Systems Analysis and Simulation I pp 97-104 | Cite as

# Analytical Analysis of a Stochastic Partial Differential Equation

Conference paper

## Abstract

This paper investigates the following stochastic partial differential equation where and with \({\eta _n} = \lambda _n^\alpha {\xi _n},\,n = 0,1,2, \cdots \) and

$$d{X_t} = - A{X_t}dt + G\left( {{X_t}} \right)d{B_t}$$

(a1)

*A*is a self-adjoint, positive definite operator with eigensystem$$\left\{ {{\lambda _n};{\xi _n};n = 0,1,2, \ldots } \right\}$$

*G*(·):*H*_{−α}→ σ_{2}(*H, H*_{−α}) (α≥0) is defined by$$\matrix{ {G\left( h \right){\xi _n} = {{\left( {h,{\eta _n}} \right)}_{ - \alpha }}{\eta _n}} & {n = 0,1,2, \cdots } & {h \in {H_{ - \alpha }}} \cr } $$

*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 B_{t}> 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 where \({X_0} = \sum\limits_{n = 0}^\infty {{c_n}{\eta _n} \in {H_{ - \alpha }}} \), and

$${X_t} = {e^{ - {t \over 2}}}\sum\limits_{n = 0}^\infty {{c_n}{e^{ - {\lambda _n}t}}{\eta _n}{e^{{w_n}\left( t \right)}}} $$

(a2)

*w*_{ n }(*t*) =*B*_{ t }(ξ_{ 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.

## Keywords

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

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## References

- [1]A. V. Balakrishnan, “Stochastic bilinear partial differential equations”,
*Proc. of the 2nd US.-Italy Seminar on Variable Structure Systems*, May, 1974;*Lecture Notes in Systems Theory 111*, Springer-Verlag, 1975, ed. A. Ruberti and R. R. Mohler.Google Scholar - [2]A. T. Bharucha-Reid,
*Random Integral Equations*, Academic Press, New York, 1972MATHGoogle Scholar - [3]D. A. Dawson, “Stochastic evolution equation”,
*Mathematical Biosiences*, Vol 15, 1972, pp 287–316.MATHCrossRefGoogle Scholar - [4]Y. Miyahara, “Stability of linear stochastic differential equations in Hubert space”,
*Information, Decision and Control in Dynamic Socio-Economics*, ed. H. Myoken, Bunshindo/Kinokuniyo, Tokyo, 1978, pp 237–252Google Scholar - [5]Y. Miyahara, “Infinite dimensional Langevin equation and Fokker-Planck equation”,
*Nagoya Math. J*. Vol. 81, 1981, pp 177–223MathSciNetMATHGoogle Scholar - [6]Y. Miyahara, “Stochastic differential equations in Hubert space”,
*Oikonomika*, Vol. 14, No. 1, June, 1977, pp 37–47Google Scholar - [7]A. Shimizu, “Construction of a solution of linear stochastic evolution equations on a Hilbert space”,
*Proc. of the Interna. Symp. on Stoch. Differen. eq*, Kyoto, 1976, pp 385–395Google Scholar

## Copyright information

© Akademie-Verlag Berlin 1988