In this last decade, an important stochastic model emerged: the Brownian map. It is the limit of various models of random combinatorial maps after rescaling: it is a random metric space with Hausdorff dimension 4, almost surely homeomorphic to the 2-sphere, and possesses some deep connections with
Liouville quantum gravity in 2D. In this paper, we present a sequence of random objects that we call $D$th-random feuilletages (denoted by ${\bf r}[D]$), indexed by a parameter $D\geq 0$ and which are candidate to play the role of the Brownian map in dimension $D$. The construction relies on some objects that we name iterated Brownian snakes, which are branching analogues of iterated Brownian motions, and which are moreover limits of iterated discrete snakes. In the planar $D=2$ case, the family of discrete snakes considered coincides with some family of (random) labeled trees known to encode planar quadrangulations.
Iterating snakes provides a sequence of random trees $({\bf t}^{(j)}, j\geq 1)$. The $D$th-random feuilletage ${\bf r}[D]$ is built using $({\bf t}^{(1)},\cdots,{\bf t}^{(D)})$: ${\bf r}[0]$ is a deterministic circle, ${\bf r}[1]$ is Aldous' continuum random tree, ${\bf r}[2]$ is the Brownian map, and somehow, ${\bf r}[D]$ is obtained by quotienting ${\bf t}^{(D)}$ by ${\bf r}[D-1]$.
A discrete counterpart to ${\bf r}[D]$ is introduced and called the $D$th random discrete feuilletage with $n+D$ nodes (${\bf r}_n[D]$). The proof of the convergence of ${\bf r}_n[D]$ to ${\bf r}[D]$ after appropriate rescaling in some functional space is provided (however, the convergence obtained is too weak to imply the Gromov-Hausdorff convergence). An upper bound on the diameter of ${\bf r}_{n}[D]$ is $n^{1/2^{D}}$. Some elements allowing to conjecture that the Hausdorff dimension of ${\bf r}[D]$ is $2^D$ are given.
In this paper we prove several inequalities by means of diagrammatic expansions, a technique already used in [BG13]. This time we show that iterations of the folding of a probability leads to the proof of some in- equalities by means of a generalized random cluster representation of the iterated foldings. One of the inequalities is the well known FKG inequal- ity, which ends up being proven, quite unexpectedly, by means of the (generalized) FK representation. Although most of the results are not new, we hope that the techniques will find applications in other contexts.
A small ball problem and Chung's law of iterated logarithm for a hypoelliptic Brownian motion in Heisenberg group are proven. In addition, bounds on the limit in Chung's law are established.
We extend generalized isoperimetric-type inequalities to iterated Brownian motion over several domains in $\RR{R}^{n}$. These kinds of inequalities imply in particular that for domains of finite volume, the exit distribution and moments of the first exit time for iterated Brownian motion are maximized with the ball $D^{*}$ centered at the origin, which has the same volume as $D$
We study the distribution of the exit place of iterated Brownian motion in a cone, obtaining information about the chance of the exit place having large magnitude. Along the way, we determine the joint distribution of the exit time and exit place of Brownian motion in a cone. This yields information on large values of the exit place (harmonic measure) for Brownian motion. The harmonic measure for cones has been studied by many authors for many years. Our results are sharper than any previously obtained
By taking a functional analytic point of view, we consider a family of distributions (continuous linear functionals on smooth functions), denoted by $\{μ_t,t>0\}$, associated to the law of iterated logarithm for Brownian motion on a compact manifold. We give a complete characterization of the collection of limiting distributions of $\{μ_t, t>0\}$.
Suppose a solid has a crack filled with a gas. If the crack reaches the surrounding medium, how long does it take the gas to diffuse out of the crack? Iterated Brownian motion serves as a model for diffusion in a crack. If τis the first exit time of iterated Brownian motion from the solid, then P(τ>t) can be viewed as a measurement of the amount of contaminant left in the crack at time t. We determine the large time asymptotics of P(τ>t) for both bounded and unbounded sets. We also discuss a strange connection between iterated Brownian motion and the parabolic operator {1/8}Δ^2-\frac{\partial}…
The Levy transform of a Brownian motion B is the Brownian motion B't, the integral over (O,t) of sign of Bs with respect to dBs. Call T the corresponding transformation on the Wiener space W. We establish that a.s. the orbit of w in W under T is dense in W for the compact uniform convergence topology.
