arXiv subjects
Publications and source records attributed to Alexander Shaposhnikov.
We present self-contained martingale proofs of Talagrand's convolution conjecture in the Gaussian and Boolean settings with explicit dimension-free constants. The argument is based on the analysis of the stopped density martingale and a weighted It\^o isometry estimate. The Gaussian case was settled by Eldan and Lee, and Lehec; the Boolean case was proved up to a $\log\log$ factor by Chen and recently settled by Lu, Guo and Fang.
We present a versatile framework to study strong existence and uniqueness for stochastic differential equations (SDEs) in Hilbert spaces with irregular drift. We consider an SDE in a separable Hilbert space $H$ \begin{equation*} dX_t= (A X_t + b(X_t))dt +(-A)^{-\gamma/2}dW_t,\quad X_0=x_0 \in H, \end{equation*} where $A$ is a self-adjoint negative definite operator with purely atomic spectrum, $W$ is a cylindrical Wiener process, $b$ is $\alpha$-H\"older continuous function $H\to H$, and a nonnegative parameter $\gamma$ such that the stochastic convolution takes values in $H$. We show that this equation has a unique strong solution provided that $\alpha > \alpha^*(\gamma)$, with an explicit function $\alpha^*$ that takes values in $(0,1)$ for all $\gamma\in[0,3)$. This substantially extends the seminal work of Da Prato and Flandoli (2010) as no structural assumption on $b$ is imposed. The range of admissible $\alpha$ is also extended. To obtain this result, we do not use infinite-dimensional Kolmogorov equations but instead develop a new technique combining L\^e's theory of stochastic sewing in Hilbert spaces, Gaussian analysis, and a method of Lasry and Lions for approximation in Hilbert spaces.
We construct a series of stochastic differential equations of the form $dX_t = b(t, X_t) dt + dB_t$ which exhibit nonuniqueness in the path-by-path sense while having a unique adapted solution in the sense of stochastic processes, i.e. pathwise uniqueness holds.
We establish new approximation results in the sense of Lusin for Sobolev functions $f$ with $|\nabla f| \in L\log L$ on infinite-dimensional spaces equipped with Gaussian measures. The proof relies on some new pointwise estimate for the approximations based on the corresponding semigroup which can be of independent interest.
We present a new approach to Davie's theorem on the uniqueness of solutions to the equation $dX_t = b(t, X_t)\,dt + dW_t$ for almost all Brownian paths. A generalization of this result and a discussion of some close problems are given.