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Davar Khoshnevisan

Publications and source records attributed to Davar Khoshnevisan.

At least 73 records · Page 4Linked to original sources
arXiv2010

Dynkin's isomorphism theorem and the stochastic heat equation

Consider the stochastic heat equation $\partial_t u = \sL u + \dot{W}$, where $\sL$ is the generator of a [Borel right] Markov process in duality. We show that the solution is locally mutually absolutely continuous with respect to a smooth perturbation of the Gaussian process that is associated, via Dynkin's isomorphism theorem, to the local times of the replica-symmetric process that corresponds to $\sL$.In the case that $\sL$ is the generator of a Lévy process on $\R^d$, our result gives a probabilistic explanation of the recent findings of Foondun et al.

math.PR↗
arXiv2009

Strong approximations in a charged-polymer model

We study the large-time behavior of the charged-polymer Hamiltonian $H_n$ of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths, and Higgs [Gaussian case], using strong approximations to Brownian motion. Our results imply, among other things, that in one dimension the process $\{H_{[nt]}\}_{0\le t\le 1}$ behaves like a Brownian motion, time-changed by the intersection local-time process of an independent Brownian motion. Chung-type LILs are also discussed.

math.PR↗
arXiv2009

Zeros of a two-parameter random walk

We prove that the number gamma(N) of the zeros of a two-parameter simple random walk in its first N-by-N time steps is almost surely equal to N to the power 1+o(1) as N goes to infinity. This is in contrast with our earlier joint effort with Z. Shi [4]; that work shows that the number of zero crossings in the first N-by-N time steps is N to the power (3/2)+o(1) as N goes to infinity. We prove also that the number of zeros on the diagonal in the first N time steps is (c+o(1)) log N as N goes to infinity, where c is 2π.

math.PR↗
arXiv2009

On dynamical bit sequences

Let X^{(k)}(t) = (X_1(t), ..., X_k(t)) denote a k-vector of i.i.d. random variables, each taking the values 1 or 0 with respective probabilities p and 1-p. As a process indexed by non-negative t, $X^{(k)}(t)$ is constructed--following Benjamini, Haggstrom, Peres, and Steif (2003)--so that it is strong Markov with invariant measure ((1-p)δ_0+pδ_1)^k. We derive sharp estimates for the probability that ``X_1(t)+...+X_k(t)=k-\ell for some t in F,'' where F \subset [0,1] is nonrandom and compact. We do this in two very different settings: (i) Where \ell is a constant; and (ii) Where \ell=k/2, k is even, and p=q=1/2. We prove that the probability is described by the Kolmogorov capacitance of F for case (i) and Howroyd's 1/2-dimensional box-dimension profiles for case (ii). We also present sample-path consequences, and a connection to capacities that answers a question of Benjamini et. al. (2003)

math.PR↗
arXiv2009

On the global maximum of the solution to a stochastic heat equation with compact-support initial data

Consider a stochastic heat equation $\partial_t u = κ\partial^2_{xx}u+σ(u)\dot{w}$ for a space-time white noise $\dot{w}$ and a constant $κ>0$. Under some suitable conditions on the the initial function $u_0$ and $σ$, we show that the quantity \limsup_{t\to\infty}t^{-1}\ln\E(\sup_{x\in\R} |u_t(x)|^2) is bounded away from zero and infinity by explicit multiples of $1/κ$. Our proof works by demonstrating quantitatively that the peaks of the stochastic process $x\mapsto u_t(x)$ are highly concentrated for infinitely-many large values of $t$. In the special case of the parabolic Anderson model--where $σ(u)= λu$ for some $λ>0$--this "peaking" is a way to make precise the notion of physical intermittency.

math.PR↗
arXiv2008

Level sets of the stochastic wave equation driven by a symmetric Lévy noise

We consider the solution $\{u(t,x);t\geq0,x\in\mathbf{R}\}$ of a system of $d$ linear stochastic wave equations driven by a $d$-dimensional symmetric space-time Lévy noise. We provide a necessary and sufficient condition on the characteristic exponent of the Lévy noise, which describes exactly when the zero set of $u$ is non-void. We also compute the Hausdorff dimension of that zero set when it is non-empty. These results will follow from more general potential-theoretic theorems on the level sets of Lévy sheets.

