Heat Equations with Fractional White Noise Potentials

Heat Equations with Fractional White Noise Potentials This paper is concerned with the following stochastic heat equations: $$\frac{{\partial u_t (x)}} {{\partial t}} = \frac{1} {2}\Delta u_t (x) + w^H \cdot u_t (x), x \in \mathbb{R}^d , t > 0,$$ where w H is a time independent fractional white noise with Hurst parameter H=(h 1 , h 2 ,..., h d ) , or a time dependent fractional white noise with Hurst parameter H=(h 0 , h 1 ,..., h d ) . Denote |H|=h 1 +h 2 +...+h d . When the noise is time independent, it is shown that if ½ <h i <1 for i=1, 2,..., d and if |H|>d-1 , then the solution is in L 2 and the L 2 -Lyapunov exponent of the solution is estimated. When the noise is time dependent, it is shown that if ½ <h i <1 for i=0, 1,..., d and if |H|>d- 2 /( 2h 0 -1 ) , the solution is in L 2 and the L 2 -Lyapunov exponent of the solution is also estimated. A family of distribution spaces S ρ , ρ∈ \RR , is introduced so that every chaos of an element in S ρ is in L 2 . The Lyapunov exponents in S ρ of the solution are also estimated. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Heat Equations with Fractional White Noise Potentials

Loading next page...
 
/lp/springer_journal/heat-equations-with-fractional-white-noise-potentials-W6Pevu6pDW
Publisher
Springer-Verlag
Copyright
Copyright © 2001 by Springer-Verlag New York Inc.
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-001-0001-2
Publisher site
See Article on Publisher Site

Abstract

This paper is concerned with the following stochastic heat equations: $$\frac{{\partial u_t (x)}} {{\partial t}} = \frac{1} {2}\Delta u_t (x) + w^H \cdot u_t (x), x \in \mathbb{R}^d , t > 0,$$ where w H is a time independent fractional white noise with Hurst parameter H=(h 1 , h 2 ,..., h d ) , or a time dependent fractional white noise with Hurst parameter H=(h 0 , h 1 ,..., h d ) . Denote |H|=h 1 +h 2 +...+h d . When the noise is time independent, it is shown that if ½ <h i <1 for i=1, 2,..., d and if |H|>d-1 , then the solution is in L 2 and the L 2 -Lyapunov exponent of the solution is estimated. When the noise is time dependent, it is shown that if ½ <h i <1 for i=0, 1,..., d and if |H|>d- 2 /( 2h 0 -1 ) , the solution is in L 2 and the L 2 -Lyapunov exponent of the solution is also estimated. A family of distribution spaces S ρ , ρ∈ \RR , is introduced so that every chaos of an element in S ρ is in L 2 . The Lyapunov exponents in S ρ of the solution are also estimated.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jan 1, 2001

There are no references for this article.

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off