On the density of the supremum of the solution to the linear stochastic heat equation
- PDF / 720,353 Bytes
- 48 Pages / 439.37 x 666.142 pts Page_size
- 39 Downloads / 252 Views
On the density of the supremum of the solution to the linear stochastic heat equation Robert C. Dalang1 · Fei Pu1 Received: 12 December 2018 / Revised: 10 April 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract 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 smoothness of the joint density of the random vector whose components are the solution and the supremum of an increment in time of the solution over an interval (at a fixed spatial position), and the smoothness of the density of the supremum of the solution over a space-time rectangle that touches the t = 0 axis. Applying the properties of the divergence operator, we establish a Gaussian-type upper bound on these two densities respectively, which presents a close connection with the Hölder-continuity properties of the solution. Keywords Stochastic heat equation · Supremum of a Gaussian random field · Probability density function · Gaussian-type upper bound · Malliavin calculus Mathematics Subject Classification Primary 60H15 · 60J45; Secondary: 60H07 · 60G60
1 Introduction and main results For a real-valued Gaussian random field {X (t) : t ∈ I }, where I is a parameter set, the distribution function of the supremum of this random field, or the excursion probabil-
R. C. Dalang, F. Pu: This paper is based on F. Pu’s Ph.D. thesis, written under the supervision of R. C. Dalang. Research partially supported by the Swiss National Foundation for Scientific Research.
B
Fei Pu [email protected] Robert C. Dalang [email protected]
1
Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Station 8, 1015 Lausanne, Switzerland
123
Stoch PDE: Anal Comp
ity P{supt∈I X (t) a}, has been investigated extensively; see, for example, [2,3,28] and references therein. In general, finding a formula for the distribution function of the supremum of a stochastic process is an almost impossible task, let alone for its probability density function, which is much less studied than the probability distribution function. The question of smoothness of the density of the supremum of a multiparameter Gaussian process dates back to the work of Florit and Nualart [14], in which they establish a general criterion (see Theorem 2.1) for the smoothness of the density, assuming that the random vector is locally in D∞ , and apply it to show that the maximum of the Brownian sheet on a rectangle possesses an infinitely differentiable density. Moreover, this method was applied to prove that the supremum of a fractional Brownian motion has an infinitely differentiable density; see Lanjri Zadi and Nualart [19]. Nourdin and Viens [24] use the Malliavin calculus to obtain a formula for the density of the law of any random variable which is measurable and differentiable with respect to a given isonormal Gaussian process and they apply this result to stu
Data Loading...
