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Robert C. Dalang

Publications and source records attributed to Robert C. Dalang.

At least 19 recordsLinked to original sources
arXiv2026

Stochastic Control Problems Motivated by Sailboat Trajectory Optimization

We develop a mathematical model for sailboat navigation that captures the essential features of the problem and can provide insights that might not be available otherwise. In our model, the motion of the sailboat, which would travel at speed $v>0$ in a constant wind, is the solution of a system of two stochastic differential equations driven by a Brownian motion on a circle with speed $σ> 0$. We formulate two stochastic control/optimal switching problems, in which the objective is to reach a circular upwind target of radius $η\geq 0$ as quickly as possible. In the first problem, there is a tacking cost $c > 0$, so this is an impulse control problems, while in the second problem, we assume that $c=0$ and singular controls are needed. We establish the viability of both models (assuming that $η> 0$ in the second model), that is, their value functions are finite, and we obtain bounds on these value functions related to the parameters of the problem. In the second problem, since the state equation for the optimally controlled motion has discontinuous coefficients and is driven by a degenerate diffusion, standard results on existence and uniqueness of strong solutions do not apply: we provide a proof via the Yamada-Watanabe argument.

math.OC↗
arXiv2026

Asymptotic behavior of the stochastic heat equation over large intervals

We consider a nonlinear stochastic heat equation on $[0,T]\times [-L,L]$, driven by a space-time white noise $W$, with a given initial condition $u_0: \mathbb{R} \to \mathbb{R}$ and three different types of (vanishing) boundary conditions: Dirichlet, Mixed and Neumann. We prove that as $L\to\infty$, the random field solution at any space-time position converges in the $L^p(Ω)$-norm ($p\ge 1$) to the solution of the stochastic heat equation on $\mathbb{R}$ (with the same initial condition $u_0$), and we determine the (near optimal) rate of convergence. The proof relies on estimates of differences between the corresponding Green's functions on $[-L, L]$ and the heat kernel on $\mathbb{R}$, and on a space-time version of a Gronwall-type lemma.

math.PR↗
arXiv2026

A simple derivation of the Fourier transform of the Heaviside function

We give a rigorous derivation of the Fourier transform of the Heaviside function within a framework for tempered distributions that is suitable for undergraduate engineering and mathematics students. The proofs rely on fundamental concepts typically taught in a freshman-level calculus course, including limits, generalized integrals, integration by parts and the Taylor Remainder Theorem. In passing, we examine the Principle Value of $\frac{1}{x}$ and the relationship between its derivative of order $n$ and the Principle Value of $\frac{1}{x^{n+1}}$.

math.FA↗
arXiv2026

Does 2026 AI exhibit intelligence, or can Claude outsmart Pierre or Catherine ?

Using a sequence of high-school level mathematics questions that were not available on the Internet, we compare the performance of the popular AI software Claude with that of my friends and fellow human beings Pierre and Catherine. Pierre had solid scientific training as a young man, while Catherine studied literature. All three were subjected to a simulated pre-calculus oral exam with main questions and follow-up questions. Their performances are compared and the ones with the best and worst performances are identified. The outcome is that the current version of Claude, even though it is an extremely useful tool that has probably recorded the solution to nearly all calculus questions that are available on the Internet, {\em exhibits only a very limited understanding of the subject} and {\em does not exhibit the ability to make intelligent connections} between different features of a pre-calculus mathematics problem that it has never seen before.

math.GM↗
arXiv2025

Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line

For Gaussian random fields with values in $\mathbb{R}^d$, sharp upper and lower bounds on the probability of hitting a fixed set have been available for many years. These apply in particular to the solutions of systems of linear SPDEs. For non-Gaussian random fields, the available bounds are less sharp. For nonlinear systems of stochastic heat equations, a sharp lower bound was obtained in a previous paper by two of the authors. Here, we obtain the corresponding sharp upper bound. The proof requires a bound on the joint probability density function of a two-dimensional random vector whose components are the solution to the {\em nonlinear} stochastic heat equation and the supremum over a small rectangle of the solution to the {\em linear} stochastic heat equation, in terms of the size of the rectangle. This bound makes use of a formula that expresses the density of a {\em locally nondegenerate} random vector as an iterated Skorohod integral. The main effort is to estimate, using Malliavin calculus, each of the terms that arise from this formula.

