Related papers: Weak stability and generalized weak convolution fo…

Weak values are usually associated with weak measurements of an observable on a pre- and post-selected ensemble. We show that more generally, weak values are proportional to the correlation between two pointers in a successive measurement.…

In quantum theory, a weak value is a complex number with a somewhat technical definition: it is a ratio whose numerator is the matrix element of a self-adjoint operator and whose denominator is the inner product of a corresponding pair of…

Algorithmic stability is a key characteristic to ensure the generalization ability of a learning algorithm. Among different notions of stability, \emph{uniform stability} is arguably the most popular one, which yields exponential…

For a random variable $N = 0, 1, 2, \ldots$ we study the following question: When does the sum of $N$ many independent and identically distributed copies of a random variable $X$ have the same law a a nontrivial rescaling of $X$? We show…

For each $n$, let $U_n$ be Haar distributed on the group of $n\times n$ unitary matrices. Let $\bfx_{n,1},\ldots,\bfx_{n,m} $ denote orthogonal nonrandom unit vectors in ${\Bbb C}^n$ and let $\text{\bf…

In this paper we consider weak Harnack inequality and H\"older regularity estimates for symmetric $\alpha$-stable L\'evy process in $\mathbb{R}^d$, $\alpha \in (0,2)$, $d\geq 2$. We consider a symmetric $\alpha$-stable L\'evy process $X$…

This article studies the \emph{robust covariance matrix estimation} of a data collection $X = (x_1,\ldots,x_n)$ with $x_i = \sqrt \tau_i z_i + m$, where $z_i \in \mathbb R^p$ is a \textit{concentrated vector} (e.g., an elliptical random…

Let $a$ be a finite signed measure on $[-r, 0]$ with $r \in (0, \infty)$. Consider a stochastic process $(X^{(\vartheta)}(t))_{t\in[-r,\infty)}$ given by a linear stochastic delay differential equation \[ \mathrm{d} X^{(\vartheta)}(t) =…

It is well known that any projective measurement can be decomposed into a sequence of weak measurements, which cause only small changes to the state. Similar constructions for generalized measurements, however, have relied on the use of an…

We consider a mixed moving average (MMA) process X driven by a L\'evy basis and prove that it is weakly dependent with rates computable in terms of the moving average kernel and the characteristic quadruple of the L\'evy basis. Using this…

Let $X_1, X_2,\ldots, X_n$ (resp. $Y_1, Y_2,\ldots, Y_n$) be independent random variables such that $X_i$ (resp. $Y_i$) follows generalized exponential distribution with shape parameter $\theta_i$ and scale parameter $\lambda_i$ (resp.…

New weak and strong existence and weak and strong uniqueness results for multi-dimensional stochastic McKean--Vlasov equations are established under relaxed regularity conditions. Weak existence is a variation of Krylov's weak existence for…

Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter…

We consider several integrable systems from a standpoint of the SL(2,R) invariant gauge theory. In the Drinfeld-Sokorov gauge, we get a one parameter family of nonlinear equations from zero curvature conditions. For each value of the…

We present a Lyapunov type approach to the problem of existence and uniqueness of general law-dependent stochastic differential equations. In the existing literature most results concerning existence and uniqueness are obtained under…

Consider a weakly nonlinear CGL equation on the torus~$\mathbb{T}^d$: \[u_t+i\Delta u=\epsilon [\mu(-1)^{m-1}\Delta^{m} u+b|u|^{2p}u+ ic|u|^{2q}u].\eqno{(*)}\] Here $u=u(t,x)$, $x\in\mathbb{T}^d$, $0<\epsilon<<1$, $\mu\geqslant0$,…

In this paper we consider the existence of weakly c\`adl\`ag versions of a solution to a linear equation in a Hilbert space $H$, driven by a Levy process taking values in a Hilbert space $U$. In particular we are interested in diagonal type…

In a weak measurement, the average output $\langle o\rangle$ of a probe that measures an observable $\hat{A}$ of a quantum system undergoing both a preparation in a state $\rho_i$ and a postselection in a state $E_\mathrm{f}$ is, to a good…

The weak values and weak measurement formalism were initially limited to pure states, which were later extended to mixed states, leading to intriguing applications in quantum information processing tasks. Weak values are considered to be…

Renormalization group theory does not restrict the from of continuous variation of critical exponents which occurs in presence of a marginal operator. However, the continuous variation of critical exponents, observed in different contexts,…