math.PR↗
arXiv2008

Intermittence and nonlinear parabolic stochastic partial differential equations

We consider nonlinear parabolic SPDEs of the form $\partial_t u=\sL u + σ(u)\dot w$, where $\dot w$ denotes space-time white noise, $σ:\R\to\R$ is [globally] Lipschitz continuous, and $\sL$ is the $L^2$-generator of a Lévy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when $σ$ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent,'' provided that the symmetrization of $\sL$ is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for $\sL$ in dimension $(1+1)$. When $\sL=κ\partial_{xx}$ for $κ>0$, these formulas agree with the earlier results of statistical physics \cite{Kardar,KrugSpohn,LL63}, and also probability theory \cite{BC,CM94} in the two exactly-solvable cases where $u_0=δ_0$ and $u_0\equiv 1$.

math.PR↗
arXiv2007

A Local time correspondence for stochastic partial differential equations

It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have random-field solutions only in spatial dimension one. Here we show that in many cases, where the ``spatial operator'' is the L^2-generator of a Lévy process X, a linear SPDE has a random-field solution if and only if the symmetrization of X possesses local times. This result gives a probabilistic reason for the lack of existence of random-field solutions in dimensions strictly bigger than one. In addition, we prove that the solution to the SPDE is [Hölder] continuous in its spatial variable if and only if the said local time is [Hölder] continuous in its spatial variable. We also produce examples where the random-field solution exists, but is almost surely unbounded in every open subset of space-time. Our results are based on first establishing a quasi-isometry between the linear L^2-space of the weak solutions of a family of linear SPDEs, on one hand, and the Dirichlet space generated by the symmetrization of X, on the other hand. We study mainly linear equations in order to present the local-time correspondence at a modest technical level. However, some of our work has consequences for nonlinear SPDEs as well. We demonstrate this assertion by studying a family of parabolic SPDEs that have additive nonlinearities. For those equations we prove that if the linearized problem has a random-field solution, then so does the nonlinear SPDE. Moreover, the solution to the linearized equation is [Hölder] continuous if and only if the solution to the nonlinear equation is. And the solutions are bounded and unbounded together as well. Finally, we prove that in the cases that the solutions are unbounded, they almost surely blow up at exactly the same points.

math.PR↗
arXiv2007

Harmonic analysis of additive Levy processes

Let $X_1,...,X_N$ denote $N$ independent $d$-dimensional Lévy processes, and consider the $N$-parameter random field \[\X(\bm{t}):= X_1(t_1)+...+X_N(t_N).\] First we demonstrate that for all nonrandom Borel sets $F\subseteq\R^d$, the Minkowski sum $\X(\R^N_+)\oplus F$, of the range $\X(\R^N_+)$ of $\X$ with $F$, can have positive $d$-dimensional Lebesgue measure if and only if a certain capacity of $F$ is positive. This improves our earlier joint effort with Yuquan Zhong \ycite{KXZ:03} by removing a symmetry-type condition there. Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical [non-probabilistic] harmonic analysis that might be of independent interest. As was shown in \fullocite{KXZ:03}, the potential theory of the type studied here has a large number of consequences in the theory of Lévy processes. We present a few new consequences here.

math.PR↗
arXiv2007

Dynamical percolation on general trees

Häggström, Peres, and Steif (1997) have introduced a dynamical version of percolation on a graph $G$. When $G$ is a tree they derived a necessary and sufficient condition for percolation to exist at some time $t$. In the case that $G$ is a spherically symmetric tree, Häggström, Peres, and Steif (1997) derived a necessary and sufficient condition for percolation to exist at some time $t$ in a given target set $D$. The main result of the present paper is a necessary and sufficient condition for the existence of percolation, at some time $t\in D$, in the case that the underlying tree is not necessary spherically symmetric. This answers a question of Yuval Peres (personal communication). We present also a formula for the Hausdorff dimension of the set of exceptional times of percolation.

math.PR↗
arXiv2007

Packing-Dimension Profiles and Fractional Brownian Motion

In order to compute the packing dimension of orthogonal projections Falconer and Howroyd (1997) introduced a family of packing dimension profiles ${\rm Dim}_s$ that are parametrized by real numbers $s>0$. Subsequently, Howroyd (2001) introduced alternate $s$-dimensional packing dimension profiles $\hbox{${\rm P}$-$\dim$}_s$ and proved, among many other things, that $\hbox{${\rm P}$-$\dim$}_s E={\rm Dim}_s E$ for all integers $s>0$ and all analytic sets $E\subseteq\R^N$. The goal of this article is to prove that $\hbox{${\rm P}$-$\dim$}_s E={\rm Dim}_s E$ for all real numbers $s>0$ and analytic sets $E\subseteq\R^N$. This answers a question of Howroyd (2001, p. 159). Our proof hinges on a new property of fractional Brownian motion.

