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.     

Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains

In this paper, we study nonparametric surfaces over strictly convex bounded domains in ${\mathbb{R}}^n$, which are evolving by the mean curvature flow with Neumann boundary value. We prove that solutions converge to the ones moving only by translation. And we will prove the existence and uniqueness of the constant mean curvature equation with Neumann boundary value on strictly convex bounded domains.     

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.     

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.     

Large deviations for generalized Polya urns with arbitrary urn function

We consider a generalized two-color Polya urn (black and white balls) first introduced by Hill et al. (1980), where the urn composition evolves as follows: let $\pi :\left[0,1\right]\to \left[0,1\right]$, and denote by $x_n$ the fraction of black balls after step $n$, then at step $n+1$ a black ball is added with probability $\pi \left(x_n\right)$ and a white ball is added with probability $1?\pi \left(x_n\right)$. Originally introduced to mimic attachment under imperfect information, this model has found applications in many fields, ranging from Market Share modeling to polymer physics and biology.In this work we discuss large deviations for a wide class of continuous urn functions $\pi$. In particular, we prove that this process satisfies a Sample-Path Large Deviations principle, also providing a variational representation for the rate function. Then, we derive a variational representation for the limit

$?\left(s\right)=\underset{n\to \infty }{lim}\frac{1}{n}log\mathbb{P}\left(n{x}_n=?sn?\right),s\in \left[0,1\right],$

where $n{x}_n$ is the number of black balls at time $n$, and use it to give some insight on the shape of $?\left(s\right)$. Under suitable assumptions on $\pi$ we are able to identify the optimal trajectory. We also find a non-linear Cauchy problem for the Cumulant Generating Function and provide an explicit analysis for some selected examples. In particular we discuss the linear case, which embeds the Bagchi–Pal Model [6], giving the exact implicit expression for $?$ in terms of the Cumulant Generating Function.     

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.     

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.     

On purely discontinuous additive functionals of subordinate Brownian motions

Let $A_t={\sum }_{s\le t}F(X_{s?},X_s)$ be a purely discontinuous additive functional of a subordinate Brownian motion $X=(X_t,{\mathbb{P}}_x)$. We give a sufficient condition on the non-negative function $F$ that guarantees that finiteness of $A_{\infty }$ implies finiteness of its expectation. This result is then applied to study the relative entropy of ${\mathbb{P}}_x$ and the probability measure induced by a purely discontinuous Girsanov transform of the process $X$. We prove these results under the weak global scaling condition on the Laplace exponent of the underlying subordinator.     

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.     

Small noise and long time phase diffusion in stochastic limit cycle oscillators

We study the effect of additive Brownian noise on an ODE system that has a stable hyperbolic limit cycle, for initial data that are attracted to the limit cycle. The analysis is performed in the limit of small noise – that is, we modulate the noise by a factor $\epsilon \searrow 0$ – and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and the limit cycle up to times $\exp ?\left(c{\epsilon }^{?2}\right)$, $c>0$, and we show both that on the time scale ${\epsilon }^{?2}$ the dephasing (i.e., the difference between noiseless and noisy system measured in a natural coordinate system that involves a phase) is close to a Brownian motion with constant drift, and that on longer time scales the dephasing dynamics is dominated by the drift. The natural choice of coordinates, that reduces the dynamics in a neighborhood of the cycle to a rotation, plays a central role and makes the connection with the applied science literature in which noisy limit cycle dynamics are often reduced to a diffusion model for the phase of the limit cycle.     

Viscosity solutions of obstacle problems for fully nonlinear path-dependent PDEs

In this article, we adapt the definition of viscosity solutions to the obstacle problem for fully nonlinear path-dependent PDEs with data uniformly continuous in $(t,\omega )$, and generator Lipschitz continuous in $(y,z,\gamma )$. We prove that our definition of viscosity solutions is consistent with the classical solutions, and satisfy a stability result. We show that the value functional defined via the second order reflected backward stochastic differential equation is the unique viscosity solution of the variational inequalities.     

Pointwise estimates for first passage times of perpetuity sequences

We consider first passage times ${\tau }_u=inf\{n:Y_n>u\}$ for the perpetuity sequence

$Y_n=B_1+A_1{B}_2+?+(A_1\dots A_{n?1})B_n,$

where $(A_n,B_n)$ are i.i.d. random variables with values in ${\mathbb{R}}^{+}\times \mathbb{R}$. Recently, a number of limit theorems related to ${\tau }_u$ were proved including the law of large numbers, the central limit theorem and large deviations theorems (see Buraczewski et al., in press). We obtain a precise asymptotics of the sequence $\mathbb{P}[{\tau }_u=logu∕\rho ]$, $\rho >0$, $u\to \infty$ which considerably improves the previous results of Buraczewski et al. (in press). There, probabilities $\mathbb{P}[{\tau }_u\in I_u]$ were identified, for some large intervals $I_u$ around $k_u$, with lengths growing at least as $loglogu$. Remarkable analogies and differences to random walks (Buraczewski and Ma?lanka, in press; Lalley, 1984) are discussed.     

Adaptive estimation for stochastic damping Hamiltonian systems under partial observation

The paper considers a process $Z_t=(X_t,Y_t)$ where $X_t$ is the position of a particle and $Y_t$ its velocity, driven by a hypoelliptic bi-dimensional stochastic differential equation. Under adequate conditions, the process is stationary and geometrically $\beta$-mixing. In this context, we propose an adaptive non-parametric kernel estimator of the stationary density $p$ of $Z$, based on $n$ discrete time observations with time step $\delta$. Two observation schemes are considered: in the first one, $Z$ is the observed process, in the second one, only $X$ is measured. Estimators are proposed in both settings and upper risk bounds of the mean integrated squared error (MISE) are proved and discussed in each case, the second one being more difficult than the first one. We propose a data driven bandwidth selection procedure based on the Goldenshluger and Lespki (2011) method. In both cases of complete and partial observations, we can prove a bound on the MISE asserting the adaptivity of the estimator. In practice, we take advantage of a very recent improvement of the Goldenshluger and Lespki (2011) method provided by Lacour et al. (2016), which is computationally efficient and easy to calibrate. We obtain convincing simulation results in both observation contexts.     

Denseness of volatile and nonvolatile sequences of functions

In a recent paper by Jonasson and Steif, definitions to describe the volatility of sequences of Boolean functions, $f_n:{\{?1,1\}}^n\to \{?1,1\}$ were introduced. We continue their study of how these definitions relate to noise stability and noise sensitivity. Our main results are that the set of volatile sequences of Boolean functions is a natural way “dense” in the set of all sequences of Boolean functions, and that the set of non-volatile Boolean sequences is not “dense” in the set of noise stable sequences of Boolean functions.     

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.     

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.     

Ergodic properties of generalized Ornstein–Uhlenbeck processes

We investigate ergodic properties of the solution of the SDE $d{V}_t=V_{t?}d{U}_t+d{L}_t$, where $(U,L)$ is a bivariate Lévy process. This class of processes includes the generalized Ornstein–Uhlenbeck processes. We provide sufficient conditions for ergodicity, and for subexponential and exponential convergence to the invariant probability measure. We use the Foster–Lyapunov method. The drift conditions are obtained using the explicit form of the generator of the continuous process. In some special cases the optimality of our results can be shown.