We show that the construction of a random continuum $\mathcal{C}$ from independent two-sided Brownian motions as considered in arXiv:2004.01367 almost surely yields a non-degenerate indecomposable but not-hereditary indecomposable continuum. In particular $\mathcal{C}$ is (unfortunately) not the pseudo-arc.
We characterize the asymptotic behaviour of the weighted power variation processes associated with iterated Brownian motion. We prove weak convergence results in the sense of finite dimensional distributions, and show that the laws of the limiting objects can always be expressed in terms of three independent Brownian motions X, Y and B, as well as of the local times of Y. In particular, our results involve ``weighted'' versions of Kesten and Spitzer's Brownian motion in random scenery. Our findings extend the theory initiated by Khoshnevisan and Lewis (1999), and should be compared with the re…
In this paper we consider the iterated Brownian motion $ ^{μ_1}_{μ_2}\!I(t) = B_1^{μ_1} ( | B_{2}^{μ_2} (t)|) $ where $B_j^{μ_j} , j=1,2$ are two independent Brownian motions with drift $μ_j$. Here we study the last zero crossing of $ ^{μ_1}_{μ_2}\!I(t) $ and for this purpose we derive the last zero-crossing distribution of the drifted Brownian motion. We derive also the joint distribution of the last zero crossing before $ t $ and of the first passage time through the zero level of a Brownian motion with drift $ μ$ after $ t $. All these results permit us to derive explicit formulas for ${^I_…
Let $τ_{D}(Z) $ is the first exit time of iterated Brownian motion from a domain $D \subset \RR{R}^{n}$ started at $z\in D$ and let $P_{z}[τ_{D}(Z) >t]$ be its distribution. In this paper we establish the exact asymptotics of $P_{z}[τ_{D}(Z) >t]$ over bounded domains as an extension of the result in DeBlassie \cite{deblassie}, for $z\in D$ $$ P_{z}[τ_{D}(Z)>t]\approx t^{1/2} \exp(-{3/2}π^{2/3}λ_{D}^{2/3}t^{1/3}), as t\to\infty . $$ We also study asymptotics of the life time of Brownian-time Brownian motion (BTBM), $Z^{1}_{t}=z+X(Y(t))$, where $X_{t}$ and $Y_{t}$ are independent one-dimensional…
We derive general results on the small deviation behavior for some classes of iterated processes. This allows us, in particular, to calculate the rate of the small deviations for $n$-iterated Brownian motions and, more generally, for the iteration of $n$ fractional Brownian motions. We also give a new and correct proof of some results in E. Nane, Laws of the iterated logarithm for $α$-time Brownian motion, Electron. J. Probab. 11 (2006), no. 18, 434--459.
We introduce a class of iterated processes called $α$-time Brownian motion for $0<α\leq 2$. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric $α$-stable process. We prove a Chung-type law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in \cite{hu} for iterated Brownian motion. When $α=1$ it takes the following form $$ \liminf_{T\to\infty}T^{-1/2}(\log \log T) \sup_{0\leq t\leq T}|Z_{t}|=π^{2}\sqrt{λ_{1}} a.s. $$ where $λ_{1}$ is the first eigenvalue for the Cauchy process in the interval $[-1,1].$ We also…
Consider the $n$th iterated Brownian motion $I^{(n)}=B_n \circ\cdots \circ B_1$. Curien and Konstantopoulos proved that for any distinct numbers $t_i\neq 0$, $(I^{(n)}(t_1),\dots,I^{(n)}(t_k))$ converges in distribution to a limit $I[k]$ independent of the $t_i$'s, exchangeable, and gave some elements on the limit occupation measure of $I^{(n)}$. Here, we prove under some conditions, finite dimensional distributions of $n$th iterated two-sided stable processes converge, and the same holds the reflected Brownian motions. We give a description of the law of $I[k]$, of the finite dimensional dist…
Let $τ_{D}(Z) $ be the first exit time of iterated Brownian motion from a domain $D \subset \RR{R}^{n}$ started at $z\in D$ and let $P_{z}[τ_{D}(Z) >t]$ be its distribution. In this paper we establish the exact asymptotics of $P_{z}[τ_{D}(Z) >t]$ over bounded domains as an improvement of the results in \cite{deblassie, nane2}, for $z\in D$ \begin{eqnarray} \lim_{t\to\infty} t^{-1/2}\exp({3/2}π^{2/3}λ_{D}^{2/3}t^{1/3}) P_{z}[τ_{D}(Z)>t]= C(z),\nonumber \end{eqnarray} where $C(z)=(λ_{D}2^{7/2})/\sqrt{3 π}(ψ(z)\int_{D}ψ(y)dy) ^{2}$. Here $λ_{D}$ is the first eigenvalue of the Dirichlet Laplacian …
