# 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

, Volume 43 (3) – Jan 1, 2001
23 pages

/lp/springer_journal/heat-equations-with-fractional-white-noise-potentials-W6Pevu6pDW
Publisher
Springer-Verlag
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

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