On small time asymptotics of solutions of stochastic equations in infinite dimensions

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.