arXiv subjects

Raluca Balan

Publications and source records attributed to Raluca Balan.

18 recordsLinked to original sources

arXiv2014

SPDEs with fractional noise in space with index $H<1/2$

In this article, we consider the stochastic wave and heat equations on $\mathbb{R}$ with non-vanishing initial conditions, driven by a Gaussian noise which is white in time and behaves in space like a fractional Brownian motion of index $H$, with $1/4<H<1/2$. We assume that the diffusion coefficient is given by an affine function $σ(x)=ax+b$, and the initial value functions are bounded and Hölder continuous of order $H$. We prove the existence and uniqueness of the mild solution for both equations. We show that the solution is $L^{2}(Ω)$-continuous and its $p$-th moments are uniformly bounded, for any $p \geq 2$.

math.PR↗

arXiv2014

SPDEs with $α$-stable Lévy noise: a random field approach

This article is dedicated to the study of an SPDE of the form $$Lu(t,x)=σ(u(t,x))\dot{Z}(t,x) \quad t>0, x \in \cO$$ with zero initial conditions and Dirichlet boundary conditions, where $σ$ is a Lipschitz function, $L$ is a second-order pseudo-differential operator, $\cO$ is a bounded domain in $\bR^d$, and $\dot{Z}$ is an $α$-stable Lévy noise with $α\in (0,2)$, $α\not=1$ and possibly non-symmetric tails. To give a meaning to the concept of solution, we develop a theory of stochastic integration with respect to $Z$, by generalizing the method of Giné and Marcus (1983) to higher dimensions and non-symmetric tails. The idea is to first solve the equation with "truncated" noise $\dot{Z}_{K}$ (obtained by removing from $Z$ the jumps which exceed a fixed value $K$), yielding a solution $u_{K}$, and then show that the solutions $u_L,L>K$ coincide on the event $t \leq τ_{K}$, for some stopping times $τ_K \uparrow \infty$ a.s. A similar idea was used in Peszat and Zabczyk (2007) in the setting of Hilbert-space valued processes. A major step is to show that the stochastic integral with respect to $Z_{K}$ satisfies a $p$-th moment inequality, for $p \in (α,1)$ if $α<1$, and $p \in (α,2)$ if $α>1$. This inequality plays the same role as the Burkholder-Davis-Gundy inequality in the theory of integration with respect to continuous martingales.

math.PR↗

arXiv2014

Regular variation of infinite series of processes with random coefficients

In this article, we consider a series $X(t)=\sum_{j \geq 1}Ψ_j(t) Z_j(t),t \in [0,1]$ of random processes with sample paths in the space $D=D[0,1]$ of càdlàg functions (i.e. right-continuous functions with left limits) on $[0,1]$. We assume that $(Z_j)_{j \geq 1}$ are i.i.d. processes with sample paths in $D$ and $(Ψ_j)_{j \geq 1}$ are processes with continuous sample paths. Using the notion of regular variation for $D$-valued random elements (introduced in Hult and Lindskog (2005)), we show that $X$ is regularly varying if $Z_1$ is regularly varying, $(Ψ_j)_{j \geq 1}$ satisfy some moment conditions, and a certain ``predictability assumption'' holds for the sequence $\{(Z_j,Ψ_j)\}_{j \geq 1}$. Our result can be viewed as an extension of Theorem 3.1 of Hult and Samorodnitsky (2008) from random vectors in $R^d$ to random elements in $D$. As a preliminary result, we prove a version of Breiman's lemma for $D$-valued random elements, which can be of independent interest.

math.PR↗

arXiv2013

A note on intermittency for the fractional heat equation

The goal of the present note is to study intermittency properties for the solution to the fractional heat equation $$\frac{\partial u}{\partial t}(t,x) = -(-Δ)^{β/2} u(t,x) + u(t,x)\dot{W}(t,x), \quad t>0,x \in \bR^d$$ with initial condition bounded above and below, where $β\in (0,2]$ and the noise $W$ behaves in time like a fractional Brownian motion of index $H>1/2$, and has a spatial covariance given by the Riesz kernel of index $α\in (0,d)$. As a by-product, we obtain that the necessary and sufficient condition for the existence of the solution is $α<β$.

