Stochastic Partial Differential Equations Driven by Brownian White Noise

Holden, Helge; Øksendal, Bernt; Ubøe, Jan; Zhang, Tusheng

doi:10.1007/978-0-387-89488-1_4

Cited by 109 publications

(268 citation statements)

References 0 publications

Supporting

Mentioning

260

Contrasting

Order By: Relevance

“…For additional information on the Wick product the reader is referred to the book of Holden, et al (2009), the papers Da Pelo, et al (2011), Da Pelo, et al (2013 and the references quoted there.…”

Section: This Is Called the Wick Product Of E(h) And E(k)mentioning

confidence: 99%

A note on a local limit theorem for Wiener space valued random variables

Lanconelli¹,

Stan²

2016

Bernoulli

View full text Add to dashboard Cite

We prove a local limit theorem, i.e. a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is assumed to be absolutely continuous with respect to the reference Gaussian measure. We begin by showing that the key roles of scaling operator and convolution product in this infinite dimensional Gaussian framework are played by the Ornstein-Uhlenbeck semigroup and Wick product, respectively. We proceed by establishing a necessary condition on the density of the random variables for the local limit theorem to hold true. We then reverse the implication and prove under an additional assumption the desired L 1 -convergence of the density of X 1 +···+Xn √ n . We close the paper comparing our result with certain Berry-Esseen bounds for multidimensional central limit theorems.

show abstract

Section: This Is Called the Wick Product Of E(h) And E(k)mentioning

confidence: 99%

A note on a local limit theorem for Wiener space valued random variables

Lanconelli¹,

Stan²

2016

Bernoulli

View full text Add to dashboard Cite

show abstract

“…In terms of well-posedness, the technology at the present time [23] can only handle far smoother noise terms than the white noise. An unusual type of Wick product version of the problem has been introduced [12]. But besides requiring fairly smooth noises, this does not have the scaling expected [9], and is therefore believed not to be physically relevant.…”

Section: (S B(s))ds}z(0 B(t))]mentioning

confidence: 99%

Fluctuation exponent of the KPZ/stochastic Burgers equation

Balázs

Quastel

Seppäläinen

2011

J. Amer. Math. Soc.

106

View full text Add to dashboard Cite

We consider the stochastic heat equation \[ ∂ t Z = ∂ x 2 Z − Z W ˙ \partial _tZ= \partial _x^2 Z - Z \dot W \] on the real line, where W ˙ \dot W is space-time white noise. h ( t , x ) = − log ⁡ Z ( t , x ) h(t,x)=-\operatorname {log} Z(t,x) is interpreted as a solution of the KPZ equation, and u ( t , x ) = ∂ x h ( t , x ) u(t,x)=\partial _x h(t,x) as a solution of the stochastic Burgers equation. We take Z ( 0 , x ) = exp ⁡ { B ( x ) } Z(0,x)=\exp \{B(x)\} , where B ( x ) B(x) is a two-sided Brownian motion, corresponding to the stationary solution of the stochastic Burgers equation. We show that there exist 0 > c 1 ≤ c 2 > ∞ 0> c_1\le c_2 >\infty such that \[ c 1 t 2 / 3 ≤ Var ⁡ ( log ⁡ Z ( t , x ) ) ≤ c 2 t 2 / 3 . c_1t^{2/3}\le \operatorname {Var}(\operatorname {log} Z(t,x) )\le c_2 t^{2/3}. \] Analogous results are obtained for some moments of the correlation functions of u ( t , x ) u(t,x) . In particular, it is shown there that the bulk diffusivity satisfies \[ c 1 t 1 / 3 ≤ D bulk ( t ) ≤ c 2 t 1 / 3 . c_1t^{1/3}\le D_\textrm {bulk}(t) \le c_2 t^{1/3}. \] The proof uses approximation by weakly asymmetric simple exclusion processes, for which we obtain the microscopic analogies of the results by coupling.

show abstract

“…It is known [18] that, with standard Brownian motion, the Wick product and the usual calculus lead to the same results as the usual product and the Itô calculus. While the use of the Wick product has been questioned as a modeling tool for certain applications in economics and finance [8], it is still an effective tool for theoretical investigations, corresponding to the Itô-Skorokhod integral in the white noise analysis.…”

Section: S V Lototsky and K Stemmannmentioning

confidence: 94%

“…Traditionally, a Gaussian process is defined by its mean and covariance functions, but then the definition of the integral immediately leads to a number of technical conditions on these functions [2]. An alternative definition is possible [29] by combining the ideas from the theory of generalized Gaussian fields [15,30], the white noise theory [17,18], and the Malliavin calculus [31,34]. This approach to stochastic integration is used in this paper and is outlined in Section 2 Numerical methods for stochastic ordinary differential equations driven by white noise are a well-developed subject [24,32].…”

Section: S V Lototsky and K Stemmannmentioning

confidence: 99%

Solving SPDEs driven by colored noise: A chaos approach

Lototsky¹,

Stemmann²

2008

Quart. Appl. Math.

View full text Add to dashboard Cite

Abstract. An Itô-Skorokhod bilinear equation driven by infinitely many independent colored noises is considered in a normal triple of Hilbert spaces. The special feature of the equation is the appearance of the Wick product in the definition of the Itô-Skorokhod integral, requiring innovative approaches to computing the solution. A chaos expansion of the solution is derived and several truncations of this expansion are studied. A recursive approximation of the solution is suggested and the corresponding approximation error bound is computed.1. Introduction. Stochastic differential equations driven by Gaussian white noise are well-studied; see, for example, the book [35] for ordinary differential equations and the book [37] for equations with partial derivatives. The underlying stochastic process in these equations is the standard Brownian motion W , which is a square-integrable Gaussian martingale with continuous trajectories and independent increments. A lot less is known about equations driven by colored noise, when the underlying process is still Gaussian, but no longer has independent increments. An important example is the fractional Brownian motion W H , H ∈ (0, 1), which coincides with the standard Brownian motion W for H = 1/2 and is not a semi-martingale for all H = 1/2. It is the lack of the semi-martingale property that makes the analysis difficult at the very basic level, the definition of the corresponding stochastic integral. Several versions of the stochastic integral with respect to W H have been proposed [1,12,13,14,23,26,38]. Unlike the standard Brownian motion, different approaches such as Itô-type vs. Stratonovichtype integral or path-wise vs. mean-square definition, become much more difficult to reconcile. The paper by V. Pipiras and M. Taqqu [36] describes the main difference

show abstract

Stochastic Partial Differential Equations Driven by Brownian White Noise

Cited by 109 publications

References 0 publications

A note on a local limit theorem for Wiener space valued random variables

A note on a local limit theorem for Wiener space valued random variables

Fluctuation exponent of the KPZ/stochastic Burgers equation

Solving SPDEs driven by colored noise: A chaos approach

Contact Info

Product

Resources

About