Iterated Brownian motion $Z_{t}$ serves as a physical model for diffusions in a crack. If $τ_{D}(Z) $ is the first exit time of this processes from a domain $D \subset \RR{R}^{n}$, started at $z\in D$, then $P_{z}[τ_{D}(Z)>t]$ is the distribution of the lifetime of the process in $D$. In this paper we determine the large time asymptotics of $P_{z}[τ_{P_α}(Z) > t]$ which gives exponential integrability of $τ_{P_α}(Z) $ for parabola-shaped domains of the form $ P_α=\{(x,Y)\in \RR{R} \times \RR{R}^{n-1}: x>0, |Y|<Ax^α \}$, for $ 0<α<1$, $A>0.$ We also obtain similar results for twisted domains in…
The classical law of the iterated logarithm (LIL for short)as fundamental limit theorems in probability theory play an important role in the development of probability theory and its applications. Strassen (1964) extended LIL to large classes of functional random variables, it is well known as the invariance principle for LIL which provide an extremely powerful tool in probability and statistical inference. But recently many phenomena show that the linearity of probability is a limit for applications, for example in finance, statistics. As while a nonlinear expectation--- G-expectation has att…
Let $X=\{X(t), t\geq 0\}$ be a Brownian motion or a spectrally negative stable process of index $1<\a<2$. Let $E=\{E(t),t\geq 0\}$ be the hitting time of a stable subordinator of index $0<β<1$ independent of $X$. We use a connection between $X(E(t))$ and the stable subordinator of index $β/\a$ to derive information on the path behavior of $X(E_t)$. This is an extension of the connection of iterated Brownian motion and (1/4)-stable subordinator due to Bertoin \cite{bertoin}. Using this connection, we obtain various laws of the iterated logarithm for $X(E(t))$. In particular, we establish law of…
Let $(B(t))_{t\in [0,1]}$ be the linear Brownian motion and $(X_n(t))_{t\in [0,1]}$ be the $(n-1)$-fold integral of Brownian motion, $n$ being a positive integer: $$ X_n(t)=\int_0^t \frac{(t-s)^{n-1}}{(n-1)!} \,\dd B(s) for any $t\in[0,1]$. $$ In this paper we construct several bridges between times 0 and 1 of the process $(X_n(t))_{t\in [0,1]}$ involving conditions on the successive derivatives of $X_n$ at times 0 and 1. For this family of bridges, we make a correspondance with certain boundary value problems related to the one-dimensional polyharmonic operator. We also study the classical pr…
We construct in this article a rough path over fractional Brownian motion with arbitrary Hurst index by (i) using the Fourier normal ordering algorithm introduced in \cite{Unt-Holder} to reduce the problem to that of regularizing tree iterated integrals and (ii) applying the Bogolioubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization algorithm to Feynman diagrams representing tree iterated integrals.
A new algorithm for the approximation and simulation of twofold iterated stochastic integrals together with the corresponding Lévy areas driven by a multidimensional Brownian motion is proposed. The algorithm is based on a truncated Fourier series approach. However, the approximation of the remainder terms differs from the approach considered by Wiktrosson (2001). As the main advantage, the presented algorithm makes use of a diagonal covariance matrix for the approximation of one part of the remainder term and has a higher accuracy due to an exact approximation of the other part of the remaind…
{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $α<1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Ga…
We construct an iterated stochastic integral with fractional Brownian motion with H > 1/2. The first integrand is a deterministic function, and each successive integral is with respect to an independent fBm. We show that this symmetric stochastic integral is equal to the Malliavin divergence integral. By a version of the Fourth Moment theorem of Nualart and Peccati, we show that a family of such integrals converges in distribution to a scaled Brownian motion. An application is an approximation to the windings for a planar fBm, previously studied by Baudoin and Nualart.
We describe a probabilistic model involving iterated Brownian motion for constructing a random chainable continuum. We show that this random continuum is indecomposable.
For a random walk defined for a doubly infinite sequence of times, we let the time parameter itself be an integer-valued process, and call the orginal process a random walk at random time. We find the scaling limit which generalizes the so-called iterated Brownian motion. Khoshnevisan and Lewis [Ann. Appl. Probab. 9 (1999) 629-667] suggested "the existence of a form of measure-theoretic duality" between iterated Brownian motion and a Brownian motion in random scenery. We show that a random walk at random time can be considered a random walk in "alternating" scenery, thus hinting at a mechanism…
We consider a standard one-dimensional Brownian motion on the time interval $[0,1]$ conditioned to have vanishing iterated time integrals up to order $N$. We show that the resulting processes can be expressed explicitly in terms of shifted Legendre polynomials and the original Brownian motion, and we use these representations to prove that the processes converge weakly as $N\to\infty$ to the zero process. This gives rise to a polynomial decomposition for Brownian motion. We further study the fluctuation processes obtained through scaling by $\sqrt{N}$ and show that they converge in finite dime…
We present an extreme-value analysis of the classical law of the iterated logarithm (LIL) for Brownian motion. Our result can be viewed as a new improvement to the LIL.
We study the Taylor expansion for the solution of a differential equation driven by a multidimensional Holder path with exponent β> 1/2. We derive a convergence criterion that enables us to write the solution as an infinite sum of iterated integrals on a nonempty interval. We apply our deterministic results to stochastic differential equations driven by fractional Brownian motions with Hurst parameter H > 1\2. We also prove that by using L_2 estimates of iterated integrals, the criterion and the speed of convergence for the stochastic Taylor expansion can be improved using Borel-Cantelli type…
Let $B=(B_1(t),\ldots,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $α<1/4$. Defining properly iterated integrals of $B$ is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We show in this paper how to obtain second-order iterated integrals as the limit when the ultra-violet cut-off goes to infinity of iterated integrals of weakly interacting fields defined using the tools of construc…
Iterated Bessel processes R^γ(t), t>0, γ>0 and their counterparts on hyperbolic spaces, i.e. hyperbolic Brownian motions B^{hp}(t), t>0 are examined and their probability laws derived. The higher-order partial differential equations governing the distributions of I_R(t)=_1R^γ(_2R^γ(t)), t>0 and J_R(t) =_1R^γ(|_2R^γ(t)|^2), t>0 are obtained and discussed. Processes of the form R^γ(T_t), t>0, B^{hp}(T_t), t>0 where T_t=\inf{s: B(s)=t} are examined and numerous probability laws derived, including the Student law, the arcsin laws (also their asymmetric versions), the Lamperti distribution of the r…
We introduce a class of stochastic processes based on symmetric $α$-stable processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric $α$-stable process. We call them $α$-time processes. They generalize Brownian time processes studied in \cite{allouba1, allouba2, allouba3}, and they introduce new interesting examples. We establish the connection of $α-$time processes to some higher order PDE's for $α$ rational. We also study the exit problem for $α$-time processes as they exit regular domains and connect them to elliptic PDE's. We …
Let $B_H=\{B_H(t):t\in\mathbb R\}$ be a fractional Brownian motion with Hurst parameter $H\in(0,1)$. For the stationary storage process $Q_{B_H}(t)=\sup_{-\infty<s\le t}(B_H(t)-B_H(s)-(t-s))$, $t\ge0$, we provide a tractable criterion for assessing whether, for any positive, non-decreasing function $f$, $\mathbb P(Q_{B_H}(t) > f(t)\, \text{ i.o.})$ equals 0 or 1. Using this criterion we find that, for a family of functions $f_p(t)$, such that $z_p(t)=\mathbb P(\sup_{s\in[0,f_p(t)]}Q_{B_H}(s)>f_p(t))/f_p(t)=\mathscr C(t\log^{1-p} t)^{-1}$, for some $\mathscr C>0$, $\mathbb P(Q_{B_H}(t) > f_p(t)…
We consider different types of processes obtained by composing Brownian motion $B(t)$, fractional Brownian motion $B_{H}(t)$ and Cauchy processes $% C(t)$ in different manners. We study also multidimensional iterated processes in $\mathbb{R}^{d},$ like, for example, $\left( B_{1}(|C(t)|),...,B_{d}(|C(t)|)\right) $ and $\left( C_{1}(|C(t)|),...,C_{d}(|C(t)|)\right) ,$ deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equa…
We prove a general functional limit theorem for multiparameter fractional Brownian motion. The functional law of the iterated logarithm, functional Lévy's modulus of continuity and many other results are its particular cases. Applications to approximation theory are discussed.