math.PR↗
arXiv2007

Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise

We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish upper and lower bounds on the one-point density of the solution u(t,x), and upper bounds of Gaussian-type on the two-point density of (u(s,y),u(t,x)). In particular, this estimate quantifies how this density degenerates as (s,y) converges to (t,x). From these results, we deduce upper and lower bounds on hitting probabilities of the process {u(t,x)}_{t \in \mathbb{R}_+, x \in [0,1]}, in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d >6 and are not polar when d<6. We also show that the Hausdorff dimension of the range of the process is 6 when d>6, and give analogous results for the processes t \mapsto u(t,x) and x \mapsto u(t,x). Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes.

math.PR↗
arXiv2007

Hitting Probabilities for Systems of Non-Linear Stochastic Heat Equations with Additive Noise

We consider a system of $d$ coupled non-linear stochastic heat equations in spatial dimension 1 driven by $d$-dimensional additive space-time white noise. We establish upper and lower bounds on hitting probabilities of the solution $\{u(t, x)\}_{t \in \mathbb{R}_+, x \in [0, 1]}$, in terms of respectively Hausdorff measure and Newtonian capacity. We also obtain the Hausdorff dimensions of level sets and their projections. A result of independent interest is an anisotropic form of the Kolmogorov continuity theorem.

math.PR↗
arXiv2005

Levy processes: Capacity and Hausdorff dimension

We use the recently-developed multiparameter theory of additive Levy processes to establish novel connections between an arbitrary Levy process $X$ in $\mathbf{R}^d$, and a new class of energy forms and their corresponding capacities. We then apply these connections to solve two long-standing problems in the folklore of the theory of Levy processes. First, we compute the Hausdorff dimension of the image $X(G)$ of a nonrandom linear Borel set $G\subset \mathbf{R}_+$, where $X$ is an arbitrary Levy process in $\mathbf{R}^d$. Our work completes the various earlier efforts of Taylor [Proc. Cambridge Phil. Soc. 49 (1953) 31-39], McKean [Duke Math. J. 22 (1955) 229-234], Blumenthal and Getoor [Illinois J. Math. 4 (1960) 370-375, J. Math. Mech. 10 (1961) 493-516], Millar [Z. Wahrsch. verw. Gebiete 17 (1971) 53-73], Pruitt [J. Math. Mech. 19 (1969) 371-378], Pruitt and Taylor [Z. Wahrsch. Verw. Gebiete 12 (1969) 267-289], Hawkes [Z. Wahrsch. verw. Gebiete 19 (1971) 90-102, J. London Math. Soc. (2) 17 (1978) 567-576, Probab. Theory Related Fields 112 (1998) 1-11], Hendricks [Ann. Math. Stat. 43 (1972) 690-694, Ann. Probab. 1 (1973) 849-853], Kahane [Publ. Math. Orsay (83-02) (1983) 74-105, Recent Progress in Fourier Analysis (1985b) 65-121], Becker-Kern, Meerschaert and Scheffler [Monatsh. Math. 14 (2003) 91-101] and Khoshnevisan, Xiao and Zhong [Ann. Probab. 31 (2003a) 1097-1141], where $\dim X(G)$ is computed under various conditions on $G$, $X$ or both.

math.PR↗
arXiv2005

A Coupling, and the Darling-Erdos Conjectures

We derive a new coupling of the running maximum of an Ornstein-Uhlenbeck process and the running maximum of an explicit i.i.d. sequence. We use this coupling to verify a conjecture of Darling and Erdos (1956).

math.PR↗
arXiv2005

Slices of Brownian Sheet: New Results, and Open Problems

We can view Brownian sheet as a sequence of interacting Brownian motions or slices. Here we present a number of results about the slices of the sheet. A common feature of our results is that they exhibit phase transition. In addition, a number of open problems are presented.

math.PR↗
arXiv2005

Schoenberg's Theorem via the law of large numbers

A classical theorem of S. Bochner states that a function $f:R^n \to C$ is the Fourier transform of a finite Borel measure if and only if $f$ is positive definite. In 1938, I. Schoenberg found a beautiful complement to Bochner's theorem. We present a non-technical derivation of of Schoenberg's theorem that relies chiefly on the de Finneti theorem and the law of large numbers of classical probability theory.

math.PR↗