In this thesis, an integral representation theorem is obtained for non-negative solutions of the heat equation on X = (//R)('n-1) x (0,(INFIN)) x (0,T) and their boundary behaviour is investigated by using the abstract Fatou-Naim-Doob theorem. The boundary behaviour of positive solutions of the equation Lu = 0 on Y = (//R)('n) x (0,T), where L is a uniformly parabolic second-order differential operator in divergence form is also studied. / In particular, the notion of semi-thinness is introduced for the corresponding potential theories on X and Y and relationships between fine, semi-fine and parabolic limits are obtained. / Results of Kemper specialised to X are obtained by means of fine convergence and a Carleson-type local Fatou theorem is obtained for solutions of Lu = 0 on a union of parabolic regions.

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This thesis investigates the small time asymptotics of solutions of stochastic equations in infinite dimensions. In this abstract H denotes a separable Hilbert space, A denotes a linear operator on H generating a strongly continuous semigroup and (W(t))t???0 denotes a separable Hilbert space-valued Wiener process. In chapter 2 we consider the mild solution (Xx(t))t???[0,1] of a stochastic initial value problem dX = AX dt + dW t ??? (0, 1] X(0) = x ??? H , where the equation has an invariant measure ??. Under some conditions L(Xx(t)) has a density k(t, x, ??) with respect to ?? and we can find the limit limt???0 t ln k(t, x, y). For infinite dimensional H this limit only provides the lower bound of a large deviation principle (LDP) for the family of continuous trajectory-valued random variables { t ??? [0, 1] ??? Xx(??t) : ?? ??? (0, 1]}. In each of chapters 3, 4 and 5 we find an LDP which describes the small time asymptotics of the continuous trajectories of the solution of a stochastic initial value problem. A crucial role is played by the LDP associated with the Gaussian trajectory-valued random variable of the noise. Chapter 3 considers the initial value problem dX(t) = (AX(t) + F(t,X(t))) dt + G(X(t)) dW(t) t ??? (0, 1] X(0) = x ??? H, where drift function F(t, ??) is Lipschitz continuous on H uniformly in t ??? [0, 1] and diffusion function G is Lipschitz continuous, taking values that are Hilbert-Schmidt operators. Chapter 4 considers an equation with dissipative drift function F defined on a separable Banach space continuously embedded in H; the solution has continuous trajectories in the Banach space. Chapter 5 considers a linear initial value problem with fractional Brownian motion noise. In chapter 6 we return to equations with Wiener process noise and find a lower bound for liminft???0 t ln P{X(0) ??? B,X(t) ??? C} for arbitrary L(X(0)) and Borel subsets B and C of H. We also obtain an upper bound for limsupt???0 t ln P{X(0) ??? B,X(t) ??? C} when the equation has an invariant measure ??, L(X(0)) is absolutely continuous with respect to ?? and the transition semigroup is holomorphic.

Bibliography: leaves 110-115. / iv, 115 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1999

vii, 159 leaves : ill., graphs, tables ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1979

Thesis (Ph. D.)--University of Adelaide, 1987. / Errata inserted. Includes bibliographical references (leaves 170-175).

Thesis (Ph.D.)--York University, 2005. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 69-75). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR11585

Thesis (Ph. D.)--University of Oregon, 2006. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 90-99). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis ( Ph.D. ) -- McMaster University, 1997. / Includes bibliographical references (leaves 63-66). Also available via World Wide Web.

Thesis (Ph.D.)--McMaster University (Canada), 1996. / Source: Dissertation Abstracts International, Volume: 57-10, Section: B, page: 6300. Adviser: H. P. Heinig.