The adapted weak topology is an extension of the weak topology for stochastic processes designed to adequately capture properties of underlying filtrations. With the recent work of Bart–Beiglböck–P. [7] as starting point, the purpose of this note is to recover with topological arguments the intriguing result by Backhoff–Bartl–Beiglböck–Eder [3] that all adapted topologies in discrete time coincide. We also derive new characterizations of this topology, including descriptions of its trace on the sets of Markov processes and processes equipped with their natural filtration. To emphasize the generality of the argument, we also describe the classical weak topology for measures on ${\mathbb{R}}^d$ by a weak Wasserstein metric based on the theory of weak optimal transport that was initiated by Gozlan–Roberto–Samson–Tetali [11].

This paper considers the $(n,k)$-Bernoulli–Laplace urn model in the case when there are two urns containing n balls each, with two different colors of balls (red and white). In our setting, the total number of red and white balls is the same. Our focus is on the large-time behavior of the corresponding Markov chain tracking the number of red balls in a given urn assuming that the number of selections k at each step obeys $\mathit{\alpha }\le k∕n\le \mathit{\beta }$, where $\mathit{\alpha },\mathit{\beta }$ are constants satisfying $0<\mathit{\alpha }<\mathit{\beta }<\frac{1}{2}$. Under this assumption, cutoff in the total variation distance is established and a cutoff window is provided. The results in this paper solve an open problem posed by Eskenazis and Nestoridi in [8].

We give a description of the high-dimensional limit of one-pass single-batch stochastic gradient descent (SGD) on a least squares problem. This limit is taken with non-vanishing step-size, and with proportionally related number of samples to problem-dimensionality. The limit is described in terms of a stochastic differential equation in high dimensions, which is shown to approximate the state evolution of SGD. As a corollary, the statistical risk is shown to be approximated by the solution of a convolution-type Volterra equation with vanishing errors as dimensionality tends to infinity. The sense of convergence is the weakest that shows that statistical risks of the two processes coincide. This is distinguished from existing analyses by the type of high-dimensional limit given as well as generality of the covariance structure of the samples.

Bougerol’s identity in law is one of the most infamous in the study of Brownian motion. To that end, in this paper, we present a further extension of its study through a proof in the third dimension. Second-dimension proofs have been extensively explored, as have those of the third dimension. However, here we will extrapolate a previously two-dimensional proof into the third dimension through the use of a three-Brownian joint distribution.

We prove that the solution to the singular-degenerate stochastic fast-diffusion equation with parameter $m\in (0,1)$, with zero Dirichlet boundary conditions on a bounded domain in any spatial dimension, and driven by linear multiplicative Wiener noise, exhibits improved regularity in the Sobolev space $W_0^{1,m+1}$ for initial data in $L^2$.

Motivated by a problem posed by Aldous [2, 1] our goal is to find the maximal-entropy win-martingale:

In a sports game between two teams, the chance the home team wins is initially $x_0\in (0,1)$ and finally 0 or 1. As an idealization we take a continuous time interval $[0,1]$ and let $M_t$ be the probability at time t that the home team wins. Mathematically, $M={(M_t)}_{t\in [0,1]}$ is modelled as a continuous martingale. We consider the problem to find the most random martingale M of this type, where ‘most random’ is interpreted as a maximal entropy criterion. In discrete time this is equivalent to the minimization of relative entropy w.r.t. a Gaussian random walk. The continuous time analogue is that the max-entropy win-martingale M should minimize specific relative entropy with respect to Brownian motion in the sense of Gantert [20]. We use this to prove that M is characterized by the stochastic differential equation

$$d{M}_t=\frac{sin(\mathrm{\pi }{M}_t)}{\mathrm{\pi }\sqrt{1-t}}d{B}_t.$$

To derive the form of the optimizer we use a scaling argument together with a new first order condition for martingale optimal transport, which may be of interest in its own right.

