# Existence and Uniqueness Theorems for Some White Noise Integral Equations.

1996

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

Mathematics

Hui-Hsiung Kuo

## Abstract

Let $({\cal S})\sbsp{\beta}{*},0\le\beta<1,$ be the Kondratiev-Streit spaces of generalized functions. Let $f:\lbrack 0,T\rbrack\times ({\cal S})\sbsp{\beta}{*}\to ({\cal S})\sbsp{\beta}{*},$ be weakly measurable, and satisfy a growth condition and a Lipschitz condition. Let $\theta :\lbrack 0, T\rbrack\to ({\cal S})\sbsp{\beta}{*},$ be weakly measurable and satisfy a growth condition. Then it is shown that the white noise integral equation $X\sb{t}=\theta\sb{t}+\int\sbsp{0}{t}f(s, X\sb{s})ds,0\le t\le T,$ has a unique solution in $({\cal S})\sbsp{\beta}{*}$, where the integral is a white noise integral in the Pettis or Bochner sense. This result is extended to ${\cal M}\sp*$, the Meyer-Yan distribution space. Some special equations are also solved explicitly. For $F\in L\sp2({\bf R}\sp+)$, let $A\sb{s}=\int\sbsp{-\infty}{s} F(s-u)\partial\sb{u}du,\ E\sb{s}= {\rm exp}(A\sb{s}),$ and $A\sbsp{s}{*},\ E\sbsp{s}{*}$ be their duals, respectively. The equation $X\sb{t}=\theta\sb{t}+\int\sbsp{0}{t} A\sbsp{s}{*}X\sb{s}ds, t\in\lbrack 0,T\rbrack,$ is solved in $({\cal S})\sp*$ or $(L\sp2)$, and the equation $X\sb{t}=\theta\sb{t}+\int\sbsp{0}{t} E\sbsp{s}{*}X\sb{s}ds, t\in\lbrack 0, T\rbrack,$ is solved in ${\cal M}\sp*,$ where $\theta$ is as above. Moreover, under certain conditions on $\theta,\Phi:\lbrack 0,T\rbrack\to ({\cal S})\sp*$ and $\sigma:\lbrack 0,T\rbrack\sp2\to{\bf R},$ the Volterra equation $X\sb{t}=\theta\sb{t}+\int\sbsp{0}{t}\sigma(t,s)\Phi\sb{s}\ {\rm o}\ X\sb{s}ds, t\in\lbrack 0,T\rbrack,$ is also solved, and its solution is in ${\cal M}\sp*, ({\cal S})\sbsp{\beta}{*},$ or $(L\sp2),$ depending on the growth conditions for $\theta$ and $\Phi.$ Finally, for a suitable deterministic function f, the white noise partial differential equation ${\partial u\over\partial t}=\Delta u+:e\sp{\dot B\sb{x}}: {\rm o}\thinspace u, u(0,x)=f(x),x\in{\bf R}\sp{n}, t\in\lbrack 0,\infty),$ is solved in ${\cal M}\sp*$.

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