuating ordered phases are impossible. In this paper we prove rigorously that the phase transitions are discontinuous in both the pair and triplet models when the stirring rate is large. At the end of Section 2 we state some open problems and suggest reasons that the results in the physics literature differ from those presented here,

We prove local laws for the resolvents of separable covariance matrices of the form $\mathcal Q=A^{1/2}XBX^*A^{1/2}$, where $X=(x_{ij})$ is a $p\times n$ random matrix whose entries $x_{ij}$ are i.i.d.~random variables with mean 0 and variance $n^{-1}$, and $A,B$ are deterministic non-negative definite symmetric (or Hermitian) matrices. Following the method developed in arXiv:1611.05364, we first establish a self-consistent equation for the resolvent of $\mathcal Q$ and use it to prove optimal local laws without the technical assumption $\mathbb{E}[x_{ij}^{3}]=0$, which was essential in the previous derivation of the local laws in arXiv:1809.04572. As an application of our local law, we compute the asymptotic distribution of the outlier eigenvalues for spiked separable covariance matrices, extending the corresponding result in arXiv:2008.11903.

Let $X$ be a time-homogeneous one-dimensional diffusion whose coefficients satisfy the Engelbert–Schmidt conditions, and let $q$ be a bounded potential. We show that the associated Feynman–Kac semigroup is smooth in time and $C^1$ in space, with a locally absolutely continuous first spatial derivative; the Kolmogorov equation holds almost everywhere. If the drift, second-order coefficient, and potential are continuous, the Feynman–Kac value is $C^{1,2}$ in the interior. Thus, in the time-homogeneous one-dimensional setting, continuity suffices for interior $C^{1,2}$ regularity. We identify the corresponding semigroups with the killed and reflected Feynman–Kac functionals, treat nonzero, time-independent Dirichlet data, and give examples showing the sharpness of the continuity assumptions and the failure of the corresponding statement in two dimensions.

In this paper, we study the long-time behavior of the viscous conservation law equation on the whole real line, subject to a white-in-time force and arbitrarily small damping and viscosity constants. We establish polynomial mixing rates in two settings: (a) a general super-quadratic flux with hyperviscous dissipation, where the growth of the nonlinearity is controlled by the order of hyperviscosity, and the force is sufficiently non-degenerate on a generic L2-basis; and (b) a general quadratic flux with standard second-order dissipation, and the force is non-degenerate on some unconditional basis. Both settings include the classical Burgers-type nonlinearity. The proof is based on an abstract polynomial coupling criterion developed in arXiv:2408.00592 and a Foiaş–Prodi estimate.

We prove uniform-in-time quantitative relative entropy estimates for Kac's particle approximation of the three-dimensional spatially homogeneous Landau equation for Maxwellian molecules. Using the relative entropy method and logarithmic derivative estimates, we obtain a finite-time normalized entropy bound of order $N^{-1/2}$. The proof relies on explicit covariance identities and a new law-of-large-numbers estimate for the particle system using a new duality method. The uniform-in-time propagation of chaos is derived from an interpolation with an algebraic relaxation estimate toward equilibrium.

We study branching SDEs with law-dependent immigration and their mean-field particle approximations. Under a dissipativity condition and sufficiently weak interaction, a uniform propagation-of-chaos bound in time of order $N^{-1/2}$ is established. On every fixed finite time horizon, the same order of propagation of chaos holds for arbitrary finite interaction strength. A two-stage scaling limit connects continuous-time discrete-state mean-field birth–death processes to interacting branching diffusions and then to the nonlinear equation. For a logistic mean-field diffusion we prove a sharp criterion for extinction/non-extinction, and further show that weak enough interaction strength is necessary for a uniform-in-time approximation.