By using a change of scale and space, we study a class of stochastic differential equations (SDEs) whose solutions are drift--perturbed and exhibit behaviour analogous to standard Brownian motion including to the Law of the Iterated Logarithm (LIL). Sufficient conditions ensuring that these processes obey the LIL are given.
New Simple Method of Expansion of Iterated Ito Stochastic integrals of Multiplicity 2 Based on Expansion of the Brownian Motion Using Legendre Polynomials and Trigonometric Functions
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
We describe a measurement device principle based on discrete iterations of Bayesian updating of system state probability distributions. Although purely classical by nature, these measurements are accompanied with a progressive collapse of the system state probability distribution during each complete system measurement. This measurement scheme finds applications in analysing repeated non-demolition indirect quantum measurements. We also analyse the continuous time limit of these processes, either in the Brownian diffusive limit or in the Poissonian jumpy limit. In the quantum mechanical framew…
When a Brownian motion is scaled according to the law of the iterated logarithm, its supremum converges to one as time tends to zero. Upper large deviations of the supremum process can be quantified by writing the problem in terms of hitting times and applying a result of Strassen (1967) on hitting time densities. We extend this to a small-time large deviations principle for the supremum of scaled Ito diffusions, using as our main tool a refinement of Strassen's result due to Lerche (1986).
We prove a Chung-type law of the iterated logarithm for a multiparameter extension of the fractional Brownian motion which is not increment stationary. This multiparameter fractional Brownian motion behaves very differently at the origin and away from the axes, which also appears in the Hausdorff dimension of its range and in the measure of its pointwise Hölder exponents. A functional version of this Chung-type law is also provided.
A famous result of Orey and Taylor gives the Hausdorff dimension of the set of fast times, that is the set of points where linear Brownian motion moves faster than according to the law of iterated logarithm. In this paper we examine what happens to the set of fast times if a variable drift is added to linear Brownian motion. In particular, we will show that the Hausdorff dimension of the set of fast times cannot be decreased by adding a function to Brownian motion.
If β_t is renormalized self-intersection local time for planar Brownian motion, we characterize when Ee^{γβ_1} is finite or infinite in terms of the best constant of a Gagliardo-Nirenberg inequality. We prove large deviation estimates for β_1 and -β_1. We establish lim sup and lim inf laws of the iterated logarithm for β_t as t\to\infty.
We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron--Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon--Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.
