Publications by Robert Dalang | EPFL Graph Search

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This open access book provides a comprehensive introduction to the theory of stochastic partial differential equations (SPDEs). The focus is on SPDEs driven by Gaussian space-time white noise. The book covers both linear and nonlinear SPDEs, with Lipschitz and locally Lipschitz coefficients and multiplicative noise. It provides a modern presentation of the theory of stochastic integration with respect to space-time white noise and unifies many results in the literature. The book discusses fundamental topics

In the classical quickest detection problem, one must detect as quickly as possible when a Brownian motion without drift "changes" into a Brownian motion with positive drift. The change occurs at an unknown "disorder" time with exponential distribution. Th ...

We study the nonlinear stochastic heat equation in the spatial domain R, driven by space-time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on R, such as the Dirac delta function, but th ...

Randomized stopping points form a convex set associated with the information structure that arises in the context of the optimal stopping problem for two-parameter processes. We study combinatorial properties of this structure when the underlying space is finite, in which case this convex set can be identified with a bounded polyhedron. Study of the extreme points of this polytope motivates the definition of an apparently new class of perfectly orderable graphs. Properties of this class of graphs are examin

We present the Walsh theory of stochastic integrals with respect to martingale measures, and various extensions of this theory, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes, and ...

Given a stable Levy process X = (X-t)(0

We study the regularity of the probability density function of the supremum of the solution to the linear stochastic heat equation. Using a general criterion for the smoothness of densities for locally nondegenerate random variables, we establish the smoot ...

We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Levy white noise, with symmetric alpha-stable Levy white noise as an important special case. We identify con ...

We study the hitting probabilities of the solution to a system of d stochastic heat equations with additive noise subject to Dirichlet boundary conditions. We show that for any bounded Borel set with positive (d-6)\documentclass[12pt]{minimal} \usepackage{ ...

We derive a decoupling formula for the Brownian sheet which has the following ready consequence: An N-parameter Brownian sheet in R-d has double points if and only if d < 4N. In particular, in the critical case where d = 4N, the Brownian sheet does not hav ...

In a sailboat race, the navigator's attempts to plot the fastest possible course are hindered by shifty winds. We present mathematical models appropriate for this situation, which use statistical analysis of wind fluctuations and are amenable to stochastic ...

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel-Riesz capacity, respectively. We apply these ...

We consider a system of d non-linear stochastic fractional heat equations in spatial dimension 1 driven by multiplicative d-dimensional space-time white noise. We establish a sharp Gaussian-type upper bound on the two-point probability density function of ...

We study a class of linear hyperbolic stochastic partial differential equations in bounded domains, that includes the wave equation and the telegraph equation, driven by Gaussian noise that is white in time but not in space. We give necessary and sufficient conditions on the spatial correlation of the noise for the existence (and uniqueness) of square-integrable solutions. In the particular case where the domain is a ball and the noise is concentrated on a sphere, we characterize the isotropic Gaussian nois

We solve two stochastic control problems in which a player tries to minimize or maximize the exit time from an interval of a Brownian particle, by controlling its drift. The player can change from one drift to another but is subject to a switching cost. In ...

We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix gamma z of Z := (u(s, y), u(t , x) - u(s, y)), where u is the solution to a system of d non-linear stochastic heat equations in spatial dimension k >= ...

Let xi(t, x) denote space-time white noise and consider a reaction-diffusion equation of the form ...

We study a class of hyperbolic stochastic partial di®erential equations in Euclidean space, that includes the wave equation and the telegraph equation, driven by Gaussian noise concentrated on a hyperplane. The noise is assumed to be white in time but spat ...

These notes give an overview of recent results concerning the non-linear stochastic wave equation in spatial dimensions d >= 1, in the case where the driving noise is Gaussian, spatially homogeneous and white in time. We mainly address issues of existence, uniqueness and Holder-Sobolev regularity. We also present an extension of Walsh's theory of stochastic integration with respect to martingale measures that is useful for spatial dimensions d >= 3.

We consider a class of parabolic stochastic PDEs on bounded domains D c Rd that includes the stochastic heat equation but with a fractional power gamma of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces ...

Robert Dalang

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Stochastic Partial Differential Equations, Space-Time White Noise and Random Fields

A Quickest Detection Problem With An Observation Cost

Moments And Growth Indices For The Nonlinear Stochastic Heat Equation With Rough Initial Conditions

On randomized stopping points and perfect graphs

Stochastic integrals for spde's: A comparison

Predicting The Ultimate Supremum Of A Stable Levy Process With No Negative Jumps

On the density of the supremum of the solution to the linear stochastic heat equation

Random field solutions to linear SPDEs driven by symmetric pure jump Levy space-time white noises

Hitting with Probability One for Stochastic Heat Equations with Additive Noise

Critical Brownian Sheet Does Not Have Double Points

Stochastic optimization of sailing trajectories in an upwind regatta

Criteria for hitting probabilities with applications to systems of stochastic wave equations

Optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations

Second-order linear hyperbolic SPDEs driven by isotropic Gaussian noise on a sphere

Optimal expulsion and optimal confinement of a Brownian particle with a switching cost

Optimal lower bounds on hitting probabilities for stochastic heat equations in spatial dimension k >= 1

Global Solutions To Stochastic Reaction-Diffusion Equations With Super-Linear Drift And Multiplicative Noise

Second-order hyperbolic S.P.D.E.'s driven by homogeneous Gaussian noise on a hyperplane

The stochastic wave equation

Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs