Analytical Analysis of a Stochastic Partial Differential Equation

  • Weijian Zhang
Conference paper
Part of the Advances in Simulation book series (ADVS.SIMULATION, volume 1)


This paper investigates the following stochastic partial differential equation
$$d{X_t} = - A{X_t}dt + G\left( {{X_t}} \right)d{B_t}$$
where A is a self-adjoint, positive definite operator with eigensystem
$$\left\{ {{\lambda _n};{\xi _n};n = 0,1,2, \ldots } \right\}$$
and G(·): H −α → σ2(H, H −α) (α≥0) is defined by
$$\matrix{ {G\left( h \right){\xi _n} = {{\left( {h,{\eta _n}} \right)}_{ - \alpha }}{\eta _n}} & {n = 0,1,2, \cdots } & {h \in {H_{ - \alpha }}} \cr } $$
with \({\eta _n} = \lambda _n^\alpha {\xi _n},\,n = 0,1,2, \cdots \) and H β,−∞<β<∞ is a Hilbert scale. with generating operator A. And B t , t ≥ 0 is a Hilben scale with generating operator A. And Bt > 0 is a cylindrical Brownian motion on H = L 2 (D) where D is a domain in R d.
The existence and the uniqueness of the solution of (al) is proved and its stability is discussed. By computing the kernels of the integral representation of the solution, the analytical expression of the solution is obtained as
$${X_t} = {e^{ - {t \over 2}}}\sum\limits_{n = 0}^\infty {{c_n}{e^{ - {\lambda _n}t}}{\eta _n}{e^{{w_n}\left( t \right)}}} $$
where \({X_0} = \sum\limits_{n = 0}^\infty {{c_n}{\eta _n} \in {H_{ - \alpha }}} \), and w n (t) = B t n , n = 0,1,2, … is a sequence of independent one-dimensional Brownian motion. The second order statistics of the solution is fully studied by determining the covariance operator.

The approach is along the line of the Wiener-Ito direct sum decomposition of L 2 (E* → H−α) by generalizing the notions of multiple Wiener integral and iterated stochastic integral.


Covariance Operator Stochastic Partial Differential Equation Stochastic Evolution Equation Variable Structure System Positive Definite Operator 


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Copyright information

© Akademie-Verlag Berlin 1988

Authors and Affiliations

  • Weijian Zhang
    • 1
  1. 1.Department of Electrical EngineeringUniversity of California, Los AngelesLos AngelesUSA

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