Max-convolution semigroups and extreme values in limit theorems for the free multiplicative convolution

Ergodicity of the 2D Navier-Stokes Equations with Degenerate Multiplicative Noise

Consider the two-dimensional, incompressible Navier-Stokes equations on the torus We prove that the semigroup P_t generated by the solutions to stochastic Navier-stokes equations is asymptotically strong Feller. Moreover, we also prove that semigroup P_t is exponentially ergodic in some sense

On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution

Let M denote the space of Borel probability measures on the real line. For every nonnegative t we consider the transformation $\mathbb B_t : M \to M$ defined for any given element in M by taking succesively the the (1+t) power with respect to free additive convolution and then the 1/(1+t) power with respect to Boolean convolution of the given element. We show that the family of maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the operation of composition and that, quite surprisingly, every $\mathbb B_t$ is a homomorphism for the operation of free multiplicative convolution. We prove t…

连续流驱动的 Markov 过程与乘性 $L^p$-半群

Continuous flows driving Markov processes and multiplicative $L^p$-semigroups

本文提出通过连续流驱动 Markov 过程的方法，研究生成元可交换时一阶算子的添加，证明流的连续性并展示 Lp 空间上的 Markov 乘性半群由连续流生成，同时将弱生成元与鞅问题扩展至无界函数。

Gradient estimates for SDEs Driven by Multiplicative Lévy Noise

Gradient estimates are derived, for the first time, for the semigroup associated to a class of stochastic differential equations driven by multiplicative Lévy noise. In particular, the estimates are sharp for $α$-stable type noises. To derive these estimates, a new derivative formula of Bismut-Elworthy-Li's type is established for the semigroup by using the Malliavin calculus and a finite-jump approximation argument.

Semigroups for One-Dimensional Schrödinger Operators with Multiplicative Gaussian Noise

Let $ H:=-\tfrac12Δ+V$ be a one-dimensional continuum Schrödinger operator. Consider ${\hat H}:= H+ξ$, where $ξ$ is a translation invariant Gaussian noise. Under some assumptions on $ξ$, we prove that if $V$ is locally integrable, bounded below, and grows faster than $\log$ at infinity, then the semigroup $\mathrm e^{-t {\hat H}}$ is trace class and admits a probabilistic representation via a Feynman-Kac formula. Our result applies to operators acting on the whole line $\mathbb R$, the half line $(0,\infty)$, or a bounded interval $(0,b)$, with a variety of boundary conditions. Our method of p…

Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Dini Continuous Drift

Consider the stochastic evolution equation in a separable Hilbert space with a nice multiplicative noise and a locally Dini continuous drift. We prove that for any initial data the equation has a unique (possibly explosive) mild solution. Under a reasonable condition ensuring the non-explosion of the solution, the strong Feller property of the associated Markov semigroup is proved. Gradient estimates and log-Harnack inequalities are derived for the associated semigroup under certain global conditions, which are new even in finite-dimensions.

The Arithmetic of Distributions in Free Probability Theory

We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and of Schur functions. We consider the set of probability distributions as a semigroup $\bold M$ equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of $\bold M$ contains either indecomposable ("prime") factors or it belongs to a class, say $I_0$, of distributions without indecomposable factors. In contrast to the classical convolution semigroup in the free ad…

Harnack Inequality for Semilinear spdes with multiplicative noise

By a new approximate method, dimensional free Harnack inequalities are established for a class of semilinear stochastic differential equations in Hilbert space with multiplicative noise. These inequalities are applied to study the strong Feller property for the semigroup and some properties of invariant measure.

Multiplicative monotone convolutions

Recently, Bercovici has introduced multiplicative convolutions based on Muraki's monotone independence and shown that these convolution of probability measures correspond to the composition of some function of their Cauchy transforms. We provide a new proof of this fact based on the combinatorics of moments. We also give a new characterisation of the probability measures that can be embedded into continuous monotone convolution semigroups of probability measures on the unit circle and briefly discuss a relation to Galton-Watson processes.

具乘性噪声的非局部随机Swift-Hohenberg方程随机吸引子的上半连续性及不变测度的存在性

Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise

本文研究了具乘性噪声的2D非局部随机Swift-Hohenberg方程的长时间动力学行为，利用解析半群理论证明了随机吸引子的上半连续性，并推广随机Gronwall引理以建立遍历不变测度的存在性。

The exponential map in non-commutative probability

The wrapping transformation $W$ is a homomorphism from the semigroup of probability measures on the real line, with the convolution operation, to the semigroup of probability measures on the circle, with the multiplicative convolution operation. We show that on a large class $\mathcal{L}$ of measures, $W$ also transforms the three non-commutative convolutions---free, Boolean, and monotone---to their multiplicative counterparts. Moreover, the restriction of $W$ to $\mathcal{L}$ preserves various qualitative properties of measures and triangular arrays. We use these facts to give short proofs of…

自由乘法卷积与实线上任意测度

Free multiplicative convolution with an arbitrary measure on the real line

本文研究实线上任意测度与非负实线上任意测度的自由乘法卷积，通过构造子ordination函数和S变换，证明了S变换的乘法性，并应用于稳定分布的卷积恒等式，扩展了Belinschi-Nica半群并建立了自由乘法卷积的正则性性质。

Limit Theorems and Wrapping Transforms in Bi-free Probability Theory

In this paper, we characterize idempotent distributions with respect to the bi-free multiplicative convolution on the bi-torus. Also, the bi-free analogous Levy triplet of an infinitely divisible distribution on the bi-torus without non-trivial idempotent factors is obtained. This triplet is unique and generates a homomorphism from the bi-free multiplicative semigroup of infinitely divisible distributions to the classical one. The relevances of the limit theorems associated with four convolutions, classical and bi-free additive convolutions and classical and bi-free multiplicative convolutions…