math.PR↗
arXiv2025

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

This book is an introduction to the theory of stochastic partial differential equations (SPDEs), using the random field approach pioneered by J.B. Walsh (1986). It consists of two blocks: the core matter (Chapters 1 to 6) and the appendices (A to C). Chapter 1 introduces the subject, with a discussion of isonormal Gaussian processes, space-time white noise, and motivating examples of SPDEs. Chapter 2 presents a theory of stochastic integration with respect to space-time white noise. Chapter 3 deals with SPDEs with additive noise. In Chapter 4, we study a general class of SPDEs, in which additive and multiplicative nonlinearities appear. Chapter 5 discusses asymptotic properties of the solution to the stochastic heat equation such as existence of invariant and reversible measures, convergence in law to the invariant distribution, mixing and irreducibility. In Chapter 6, we prove a theorem on existence and uniqueness of solutions in the weak sense. Then we present a selection of important topics in the theory of SPDEs: the Markov field property, asymptotic bounds on moments of solutions that are useful for studying long-time behavior of the solutions, a comparison theorem for the stochastic heat equation, an introduction to potential theory for SPDEs, and a study of SPDEs with rough initial conditions. Appendix A summarises the main results from the theory of stochastic processes and stochastic analysis that are used throughout the book. Appendix B is devoted to a systematic presentation of properties of fundamental solutions and Green's functions associated to the classical linear differential operators (heat, fractional heat and wave operators). Appendix C is a toolbox section. Each chapter is followed by a "Notes" section, which gives historically important references, original sources and points towards other related important contributions.

math.PR↗
arXiv2024

Sample path properties of parabolic SPDEs with non constant coefficients

We consider an SPDE driven by a parabolic second order partial differential operator with a nonlinear random external forcing defined by a Gaussian noise that is white in time and has a spatially homogeneous covariance. We prove existence and uniqueness of a random field solution to this SPDE. Our main result concerns the space-time sample path regularity of its solution.

math.PR↗
arXiv2021

Multiple Points of Gaussian Random Fields

This paper is concerned with the existence of multiple points of Gaussian random fields. Under the framework of Dalang et al. (2017), we prove that, for a wide class of Gaussian random fields, multiple points do not exist in critical dimensions. The result is applicable to fractional Brownian sheets and the solutions of systems of stochastic heat and wave equations.

math.PR↗
arXiv2020

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

We consider a class of parabolic stochastic PDEs on bounded domains $D\subseteq\mathbb{R}^d$ that includes the stochastic heat equation, but with a fractional power $γ$ of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces $H_r$, with $r < γ- d/2$, we study its power variations in $H_r$ along regular partitions of the time-axis. As the mesh size tends to zero, we find a phase transition at $r=-d/2$: the solutions have a nontrivial quadratic variation when $r<-d/2$ and a nontrivial $p$th order variation for $p= 2γ/(γ-d/2-r)>2$ when $r>-d/2$. More generally, suitably normalized power variations of any order satisfy a genuine law of large numbers in the first case and a degenerate limit theorem in the second case. When $r<-d/2$, the quadratic variation is given explicitly via an expression that involves the spectral zeta function, which reduces to the Riemann zeta function when $d=1$ and $D$ is an interval.

math.PR↗
arXiv2020

Polarity of almost all points for systems of non-linear stochastic heat equations in the critical dimension

We study vector-valued solutions $u(t,x)\in\mathbb{R}^d$ to systems of nonlinear stochastic heat equations with multiplicative noise: \begin{equation*} \frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)+σ(u(t,x))\dot{W}(t,x). \end{equation*} Here $t\geq 0$, $x\in\mathbb{R}$ and $\dot{W}(t,x)$ is an $\mathbb{R}^d$-valued space-time white noise. We say that a point $z\in\mathbb{R}^d$ is polar if \begin{equation*} P\{u(t,x)=z\text{ for some $t>0$ and $x\in\mathbb{R}$}\}=0. \end{equation*} We show that in the critical dimension $d=6$, almost all points in $\mathbb{R}^d$ are polar.

math.PR↗
arXiv2018

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

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 $(u(s, y), u (t, x))$. From this result, we deduce optimal lower bounds on hitting probabilities of the process $\{u(t, x): (t, x) \in [0, \infty[ \times \mathbb{R}\}$ in the non-Gaussian case, in terms of Newtonian capacity, which is as sharp as that in the Gaussian case. This also improves the result in Dalang, Khoshnevisan and Nualart [\textit{Probab. Theory Related Fields} \textbf{144} (2009) 371--424] for systems of classical stochastic heat equations. We also establish upper bounds on hitting probabilities of the solution in terms of Hausdorff measure.

