It is known that real Non-Gaussian stable distributions are unimodal, not additive strongly unimodal, and multiplicative strongly unimodal in the symmetric case. By a theorem of Cuculescu-Theodorescu, the only remaining relevant situation for the multiplicative strong unimodality of stable laws is the one-sided. In this paper, we show that positive $α-$stable laws are multiplicative strongly unimodal iff $α\le 1/2.$
We prove that the marginal law $σ_{t}\boxtimesν$ of free positive multiplicative Brownian motion is log-unimodal for all $t>0$ if $ν$ is a multiplicatively symmetric log-unimodal distribution, and that $σ_{t}\boxtimesν$ is log-unimodal for sufficiently large $t$ if $ν$ is supported on a suitably chosen finite interval. Counterexamples are given when $ν$ is not assumed to be symmetric or having a bounded support.
We study unimodality for free multiplicative convolution with free normal distributions $\{λ_t\}_{t>0}$ on the unit circle. We give four results on unimodality for $μ\boxtimesλ_t$: (1) if $μ$ is a symmetric unimodal distribution on the unit circle then so is $μ\boxtimes λ_t$ at any time $t>0$; (2) if $μ$ is a symmetric distribution on $\mathbb{T}$ supported on $\{e^{iθ}: θ\in [-φ,φ]\}$ for some $φ\in (0,π/2)$, then $μ\boxtimes λ_t$ is unimodal for sufficiently large $t>0$; (3) ${\bf b} \boxtimes λ_t$ is not unimodal at any time $t>0$, where ${\bf b}$ is the equally weighted Bernoulli distribut…
We show that any freely selfdecomposable probability law is unimodal. This is the free probabilistic analog of Yamazato's result in [Ann. Probab. 6 (1978), 523-531].
We show that the hitting times for points of real $α-$stable Lévy processes ($1<α\le 2$) are unimodal random variables. The argument relies on strong unimodality and several recent multiplicative identities in law. In the symmetric case we use a factorization of Yano et al., whereas in the completely asymmetric case we apply an identity of the second author. The method extends to the general case thanks to a fractional moment evaluation due to Kuznetsov et al.
We proved that there exists a unique invariant measure for solutions of stochastic conservation laws with Dirichlet boundary condition driven by multiplicative noise. Moreover, a polynomial mixing property is established. This is done in the setting of kinetic solutions taking values in an L^1-weighted space.
We determine the rate of decay of the expectation Z(t) of some multiplicative functional related to Brownian motion up to time t. This permits to prove that the Wiener measure, penalized by this multiplicative functional, converges as t goes to infinity to a probability measure (p.m.) . We obtain the law of the canonical process under this new p.m.
We study Boolean stable laws, $\mathbf{b}_{α,ρ}$, with stability index $α$ and asymmetry parameter $ρ$. We show that the classical scale mixture of $\mathbf{b}_{α,ρ}$ coincides with a free mixture and also a monotone mixture of $\mathbf{b}_{α,ρ}$. For this purpose we define the multiplicative monotone convolution of probability measures, one is supported on the positive real line and the other is arbitrary. We prove that any scale mixture of $\mathbf{b}_{α,ρ}$ is both classically and freely infinitely divisible for $α\leq1/2$ and also for some $α>1/2$. Furthermore, we show the multiplicative i…
We consider different limit theorems for additive and multiplicative free Lévy processes. The main results are concerned with positive and unitary multiplicative free Lévy processes at small time, showing convergence to log free stable laws for many examples. The additive case is much easier, and we establish the convergence at small or large time to free stable laws. During the investigation we found out that a log free stable law with index $1$ coincides with the Dykema-Haagerup distribution. We also consider limit theorems for positive multiplicative Boolean Lévy processes at small time, ob…
Convergence of the Finite Volume Method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation
Aix-Marseille Université, CNRS, Centrale Marseille, I2M（艾克斯-马赛大学）；Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan（里昂大学，克莱贝尔·里昂第一大学）
We consider a general multidimensional affine recursion with corresponding Markov operator $P$ and a unique $P$-stationary measure. We show spectral gap properties on Hölder spaces for the corresponding Fourier operators and we deduce convergence to stable laws for the Birkhoff sums along the recursion. The parameters of the stable laws are expressed in terms of basic quantities depending essentially on the matricial multiplicative part of $P$. Spectral gap properties of $P$ and homogeneity at infinity of the $P$-stationary measure play an important role in the proofs.
We develop a pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, a special case being stochastic conservation laws with quasilinear stochastic dependence. We introduce the notion of pathwise stochastic entropy solutions, which is closed with the local uniform limits of paths, and prove that it is well posed, i.e., we establish existence, uniqueness and continuous dependence, in the form of pathwise $L^1$-contraction, as well as some explicit estimates. Our approach is motivated by the theory of stochastic viscosity solutions, which was introduced…
We completely determine the free infinite divisibility for the Boolean stable law which is parametrized by a stability index $α$ and an asymmetry coefficient $ρ$. We prove that the Boolean stable law is freely infinitely divisible if and only if one of the following conditions holds: $0<α\leq\frac{1}{2}$; $\frac{1}{2}<α\leq\frac{2}{3}$ and $2-\frac{1}α\leqρ\leq \frac{1}α-1$; $α=1,~ρ=\frac{1}{2}$. Positive Boolean stable laws corresponding to $ρ=1$ and $α\leq \frac{1}{2}$ have completely monotonic densities and they are both freely and classically infinitely divisible. We also show that continu…
We continue the development of the theory of pathwise stochastic entropy solutions for scalar conservation laws in $\R^N$ with quasilinear multiplicative ''rough path'' dependence by considering inhomogeneous fluxes and a single rough path like, for example, a Brownian motion. Following our previous note where we considered spatially independent fluxes, we introduce the notion of pathwise stochastic entropy solutions and prove that it is well posed, that is we establish existence, uniqueness and continuous dependence in the form of a (pathwise) $L^1$-contraction. Our approach is motivated by t…
Let $\{T_{k}\}_{k=1}^{\infty}$ be a family of *--free identically distributed operators in a finite von Neumann algebra. In this work we prove a multiplicative version of the free central limit Theorem. More precisely, let $B_{n}=T_{1}^{*}T_{2}^{*}... T_{n}^{*}T_{n}... T_{2}T_{1}$ then $B_{n}$ is a positive operator and $B_{n}^{1/2n}$ converges in distribution to an operator $Λ$. We completely determine the probability distribution $ν$ of $Λ$ from the distribution $μ$ of $|T|^{2}$. This gives us a natural map $\mathcal{G}:\mathcal{M_{+}}\to \mathcal{M_{+}}$ with $μ\mapsto \mathcal{G}(μ)=ν.$ We…
Let $S$ be the multiplicative semigroup of $q\times q$ matrices with positive entries such that every row and every column contains a strictly positive element. Denote by $(X_n)_{n\geq1}$ a sequence of independent identically distributed random variables in $S$ and by $X^{(n)} = X_n ... X_1$, $ n\geq 1$, the associated left random walk on $S$. We assume that $(X_n)_{n\geq1}$ verifies the contraction property $¶(\bigcup_{n\geq1}[X^{(n)} \in S^\circ])>0$, where $S^\circ $ is the subset of all matrices which have strictly positive entries. We state conditions on the distribution of the random mat…
This study considers a model of the income distribution of agents whose pairwise interaction is asymmetric and price-invariant. Asymmetric transactions are typical for chain-trading groups who arrange their business such that commodities move from senior to junior partners and money moves in the opposite direction. The price-invariance of transactions means that the probability of a pairwise interaction is a function of the ratio of incomes, which is independent of the price scale or absolute income level. These two features characterize the hierarchical model. The income distribution in this …