math.PR↗

arXiv2013

Integration with respect to Lévy colored noise, with applications to SPDEs

In this article, we introduce a Lévy analogue of the spatially homogeneous Gaussian noise of Dalang (1999), and we construct a stochastic integral with respect to this noise. The spatial covariance of the noise is given by a tempered measure $μ$ on $\bR^d$, whose density is given by $|h|^2$ for a complex-valued function $h$. Without assuming that the Fourier transform of $μ$ is a non-negative function, we identify a large class of integrands with respect to this noise. As an application, we examine the linear stochastic heat and wave equations driven by this type of noise.

math.PR↗

arXiv2012

Functional Convergence of Linear Sequences in a non-Skorokhod Topology

In this article, we prove a new functional limit theorem for the partial sum sequence $S_{[nt]}=\sum_{i=1}^{[nt]}X_i$ corresponding to a linear sequence of the form $X_i=\sum_{j \in \bZ}c_j ξ_{i-j}$ with i.i.d. innovations $(ξ_i)_{i \in \bZ}$ and real-valued coefficients $(c_j)_{j \in \bZ}$. This weak convergence result is obtained in space $\bD[0,1]$ endowed with the $S$-topology introduced in Jakubowski (1992), and the limit process is a linear fractional stable motion (LFSM). One of our result provides an extension of the results of Avram and Taqqu (1992) to the case when the coefficients $(c_j)_{j \in \bZ}$ may not have the same sign. The proof of our result relies on the recent criteria for convergence in Skorokhod's $M_1$-topology (due to Louhichi and Rio (2011)), and a result which connects the weak $S$-convergence of the sum of two processes with the weak $M_1$-convergence of the two individual processes. Finally, we illustrate our results using some examples and computer simulations.

math.PR↗

arXiv2011

Some linear SPDEs driven by a fractional noise with Hurst index greater than 1/2

In this article, we identify the necessary and sufficient conditions for the existence of a random field solution for some linear s.p.d.e.'s of parabolic and hyperbolic type. These equations rely on a spatial operator $\cL$ given by the $L^2$-generator of a $d$-dimensional Lévy process $X=(X_t)_{t \geq 0}$, and are driven by a spatially-homogeneous Gaussian noise, which is fractional in time with Hurst index $H>1/2$. As an application, we consider the case when $X$ is a $β$-stable process, with $β\in (0,2]$. In the parabolic case, we develop a connection with the potential theory of the Markov process $\bar{X}$ (defined as the symmetrization of $X$), and we show that the existence of the solution is related to the existence of a "weighted" intersection local time of two independent copies of $\bar{X}$.

math.PR↗

arXiv2010

A Cluster Limit Theorem for Infinitely Divisible Point Processes

In this article, we consider a sequence $(N_n)_{n \geq 1}$ of point processes, whose points lie in a subset $E$ of $\bR \verb2\2 \{0\}$, and satisfy an asymptotic independence condition. Our main result gives some necessary and sufficient conditions for the convergence in distribution of $(N_n)_{n \geq 1}$ to an infinitely divisible point process $N$. As applications, we discuss the exceedance processes and point processes based on regularly varying sequences.

math.PR↗

arXiv2009

The Stochastic Wave Equation with Fractional Noise: a random field approach

We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which is fractional in time with index $H>1/2$. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in \cite{dalang99}, when the noise is white in time. Under this condition, we show that the solution is $L^2(Ω)$-continuous. Similar results are obtained for the heat equation. Unlike the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is {\em different} (and more general) than the one obtained for the wave equation.

math.PR↗

arXiv2009

Explicit Conditions for the Convergence of Point Processes Associated to Stationary Arrays

In this article, we consider a stationary array $(X_{j,n})_{1 \leq j \leq n, n \geq 1}$ of random variables with values in $\bR \verb2\2 \{0\}$ (which satisfy some asymptotic dependence conditions), and the corresponding sequence $(N_{n})_{n\geq 1}$ of point processes, where $N_{n}$ has the points $X_{j,n}, 1\leq j \leq n$. Our main result identifies some explicit conditions for the convergence of the sequence $(N_{n})_{n \geq 1}$, in terms of the probabilistic behavior of the variables in the array.

math.PR↗

arXiv2009

Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

We consider the stochastic heat equation with multiplicative noise $u_t={1/2}Δu+ u \diamond \dot{W}$ in $\bR_{+} \times \bR^d$, where $\diamond$ denotes the Wick product, and the solution is interpreted in the mild sense. The noise $\dot W$ is fractional in time (with Hurst index $H \geq 1/2$), and colored in space (with spatial covariance kernel $f$). We prove that if $f$ is the Riesz kernel of order $α$, or the Bessel kernel of order $α 1/2$), respectively $d<2+α$ (if $H=1/2$), whereas if $f$ is the heat kernel or the Poisson kernel, then the equation has a solution for any $d$. We give a representation of the $k$-th order moment of the solution, in terms of an exponential moment of the "convoluted weighted" intersection local time of $k$ independent $d$-dimensional Brownian motions.

math.PR↗

arXiv2009

A Note on a Fenyman-Kac-Type Formula

In this article, we establish a probabilistic representation for the second-order moment of the solution of stochastic heat equation in $[0,1] \times \bR^d$, with multiplicative noise, which is fractional in time and colored in space. This representation is similar to the one given in Dalang, Mueller and Tribe (2008) in the case of an s.p.d.e. driven by a Gaussian noise, which is white in time. Unlike the formula of Dalang, Mueller and Tribe (2008), which is based on the usual Poisson process, our representation is based on the planar Poisson process, due to the fractional component of the noise.

math.PR↗

arXiv2009

$L_p$-Theory for the Stochastic Heat Equation with Infinite-Dimensional Fractional Noise

In this article, we consider the stochastic heat equation $du=(Δu+f(t,x))dt+ \sum_{k=1}^{\infty} g^{k}(t,x) δβ_t^k, t \in [0,T]$, with random coefficients $f$ and $g^k$, driven by a sequence $(β^k)_k$ of i.i.d. fractional Brownian motions of index $H>1/2$. Using the Malliavin calculus techniques and a $p$-th moment maximal inequality for the infinite sum of Skorohod integrals with respect to $(β^k)_k$, we prove that the equation has a unique solution (in a Banach space of summability exponent $p \geq 2$), and this solution is Hölder continuous in both time and space.

math.PR↗

arXiv2008

The Stochastic Heat Equation Driven by a Gaussian Noise: germ Markov Property

Let $u=\{u(t,x);t \in [0,T], x \in {\mathbb{R}}^{d}\}$ be the process solution of the stochastic heat equation $u_{t}=Δu+ \dot F, u(0,\cdot)=0$ driven by a Gaussian noise $\dot F$, which is white in time and has spatial covariance induced by the kernel $f$. In this paper we prove that the process $u$ is locally germ Markov, if $f$ is the Bessel kernel of order $α=2k,k \in \bN_{+}$, or $f$ is the Riesz kernel of order $α=4k,k \in \bN_{+}$.

math.PR↗

arXiv2008

Convergence of Point Processes with Weakly Dependent Points

For each $n \geq 1$, let $\{X_{j,n}\}_{1 \leq j \leq n}$ be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_n=\sum_{j=1}^{n}δ_{X_{j,n}}$ to an infinitely divisible point process. From the point process convergence, we obtain the convergence in distribution of the partial sum sequence $S_n=\sum_{j=1}^{n}X_{j,n}$ to an infinitely divisible random variable, whose Lévy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model.

math.PR↗

arXiv2007

The Stochastic Heat Equation with a Fractional-Colored Noise: Existence of the Solution

In this article we consider the stochastic heat equation $u_{t}-Δu=\dot B$ in $(0,T) \times \bR^d$, with vanishing initial conditions, driven by a Gaussian noise $\dot B$ which is fractional in time, with Hurst index $H \in (1/2,1)$, and colored in space, with spatial covariance given by a function $f$. Our main result gives the necessary and sufficient condition on $H$ for the existence of the process solution. When $f$ is the Riesz kernel of order $α\in (0,d)$ this condition is $H>(d-α)/4$, which is a relaxation of the condition $H>d/4$ encountered when the noise $\dot B$ is white in space. When $f$ is the Bessel kernel or the heat kernel, the condition remains $H>d/4$.

math.PR↗

arXiv2004

A Markov property for set-indexed processes

We consider a type of Markov property for set-indexed processes which is satisfied by all processes with independent increments and which allows us to introduce a transition system theory leading to the construction of the process. A set-indexed generator is defined such that it completely characterizes the distribution of the process.

math.PR↗

arXiv2004

Q-Markov random probability measures and their posterior distributions

In this paper, we use the Markov property introduced in Balan and Ivanoff (J. Theor. Probab. 15, 2002, 553-588) for set-indexed processes and we prove that a Markov prior distribution leads to a Markov posterior distribution. In particular, by proving that a neutral to the right prior distribution leads to a neutral to the right posterior distribution, we extend a fundamental result of Doksum (Ann. Probab. 2,1974, 183-201) to arbitrary sample spaces.

math.PR↗