In this paper, we give the moderate deviation principle from the hydrodynamic limit of the simple symmetric exclusion process on the 1-dimensional torus starting from a nonequilibrium state, which extends the result given in Gao and Quastel (2003) about the case where the process starts from an equilibrium state. The exponential tightness of the scaled density field of the process and a replacement lemma play key roles in the proof of the main result. We utilize Grownwall’s inequality and the upper bound of the large deviation principle given in Kipnis, Olla and Varadhan (1989) to prove the above exponential tightness and the replacement lemma respectively in the absence of the invariance of the initial distribution.

Birkner et al. obtained necessary and sufficient conditions for the frequency between two independent and identically distributed continuous-state branching processes time-changed by a functional of the total mass process to be a Markov process. Foucart et al. extended this result to continuous-state branching processes with immigration. We generalize these results by dropping the independent and identically distributed assumption. Our result clarifies under which conditions a multi-type Λ-coalescent can be constructed from a multi-type branching process by a time change using the total mass. Finally, we address a problem formulated by Griffiths, by clarifying the relation between 2-type α-stable continuous-state branching processes and 2-type β-Fleming–Viot processes with mutation and selection.

We consider polynomial transforms (polyspectra) of Berry’s model – the Euclidean Random Wave model – and of Random Hyperspherical Harmonics. We determine the asymptotic behavior of variance for polyspectra of any order in the high-frequency limit. In particular, we are able to treat polyspectra of any odd order $q\ge 5$, whose asymptotic behavior was left as a conjecture in the case of Random Hyperspherical Harmonics by Marinucci and Wigman (Comm. Math. Phys. 2014). To this end, we exploit a relation between the variance of polyspectra and the distribution of uniform random walks on Euclidean space with finitely many steps, which allows us to rely on technical results in the latter context.

We calculate the large deviations for the length of the longest alternating subsequence and for the length of the longest increasing subsequence in a uniformly random permutation that avoids a pattern of length three. We treat all six patterns in the case of alternating subsequences. In the case of increasing subsequences, we treat two of the three patterns for which a classical large deviations result is possible. The same rate function appears in all six cases for alternating subsequences. This rate function is in fact the rate function for the large deviations of the sum of IID symmetric Bernoulli random variables. The same rate function appears in the two cases we treat for increasing subsequences. This rate function is twice the rate function for alternating subsequences.

We consider a discrete-time model for random interface growth which converges to the Polynuclear growth model in a particular limit. The height of the interface is initially flat and the evolution involves the addition of islands of height one according to a Poisson point process of nucleation events. The boundaries of these islands then spread in a stochastic manner, rather than at deterministic speed as in the Polynuclear growth model. The one-point distribution and multi-time distributions agree with point-to-line last passage percolation times in a geometric environment. An alternative interpretation for the growth model can be given through interacting particle systems experiencing pushing and blocking interactions.

Let ${\mathrm{\Lambda }}^{\ast }$ be the rate function in the large deviation principle for the sums $X_1+\cdots +X_n$ of independent identically distributed random variables $X_1,X_2,\dots$. It is shown that ${\mathrm{\Lambda }}^{\ast }(x)\sim -ln\mathsf{P}(X_1\ge x)$ (as $x\to \mathrm{\infty }$) if and only if $ln\mathsf{P}(X_1\ge x)\sim L_0(x)$ for some concave function $L_0$. The main ingredient of the proof is the general, explicit expression of a suitable quasi-minimizer in $t\ge 0$ of the Bernstein–Chernoff upper bound $e^{-tx}\mathsf{E}{e}^{t{X}_1}$ on $\mathsf{P}(X_1\ge x)$, which is amenable to analysis and, at the same time, is close enough to a true minimizer.

In this note, we establish a qualitative total variation version of Breuer–Major Central Limit Theorem for a sequence of the type $\frac{1}{\sqrt{n}}{\sum }_{1\le k\le n}f(X_k)$, where ${(X_k)}_{k\ge 1}$ is a centered stationary Gaussian process, under the hypothesis that the function f has Hermite rank $d\ge 1$ and belongs to the Malliavin space ${\mathbb{D}}^{1,2}$. This result in particular extends the recent works of [NNP21], where a quantitative version of this result was obtained under the assumption that the function f has Hermite rank $d=2$ and belongs to the Malliavin space ${\mathbb{D}}^{1,4}$. We thus weaken the ${\mathbb{D}}^{1,4}$ integrability assumption to ${\mathbb{D}}^{1,2}$ and remove the restriction on the Hermite rank of the base function. While our method is still based on Malliavin calculus, we exploit a particular instance of Malliavin gradient called the sharp operator, which reduces the desired convergence in total variation to the convergence in distribution of a bidimensional Breuer–Major type sequence.

