Davar Khoshnevisan

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.

A law of the iterated logarithm for stable processes in random scenery

We prove a law of the iterated logarithm for stable processes in a random scenery. The proof relies on the analysis of a new class of stochastic processes which exhibit long-range dependence.

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.

Chung’s law for integrated Brownian motion

The small ball problem for the integrated process of a real–valued Brownian motion is solved. In sharp contrast to more standard methods, our approach relies on the sample path properties of Brownian motion together with facts about local times and Lévy processes.

On the stochastic heat equation with spatially-colored random forcing

We consider the stochastic heat equation of the following form ∂ ∂t ut(x) = (Lut)(x) + b(ut(x)) + σ(ut(x))Ḟt(x) for t > 0, x ∈ R, where L is the generator of a Levy process and Ḟ is a spatially-colored, temporally white, gaussian noise. We will be concerned mainly with the long-term behavior of the mild solution to this stochastic PDE. For the most part, we work under the assumptions that the initial data u0 is a bounded and measurable function and σ is nonconstant and Lipschitz continuous. In this case, we find conditions under which the preceding stochastic PDE admits a unique solution which is also weakly intermittent. In addition, we study the same equation in the case that Lu is replaced by its massive/dispersive analogue Lu − λu where λ ∈ R. And we describe accurately the effect of the parameter λ on the intermittence of the solution in the case that σ(u) is proportional to u [the “parabolic Anderson model”]. We also look at the linearized version of our stochastic PDE, that is the case when σ is identically equal to one [any other constant works also]. In this case, we study not only the existence and uniqueness of a solution, but also the regularity of the solution when it exists and is unique.

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].

Intersections of Brownian motions

Abstract This article presents a survey of the theory of the intersections of Brownian motion paths. Among other things, we present a truly elementary proof of a classical theorem of A. Dvoretzky, P. Erdős and S. Kakutani. This proof is motivated by old ideas of P. Levy that were originally used to investigate the curve of planar Brownian motion.

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-generator of a Levy 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 [Holder] continuous in its spatial variable if and only if the said local time is [Holder] 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-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 [Holder] 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. Date: October 7, 2007. 2000 Mathematics Subject Classification. Primary. 60H15, 60J55; Secondary. 35R60, 35D05.

Local times on curves and uniform invariance principles

SummarySufficient conditions are given for a family of local times |Ltµ| ofd-dimensional Brownian motion to be jointly continuous as a function oft and μ. Then invariance principles are given for the weak convergence of local times of lattice valued random walks to the local times of Brownian motion, uniformly over a large family of measures. Applications included some new results for intersection local times for Brownian motions on ℝ2 and ℝ2.

Boundary crossings and the distribution function of the maximum of Brownian sheet

Our main intention is to describe the behavior of the (cumulative) distribution function of the random variable M0,1 := sup0[less-than-or-equals, slant]s,t[less-than-or-equals, slant]1 W(s,t) near 0, where W denotes one-dimensional, two-parameter Brownian sheet. A remarkable result of Florit and Nualart asserts that M0,1 has a smooth density function with respect to Lebesgues measure (cf. Florit and Nualart, 1995. Statist. Probab. Lett. 22, 25-31). Our estimates, in turn, seem to imply that the behavior of the density function of M0,1 near 0 is quite exotic and, in particular, there is no clear-cut notion of a two-parameter reflection principle. We also consider the supremum of Brownian sheet over rectangles that are away from the origin. We apply our estimates to get an infinite-dimensional analogue of Hirschs theorem for Brownian motion.