Yimin Xiao

Sample Path Properties of Anisotropic Gaussian Random Fields

Anisotropic Gaussian random flelds arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operator-scaling Gaussian flelds with stationary increments, and the solution to the stochastic heat equation. This paper is concerned with sample path properties of anisotropic Gaussian random flelds in general. Let X = fX(t); t 2 R N g be a Gaussian random fleld with values in R d and with parameters H1;:::;HN. Our goal is to characterize the anisotropic nature of X in terms of its parameters explicitly. Under some general conditions, we establish results on the modulus of continuity, small ball probabilities, fractal dimensions, hitting probabilities and local times of anisotropic Gaussian random flelds. An important tool for our study is the various forms of strong local nondeterminism.

A minicourse on stochastic partial differential equations

A Primer on Stochastic Partial Differential Equations.- The Stochastic Wave Equation.- Application of Malliavin Calculus to Stochastic Partial Differential Equations.- Some Tools and Results for Parabolic Stochastic Partial Differential Equations.- Sample Path Properties of Anisotropic Gaussian Random Fields.

Correlated continuous time random walks

Continuous time random walks impose a random waiting time before each particle jump. Scaling limits of heavy-tailed continuous time random walks are governed by fractional evolution equations. Space-fractional derivatives describe heavy-tailed jumps, and the time-fractional version codes heavy-tailed waiting times. This paper develops scaling limits and governing equations in the case of correlated jumps. For long-range dependent jumps, this leads to fractional Brownian motion or linear fractional stable motion, with the time parameter replaced by an inverse stable subordinator in the case of heavy-tailed waiting times. These scaling limits provide an interesting class of non-Markovian, non-Gaussian self-similar processes.

Lévy 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 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 C R + , where X is an arbitrary Levy process in 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, Probah. 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 dimX(G) is computed under various conditions on G, X or both. We next solve the following problem [Kahane (1983) Publ. Math. Orsay (83-02) 74-105]: When X is an isotropic stable process, what is a necessary and sufficient analytic condition on any two disjoint Borel sets F, G C R + such that with positive probability, X(F) n X(G) is nonempty? Prior to this article, this was understood only in the case that X is a Brownian motion [Khoshnevisan (1999) Trans. Amer. Math. Soc. 351 2607-2622]. Here, we present a solution to Kahanes problem for an arbitrary Levy process X, provided the distribution of X(t) is mutually absolutely continuous with respect to the Lebesgue measure on R d for all t > 0. As a third application of these methods, we compute the Hausdorff dimension and capacity of the preimage X -1 (F) of a nonrandom Borel set F ⊂ R d under very mild conditions on the process X. This completes the work of Hawkes [Probab. Theory Related Fields 112 (1998) 1-11] that covers the special case where X is a subordinator.

The packing measure of the trajectories of multiparameter fractional Brownian motion

Let X = {X(t), t ∈ RN} be a multiparameter fractional Brownian motion of index α (0 < α < 1) in R. We prove that if N < αd , then there exist positive finite constants K1 and K2 such that with probability 1, K1 ≤ φ-p(X([0, 1] )) ≤ φ-p(GrX([0, 1] )) ≤ K2 where φ(s) = s/(log log 1/s), φ-p(E) is the φ-packing measure of E, X([0, 1] ) is the image and GrX([0, 1] ) = {(t,X(t)); t ∈ [0, 1]N} is the graph of X, respectively. We also establish liminf and limsup type laws of the iterated logarithm for the sojourn measure of X. Running head: Yimin Xiao, The Packing Measure of Fractional Brownian Motion AMS Classification Numbers: Primary 60G15, 60G17; Secondary 28A78.

Local times of additive Lévy processes

Let be an additive Levy process in withwhere X1,...,XN are independent, classical Levy processes on with Levy exponents [Psi]1,...,[Psi]N, respectively. Under mild regularity conditions on the [Psi]is, we derive moment estimates that imply joint continuity of the local times in question. These results are then refined to precise estimates for the local and uniform moduli of continuity of local times when all of the Xis are strictly stable processes with the same index [alpha][set membership, variant](0,2].

Local Times and Related Properties of Multidimensional Iterated Brownian Motion

AbstractLet {W(t), t∈R} and {B(t), t≥0} be two independent Brownian motions in R with W(0) = B(0) = 0 and let

Packing dimension of the image of fractional Brownian motion

Y(t) = W(B(t)){\text{ }}(t \geqslant 0)