In this paper, we study the asymptotic behavior of supremum distribution of some classes of iterated stochastic processes $\{X(Y(t)) : t \in [0, \infty)\}$, where $\{X(t) : t \in \mathbb{R} \}$ is a centered Gaussian process and $\{Y(t): t \in [0, \infty)\}$ is an independent of $\{X(t)\}$ stochastic process with a.s. continuous sample paths. In particular, the asymptotic behavior of $\mathbb{P}\left(\sup_{s \in [0,T]} X(Y(s)) > u\right)$ as $u \to \infty$, where $T > 0$, as well as $\lim_{u\to\infty} \mathbb{P}\left(\sup_{s \in [0, h(u)]} X(Y(s)) > u\right)$, for some suitably chosen function…
Let B_1,B_2, ... be independent one-dimensional Brownian motions defined over the whole real line such that B_i(0)=0. We consider the nth iterated Brownian motion W_n(t)= B_n(B_{n-1}(...(B_2(B_1(t)))...)). Although the sequences of processes (W_n) do not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W_n converge towards a random probability measure μ_\infty. We then prove that μ_\infty almost surely has a continuous density which must be thought of as the local time process of the infi…
Lévy's stochastic area for planar Brownian motion is the difference of two iterated integrals of second rank against its component one-dimen\-sional Brownian motions. Such iterated integrals can be multiplied using the sticky shuffle product determined by the underlying Itô algebra of stochastic differentials. We use combinatorial enumerations that arise from the distributive law in the corresponding Hopf algebra structure to evaluate the moments of Lévy's area. These Lévy moments are well known to be given essentially by the Euler numbers. This has recently been confirmed in a novel combinato…
We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of strictly stationary AR(1) and INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient $α\in (0, 1)$ and idiosyncratic innovations. Assuming that $α$ has a density function of the form $ψ(x) (1 - x)^β$, $x \in (0, 1)$, with $\lim_{x\uparrow 1} ψ(x) = ψ_1 \in (0, \infty)$, different Brownian limit processes of appropriately centered and scaled aggregated partial sums are shown to exist in case $β=1$ when taking first the limit as $N \to \infty$ and then the time scale $n \to \infty…
In his, by now, classical work from 1981, Nerman made extensive use of a crucial martingale $(W_t)_{t \geq 0}$ to prove convergence in probability, in mean and almost surely, of supercritical general branching processes (a.k.a. Crump-Mode-Jagers branching processes) counted with a general characteristic. The martingale terminal value $W$ figures in the limits of his results. We investigate the rate at which the martingale, now called Nerman's martingale, converges to its limit $W$. More precisely, assuming the existence of a Malthusian parameter $α> 0$ and $W_0\in L^2$, we prove a functional c…
Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general cadlag semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by Levy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of affine iterated random maps as well as symmet…
We consider a one-dimensional diffusion process $X$ in a $(-κ/2)$-drifted Brownian potential for $κ\neq 0$. We are interested in the maximum of its local time, and study its almost sure asymptotic behaviour, which is proved to be different from the behaviour of the maximum local time of the transient random walk in random environment. We also obtain the convergence in law of the maximum local time of $X$ under the annealed law after suitable renormalization when $κ\geq 1$. Moreover, we characterize all the upper and lower classes for the hitting times of $X$, in the sense of Paul Lévy, and pro…
In a paper of Jason Swanson, a CLT for the sample median of independent Brownian motions with value 0 at 0 was proved. Here we extend this result in two ways. We prove such a result for a collection of self-similar processes which include the fractional Brownian motions, all stationary, independent increment symmetric stable processes tied down at 0 as well as iterated and integrated Brownian motions. Second, our results hold uniformly over all quantiles in a compact sub-interval of (0,1). We also examine sample function properties connected with these CLTs.
We determine the exact Hausdorff measure functions for the range and level sets of a class of Gaussian random fields satisfying sectorial local nondeterminism and other assumptions. We also establish a Chung-type law of the iterated logarithm. The results can be applied to the Brownian sheet, fractional Brownian sheets whose Hurst indices are the same in all directions, and systems of linear stochastic wave equations in one spatial dimension driven by space-time white noise or colored noise.
We study the asymptotic behaviour of the probability that a stochastic process $(Z_t)_{t \geq 0}$ does not exceed a constant barrier up to time $T$ (the so called survival probability) when Z is the composition of two independent processes $(X_t)_{t \in I}$ and $(Y_t)_{t \geq 0}$. To be precise, we consider $(Z_t)_{t \geq 0}$ defined by $Z_t = X \circ \abs{Y_t}$ when $I = [0,\infty)$ and $Z_t = X \circ Y_t$ when $I = \mathbb{R}$. For continuous self-similar processes $(Y_t)_{t \geq 0}$, the rate of decay of survival probability for $Z$ can be inferred directly from the survival probability of …
We survey the results in Nane (E. Nane, Higher order PDE's and iterated processes, Trans. American Math. Soc. (to appear)) and Baeumer, Meerschaert, and Nane (B. Baeumer, M.M. Meerschaert and E. Nane, Brownian subordinators and fractional Cauchy problems: Submitted (2007)) which deal with PDE connection of some iterated processes, and obtain a new probabilistic proof of the equivalence of the higher order PDE's and fractional in time PDE's.