The spine of the two-particle Fleming-Viot process driven by Brownian motion is not a Bessel-3 process.

We aim to set forth an extension of the result found in paper [6], which finds an explicit realisation of a reflecting Brownian motion with drift $-\mathrm{\mu }$, started at x, reflecting above zero, and its local time at zero. In this paper we find a corresponding realisation for a reflecting Brownian motion with drift $-\mathrm{\mu }$, started at x, reflected both above zero and below one, along with a corresponding expression in terms of associated local times, namely as the difference between the local time at zero and the local time at one.

We establish a central limit theorem for the eigenvalue counting function of a matrix of real Gaussian random variables.

Let $(\mathrm{\Omega },\mathcal{F})$ be a measurable space and $E\subset \mathrm{\Omega }\times \mathrm{\Omega }$. Suppose that $E\in \mathcal{F}\otimes \mathcal{F}$ and the relation on Ω defined as $x\sim y$⇔ $(x,y)\in E$ is reflexive, symmetric and transitive. Following [7], say that E is strongly dualizable if there is a sub-σ-field $\mathcal{G}\subset \mathcal{F}$ such that

$$\underset{P\in \mathrm{\Gamma }(\mathrm{\mu },\mathrm{\nu })}{min}(1-P(E))=\underset{A\in \mathcal{G}}{max}|\mathrm{\mu }(A)-\mathrm{\nu }(A)|$$

for all probabilities μ and ν on $\mathcal{F}$. This paper investigates strong duality. Essentially, it is shown that E is strongly dualizable provided some mild modifications are admitted. Let ${\mathcal{G}}_0$ be the E-invariant sub-σ-field of $\mathcal{F}$. One result is that, for all probabilities μ and ν on $\mathcal{F}$, there is a probability ${\mathrm{\nu }}_0$ on $\mathcal{F}$ such that

$${\mathrm{\nu }}_0=\mathrm{\nu }\text{on}{\mathcal{G}}_0\text{and}\underset{P\in \mathrm{\Gamma }(\mathrm{\mu },{\mathrm{\nu }}_0)}{min}(1-P(E))=\underset{A\in {\mathcal{G}}_0}{max}|\mathrm{\mu }(A)-\mathrm{\nu }(A)|.$$

In the other results, $(\mathrm{\Omega },\mathcal{F})$ is a standard Borel space and the min over $\mathrm{\Gamma }(\mathrm{\mu },\mathrm{\nu })$ is replaced by the inf over $\mathrm{\Gamma }(\mathrm{\mu },\mathrm{\nu })$ in the definition of strong duality. Then, E is strongly dualizable provided $\mathcal{G}$ is allowed to depend on $(\mathrm{\mu },\mathrm{\nu })$ or it is taken to be the universally measurable version of the E-invariant σ-field.

We consider the Sherrington-Kirkpatrick model with no external field and inverse temperature $\mathit{\beta }<1$ and prove that the expected operator norm of the covariance matrix of the Gibbs measure is bounded by a constant depending only on β. This answers an open question raised by Talagrand, who proved a bound of $C(\mathit{\beta }){(logn)}^8$. Our result follows by establishing an approximate formula for the covariance matrix which we obtain by differentiating the TAP equations and then optimally controlling the associated error terms. We complement this result by showing diverging lower bounds on the operator norm, both at the critical and low temperatures.

In this paper, we obtain the exact asymptotic behavior of Green functions of homogeneous random walks in ${\mathbb{Z}}^d$ killed at the first exit from and open cone of ${\mathbb{R}}^d$. Our approach combines methods of functional equations, integral representations of the Green function and Woess’ approach for the case of homogeneous random walks in ${\mathbb{Z}}^d$.