|
Title
|
On small time asymptotics of solutions of stochastic equations in infinite dimensions
|
| Author(s) |
Jegaraj, Terence Joseph, Mathematics & Statistics, Faculty of Science, UNSW
|
| Resource Type |
Thesis
PhD Doctorate
|
| Supervisor(s) |
Goldys, Beniamin, Mathematics & Statistics, Faculty of Science, UNSW
|
| Keyword(s) |
Asymptotics.
Spdes.
Dimensions.
Infinite.
Equations.
|
| Date |
2007 |
| School/Centre |
University of New South Wales. |
| Description/Abstract |
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 μ. Under some conditions L(Xx(t)) has a
density k(t, x, ·) with respect to μ 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(ǫt) :
ǫ ∈ (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 μ, L(X(0)) is absolutely continuous with respect
to μ and the transition semigroup is holomorphic. |
| Language |
EN |
| Rights |
Please click here to view the rights |
| Print Availability |
T/2007/112 (ask at Level 2 Information Desk, UNSW Library) |
| Citation Link |
Please use this identifier to cite or link to this item: http://handle.unsw.edu.au/1959.4/36969 |
| Full Text | 
|
| Total Attachment(s) | 2 |
List Of Titles