math.PR↗
arXiv2018

Srishti Dhar Chatterji (1935-2017): In Memoriam

This article discusses the life and work of Professor Srishti Dhar Chatterji, who passed away on September 28, 2017, in Lausanne, Switzerland, most suddenly and unexpectedly, after a very brief illness. Complete bibliographical information is included

math.HO↗
arXiv2018

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

We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise. We identify conditions for existence for these two kinds of solutions, and we identify conditions under which they are essentially equivalent. We establish a necessary condition for the existence of a random field solution to a linear SPDE, and we apply this result to the linear stochastic heat, wave and Poisson equations driven by a symmetric $α$-stable noise.

math.PR↗
arXiv2018

Path properties of the solution to the stochastic heat equation with Lévy noise

We consider sample path properties of the solution to the stochastic heat equation, in $\mathbb{R}^d$ or bounded domains of $\mathbb{R}^d$, driven by a Lévy space-time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a càdlàg modification in fractional Sobolev spaces of index less than $-\frac d 2$. Concerning the partial regularity of the solution in time or space when the other variable is fixed, we determine critical values for the Blumenthal-Getoor index of the Lévy noise such that noises with a smaller index entail continuous sample paths, while Lévy noises with a larger index entail sample paths that are unbounded on any non-empty open subset. Our results apply to additive as well as multiplicative Lévy noises, and to light- as well as heavy-tailed jumps.

math.PR↗
arXiv2017

Global solutions to reaction-diffusion equations with super-linear drift and multiplicative noise

Let $ξ(t\,,x)$ denote space-time white noise and consider a reaction-diffusion equation of the form \[ \dot{u}(t\,,x)=\tfrac12 u"(t\,,x) + b(u(t\,,x)) + σ(u(t\,,x)) ξ(t\,,x), \] on $\mathbb{R}_+\times[0\,,1]$, with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists $\varepsilon>0$ such that $\vert b(z)\vert \ge|z|(\log|z|)^{1+\varepsilon}$ for all sufficiently-large values of $|z|$. When $σ\equiv 0$, it is well known that such PDEs frequently have non-trivial stationary solutions. By contrast, Bonder and Groisman (2009) have recently shown that there is finite-time blowup when $σ$ is a non-zero constant. In this paper, we prove that the Bonder--Groisman condition is unimproveable by showing that the reaction-diffusion equation with noise is "typically" well posed when $\vert b(z) \vert =O(|z|\log_+|z|)$ as $|z|\to\infty$. We interpret the word "typically" in two essentially-different ways without altering the conclusions of our assertions.

math.PR↗
arXiv2017

Hausdorff dimension of the boundary of bubbles of additive Brownian motion and of the Brownian sheet

We first consider the additive Brownian motion process $(X(s_1,s_2),\ (s_1,s_2) \in \mathbb{R}^2)$ defined by $X(s_1,s_2) = Z_1(s_1) - Z_2 (s_2)$, where $Z_1$ and $Z_2 $ are two independent (two-sided) Brownian motions. We show that with probability one, the Hausdorff dimension of the boundary of any connected component of the random set $\{(s_1,s_2)\in \mathbb{R}^2: X(s_1,s_2) >0\}$ is equal to $$ \frac{1}{4}\left(1 + \sqrt{13 + 4 \sqrt{5}}\right) \simeq 1.421\, . $$ Then the same result is shown to hold when $X$ is replaced by a standard Brownian sheet indexed by the nonnegative quadrant.

math.PR↗
arXiv2015

Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions

We study the nonlinear stochastic heat equation in the spatial domain $\mathbb {R}$, 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 $\mathbb {R}$, such as the Dirac delta function, but this measure may also have noncompact support and even be nontempered (e.g., with exponentially growing tails). Existence and uniqueness of a random field solution is proved without appealing to Gronwall's lemma, by keeping tight control over moments in the Picard iteration scheme. Upper bounds on all $p$th moments $(p\ge2)$ are obtained as well as a lower bound on second moments. These bounds become equalities for the parabolic Anderson model when $p=2$. We determine the growth indices introduced by Conus and Khoshnevisan [Probab. Theory Related Fields 152 (2012) 681-701].

math.PR↗