Covariance of stochastic integrals with respect to fractional Brownian motion

We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a $d$-dimensional fractional Brownian motion (fBm) $B_t$ with Hurst parameter $H>\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, where the integrands are vector fields applied to $B_t$. It provides, for example, a direct alternative proof of Y. Hu and D. Nualart’s result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion.     

A regularity theory for quasi-linear Stochastic PDEs in weighted Sobolev spaces

We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on $C^1$-domains. The coefficients are random functions depending on $t,x$ and the unknown solutions. We prove the uniqueness and existence of solutions in appropriate Sobolev spaces, and in addition, we obtain $L_p$ and Hölder estimates of both the solution and its gradient.     

Distribution dependent SDEs for Landau type equations

The distribution dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated with a DDSDE solves a nonlinear PDE. Due to the distribution dependence, some standard techniques developed for SDEs do not apply. By iterating in distributions, a strong solution is constructed using SDEs with control. By proving the uniqueness, the distribution of solutions is identified with a nonlinear semigroup $P_t^1$ on the space of probability measures. The exponential contraction as well as Harnack inequalities and applications are investigated for the nonlinear semigroup $P_t^1$ using coupling by change of measures. The main results are illustrated by homogeneous Landau equations.     

A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton–Watson process and the migration of particles by a simple random walk in ${\mathbb{Z}}^d$. Denote by $Z_n\left(z\right)$ the number of particles of generation $n$ located at site $z\in {\mathbb{Z}}^d$. We give the second order asymptotic expansion for $Z_n\left(z\right)$. The higher order expansion can be derived by using our method here. As a by-product, we give the second order expansion for a simple random walk on ${\mathbb{Z}}^d$, which is used in the proof of the main theorem and is of independent interest.     

Dynamic uniqueness for stochastic chains with unbounded memory

We say that a probability kernel exhibits dynamic uniqueness (DU) if all the stochastic chains starting from a fixed past coincide on the future tail $\sigma$-algebra. Our first theorem is a set of properties that are pairwise equivalent to DU which allow us to understand how it compares to other more classical concepts. In particular, we prove that DU is equivalent to a weak-${?}^2$ summability condition on the kernel. As a corollary to this theorem, we prove that the Bramson–Kalikow and the long-range Ising models both exhibit DU if and only if their kernels are ${?}^2$ summable. Finally, if we weaken the condition for DU, asking for coincidence on the future $\sigma$-algebra for almost every pair of pasts, we obtain a condition that is equivalent to $\beta$-mixing (weak-Bernoullicity) of the compatible stationary chain. As a consequence, we show that a modification of the weak-${?}^2$ summability condition on the kernel is equivalent to the $\beta$-mixing of the compatible stationary chain.     

Stable Ulrich bundles on Fano threefolds with Picard number 2

In this paper, we consider the existence problem of rank one and two stable Ulrich bundles on imprimitive Fano 3-folds obtained by blowing-up one of ${\mathbb{P}}^3$, Q (smooth quadric in ${\mathbb{P}}^4$), $V_3$ (smooth cubic in ${\mathbb{P}}^4$) or $V_4$ (complete intersection of two quadrics in ${\mathbb{P}}^5$) along a smooth irreducible curve. We prove that the only class which admits Ulrich line bundles is the one obtained by blowing up a genus 3, degree 6 curve in ${\mathbb{P}}^3$. Also, we prove that there exist stable rank two Ulrich bundles with $c_1=3H$ on a generic member of this deformation class.     

Global fluctuations for 1D log-gas dynamics

We study in this article the hydrodynamic limit in themacroscopic regime of the coupled system of stochastic differential equations,

(0.1)$d{\lambda }_t^i=\frac{1}{\sqrt{N}}d{W}_t^i?V^{\prime }\left({\lambda }_t^i\right)dt+\frac{\beta }{2N}\sum _{j\ne i}\frac{dt}{{\lambda }_t^i?{\lambda }_t^j},i=1,\dots ,N,$

with $\beta >1$, sometimes called generalized Dyson’s Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $\beta$-ensemble, with sufficiently regular convex potential $V$. The limit $N\to \infty$ is known to satisfy a mean-field Mc-Kean–Vlasov equation. We prove that, for suitable initial conditions, fluctuations around the limit are Gaussian and satisfy an explicit PDE.The proof is very much indebted to the harmonic potential case treated in Israelsson (2001). Our key argument consists in showing that the time-evolution generator may be written in the form of a transport operator on the upper half-plane, plus a bounded non-local operator interpreted in terms of a signed jump process.     

Lp solutions of reflected BSDEs under monotonicity condition

We prove existence and uniqueness of ${\mathbb{L}}^p$ solutions, $p\in [1,2]$, of reflected backward stochastic differential equations with $p$-integrable data and generators satisfying the monotonicity condition. We also show that the solution may be approximated by the penalization method. Our results are new even in the classical case $p=2$.     

Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space

We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak–Orlicz space avoiding growth restrictions. Namely, we consider

$?\mathrm{div}A(x,\mathrm{?}u)=f\in L^1(\mathrm{\Omega }),$

on a Lipschitz bounded domain in ${\mathbb{R}}^N$. The growth of the monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N-function M. The approach does not require any particular type of growth condition of M or its conjugate $M^{?}$ (neither ${\mathrm{\Delta }}_2$, nor ${\mathrm{?}}_2$). The condition we impose is log-Hölder continuity of M, which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.     

A fully stochastic approach to limit theorems for iterates of Bernstein operators

This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator $B_n$ taking a continuous function $f\in C[0,1]$ to a degree-$n$ polynomial when the number of iterations $k$ tends to infinity and $n$ is kept fixed or when $n$ tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright–Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright–Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of $B_n$ a number of times $k=k\left(n\right)$ to a polynomial $f$ when $k\left(n\right)∕n$ tends to a constant.     

On the Komlós,Major and Tusnády strong approximation for some classes of random iterates

The famous results of Komlós, Major and Tusnády (see Komlós et al., 1976 [15] and Major, 1976 [17]) state that it is possible to approximate almost surely the partial sums of size $n$ of i.i.d. centered random variables in ${\mathbb{L}}^p$ ($p>2$) by a Wiener process with an error term of order $o\left(n^{1∕p}\right)$. Very recently, Berkes et al. (2014) extended this famous result to partial sums associated with functions of an i.i.d. sequence, provided a condition on a functional dependence measure in ${\mathbb{L}}_p$ is satisfied. In this paper, we adapt the method of Berkes, Liu and Wu to partial sums of functions of random iterates. Taking advantage of the Markovian setting, we shall give new dependent conditions, expressed in terms of a natural coupling (in ${\mathbb{L}}^{\infty }$ or in ${\mathbb{L}}^1$), under which the strong approximation result holds with rate $o\left(n^{1∕p}\right)$. As we shall see our conditions are well adapted to a large variety of models, including left random walks on $G{L}_d\left(\mathbb{R}\right)$, contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains. We also provide some examples showing that our ${\mathbb{L}}^1$-coupling condition is in some sense optimal.     

Systems of stochastic Poisson equations: Hitting probabilities

We consider a $d$-dimensional random field $u=(u\left(x\right),x\in D)$ that solves a system of elliptic stochastic equations on a bounded domain $D?{\mathbb{R}}^k$, with additive white noise and spatial dimension $k=1,2,3$. Properties of $u$ and its probability law are proved. For Gaussian solutions, using results from Dalang and Sanz-Solé (2009), we establish upper and lower bounds on hitting probabilities in terms of the Hausdorff measure and Bessel–Riesz capacity, respectively. This relies on precise estimates of the canonical distance of the process or, equivalently, on $L^2$ estimates of increments of the Green function of the Laplace equation.     

Effect of stochastic perturbations for front propagation in Kolmogorov Petrovskii Piscunov equations

This article considers equations of Kolmogorov Petrovskii Piscunov type in one space dimension, with stochastic perturbation:

where the stochastic differential is taken in the sense of Itô and $\zeta$ is a Gaussian random field satisfying $\mathbb{E}\left[\zeta \right]=0$ and $\mathbb{E}\left[\zeta (s,x)\zeta (t,y)\right]=(s\wedge t)\Gamma (x?y)$. Two situations are considered: firstly, $\zeta$ is simply a standard Wiener process (i.e. $\Gamma \equiv 1$): secondly, $\Gamma \in C^{\infty }\left(\mathbb{R}\right)$ with ${lim}_{\left|z\right|\to +\infty }\left|\Gamma \left(z\right)\right|=0$.The results are as follows: in the first situation (standard Wiener process: i.e. $\Gamma \left(x\right)\equiv 1$), there is a non-degenerate travelling wave front if and only if $\frac{{?}^2}{2}<1$, with asymptotic wave speed $max\left(\sqrt{2\kappa (1?\frac{{?}^2}{2})},\left(\frac{1}{N}(1?\frac{{?}^2}{2})+\frac{\kappa N}{2}\right){\mathbf{1}}_{\{N<\sqrt{\frac{2}{\kappa }(1?\frac{{?}^2}{2})}\}}\right)$; the noise slows the wave speed. If the stochastic integral is taken instead in the sense of Stratonovich, then the asymptotic wave speed is the classical McKean wave speed and does not depend on $?$.In the second situation (noise with spatial covariance which decays to 0 at $\pm \infty$, stochastic integral taken in the sense of Itô), a travelling front can be defined for all $?>0$. Its average asymptotic speed does not depend on $?$ and is the classical wave speed of the unperturbed KPP equation.     

Stochastic Komatu–Loewner evolutions and BMD domain constant

For Komatu–Loewner equation on a standard slit domain, we randomize the Jordan arc in a manner similar to that of Schramm (2000) to find the SDEs satisfied by the induced motion $\xi \left(t\right)$ on $?\mathbb{H}$ and the slit motion $\mathbf{s}\left(t\right)$. The diffusion coefficient $\alpha$ and drift coefficient $b$ of such SDEs are homogeneous functions.Next with solutions of such SDEs, we study the corresponding stochastic Komatu–Loewner evolution, denoted as ${\mathrm{SKLE}}_{\alpha ,b}$. We introduce a function $b_{\mathrm{BMD}}$ measuring the discrepancy of a standard slit domain from $\mathbb{H}$ relative to BMD. We show that ${\mathrm{SKLE}}_{\sqrt{6},?b_{\mathrm{BMD}}}$ enjoys a locality property.     

Discretizing Malliavin calculus

Suppose $B$ is a Brownian motion and $B^n$ is an approximating sequence of rescaled random walks on the same probability space converging to $B$ pointwise in probability. We provide necessary and sufficient conditions for weak and strong $L^2$-convergence of a discretized Malliavin derivative, a discrete Skorokhod integral, and discrete analogues of the Clark–Ocone derivative to their continuous counterparts. Moreover, given a sequence $\left(X^n\right)$ of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to $B^n$, we derive necessary and sufficient conditions for strong $L^2$-convergence to a $\sigma \left(B\right)$-measurable random variable $X$ via convergence of the discrete chaos coefficients of $X^n$ to the continuous chaos coefficients.