A local-time correspondence for stochastic partial differential equations

Mohammud Foondun, Davar Khoshnevisan, Eulalia Nualart

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have randomfield solutions only in spatial dimension one. Here we show that in many cases, where the "spatial operator" is the L2-generator of a Lévy 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 larger than one. In addition, we prove that the solution to the SPDE is [Ḧolder] continuous in its spatial variable if and only if the said local time is [Ḧolder] 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 L2- 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 mainly study 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 randomfield solution, then so does the nonlinear SPDE. Moreover, the solution to the linearized equation is [Ḧolder] 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 where the solutions are unbounded, they almost surely blow up at exactly the same points.

LanguageEnglish
Pages2481-2515
Number of pages35
JournalTransactions of the American Mathematical Society
Volume363
Issue number5
Early online date3 Dec 2010
DOIs
Publication statusPublished - 31 May 2011

Fingerprint

Stochastic Partial Differential Equations
Local Time
Partial differential equations
Correspondence
Random Field
Symmetrization
If and only if
Quasi-isometry
Dirichlet Space
Hyperbolic Partial Differential Equations
Linear partial differential equation
Lévy Process
Linear Space
Assertion
White noise
Nonlinear Partial Differential Equations
Blow-up
One Dimension
Weak Solution
Linear equation

Keywords

  • stochastic partial differential equation
  • linear equations
  • local-time correspondance

Cite this

@article{17650904c73f4c8990a7d3309e7fcb03,
title = "A local-time correspondence for stochastic partial differential equations",
abstract = "It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have randomfield solutions only in spatial dimension one. Here we show that in many cases, where the {"}spatial operator{"} is the L2-generator of a L{\'e}vy 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 larger than one. In addition, we prove that the solution to the SPDE is [Ḧolder] continuous in its spatial variable if and only if the said local time is [Ḧolder] 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 L2- 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 mainly study 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 randomfield solution, then so does the nonlinear SPDE. Moreover, the solution to the linearized equation is [Ḧolder] 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 where the solutions are unbounded, they almost surely blow up at exactly the same points.",
keywords = "stochastic partial differential equation, linear equations, local-time correspondance",
author = "Mohammud Foondun and Davar Khoshnevisan and Eulalia Nualart",
year = "2011",
month = "5",
day = "31",
doi = "10.1090/S0002-9947-2010-05017-2",
language = "English",
volume = "363",
pages = "2481--2515",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
number = "5",

}

A local-time correspondence for stochastic partial differential equations. / Foondun, Mohammud; Khoshnevisan, Davar; Nualart, Eulalia.

In: Transactions of the American Mathematical Society, Vol. 363, No. 5, 31.05.2011, p. 2481-2515.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A local-time correspondence for stochastic partial differential equations

AU - Foondun, Mohammud

AU - Khoshnevisan, Davar

AU - Nualart, Eulalia

PY - 2011/5/31

Y1 - 2011/5/31

N2 - It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have randomfield solutions only in spatial dimension one. Here we show that in many cases, where the "spatial operator" is the L2-generator of a Lévy 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 larger than one. In addition, we prove that the solution to the SPDE is [Ḧolder] continuous in its spatial variable if and only if the said local time is [Ḧolder] 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 L2- 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 mainly study 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 randomfield solution, then so does the nonlinear SPDE. Moreover, the solution to the linearized equation is [Ḧolder] 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 where the solutions are unbounded, they almost surely blow up at exactly the same points.

AB - It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have randomfield solutions only in spatial dimension one. Here we show that in many cases, where the "spatial operator" is the L2-generator of a Lévy 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 larger than one. In addition, we prove that the solution to the SPDE is [Ḧolder] continuous in its spatial variable if and only if the said local time is [Ḧolder] 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 L2- 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 mainly study 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 randomfield solution, then so does the nonlinear SPDE. Moreover, the solution to the linearized equation is [Ḧolder] 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 where the solutions are unbounded, they almost surely blow up at exactly the same points.

KW - stochastic partial differential equation

KW - linear equations

KW - local-time correspondance

UR - http://www.scopus.com/inward/record.url?scp=79951821957&partnerID=8YFLogxK

UR - http://www.ams.org/publications/journals/journalsframework/tran

U2 - 10.1090/S0002-9947-2010-05017-2

DO - 10.1090/S0002-9947-2010-05017-2

M3 - Article

VL - 363

SP - 2481

EP - 2515

JO - Transactions of the American Mathematical Society

T2 - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -