Small time asymptotics of Ornstein-Uhlenbeck densities in Hilbert spaces

We show that Varadhan's small time asymptotics for densities of the solution of a stochastic differential equation in $\mathbb{R}^n$ carries over to a Hilbert space-valued Ornstein-Uhlenbeck process whose transition semigroup is strongly Feller and symmetric. In the Hilbert space setting, densities are with respect to a Gaussian invariant measure.


Introduction
Varadhan [8] investigated the small time asymptotics of the probability densities of an n -valued diffusion process (z ζ (t)) t≥0 with initial point ζ ∈ n . Denoting the density of z ζ (t) by p(t, ζ, ·), Varadhan showed that lim t→0 t ln p(t, ζ, y) = − 1 2 d 2 (ζ, y) uniformly for ζ and y in any bounded subset of n . In equality (1) d(ζ, y) := inf 1 0 〈u(τ), a −1 (u(τ))u(τ)〉 n dτ : u : [0, 1] → n is absolutely continuous with derivativeu and u(0) = ζ and u(1) = y , where 〈·, ·〉 n is the scalar product in n and a is the diffusion matrix in the stochastic differential equation which (z ζ (t)) t≥0 solves. The small time asymptotics formula for densities (1) has been shown to hold in many different settings, for example Norris [6] showed that the formula holds in a finite dimensional Lipschitz Riemannian manifold, with the definition of the distance function d depending on the manifold.
In the setting of an infinite dimensional separable Hilbert space H, let (X x (t)) t≥0 be the mild solution of the stochastic initial value problem where A is a linear operator on H and W is a (possibly cylindrical) Wiener process on H. Only in special situations is the distribution of X x (t) absolutely continuous with respect to a natural reference measure on H at all times t > 0. We consider one such situation, namely when an invariant measure µ exists and the transition semigroup is strongly Feller and symmetric on L 2 (H, µ). Under these conditions we obtain the small time limiting behaviour of the probability density of X x (t) with respect to µ. The continuous density k(t, x, ·) of X x (t) in Proposition 1 is valid whenever the transition semigroup is strongly Feller and symmetric; we have the small time limit in Proposition 2 when Assumption 4 also holds. Assumption 4 is rather restrictive, nevertheless it is interesting that the limit in equation (10) is of the same form as that in equation (1).
In the next section we present the main results and their proofs and finish with an example.

Small time limiting behaviour of densities
. Let (W (t)) t≥0 be a Hilbert space-valued Wiener process defined on a probability space (Ω, , P) and such that the distribution of W (1) has reproducing kernel Hilbert space H W . If Q is a trace class operator then (W (t)) t≥0 is a H-valued Wiener process, otherwise it is a cylindrical Wiener process on H (see [3,Proposition 4.11]). In this article Q need not be trace class. The embedding of H W into H is denoted by i : H W → H.
We use the symbol (m, C) to denote a Gaussian measure on the Borel sets of H, with mean m and covariance operator C.

Assumption 1 A trace class operator on H is defined by
Set µ := (0, Q ∞ ). For each t > 0 the operator is trace class and ker Q t = {0}. The mild solution of the initial value problem (2) at positive times t, has distribution (S(t)x, Q t ) and µ is an invariant measure for the equation in (2). Define the strongly continuous transition semigroup (R t ) t≥0 on L 2 (H, µ) by .
defines a strongly continuous semigroup of contractions on H. Some consequences of Assumption 2 are that for each t > 0 As shown in [4, Lemma 10.3.3], it follows that for each t > 0 and each x ∈ H the distribution of X x (t), (S(t)x, Q t ), is absolutely continuous with respect to µ and its Radon-Nikodym derivative for µ a.e. y ∈ H, where I H is the identity operator on H and Θ t := S 0 (t)S * 0 (t). The second and third terms in the argument of the exponential function in equation (4) are defined for only µ a.e. y, in terms of limits (see for example [4, Proposition 1.2.10]).

Assumption 3 The operators R t are symmetric for all t ≥ 0.
Chojnowska-Michalik and Goldys [2, Lemma 2.2] have shown that symmetry of R t is equivalent to symmetry of S 0 (t) and this allows us to prove that there is a continuous version of the Radon-Nikodym derivative in equation (4).

Proposition 1. Under Assumptions 1 to 3, there is a continuous version of the Radon-Nikodym derivative
, which we denote by k(t, x, ·): for all y ∈ H.
Proof. Define the bounded linear bijections From equality (6) we have The other two terms in the argument of the exponential in equation (4) are defined in terms of limits. Let ( f k ) be an orthonormal basis of H made up of eigenvectors of Q ∞ . For each n ∈ define P n to be the orthogonal projection onto the linear span of { f 1 , . . . , f n }. In the following expressions (n k ) denotes some strictly increasing sequence of natural numbers. We have We have Substituting the expressions from equalities (7), (8) and (9) into the right hand side of equation (4), we get the formula for k(t, x, y) shown in equation (5).
When x and y belong to Q 1 2 (H) we can write k(t, x, y) in terms of the eigenvalues of A 0 , the infinitesimal generator of (S 0 (t)) t≥0 ; then it is straightforward to find lim t→0 t ln k(t, x, y). The results obtained in this way can be of interest only if µ(Q 1 2 (H)) = 1. We now introduce a further assumption to ensure that µ(Q Chojnowska-Michalik and Goldys [2, Theorem 5.1] showed that µ(Q Thus Assumption 4 is equivalent to the assumption that µ(Q 1 2 (H)) = 1. We have for t > 0:

Proposition 2. Under Assumptions 1 to 4 we have for all x and y in Q
It remains to find the limit of t times the argument of the exponential function in equation (5).
The key to this is equality (17), which we now derive.
Let t > 0. We have By [2, Proposition 2.10] therefore t is one to one and has a dense range, its adjoint (Q − 1 2 Q 1 2 t ) * has the same properties. From equation (13) we have notice that, since S Q (2t) L(H,H) < 1, (I H − S Q (2t)) is invertible and the range of the operator in equation (15) is D(A Q ). Taking inverses on both sides of equation (15) we have Let r > 0. Then since A Q is self-adjoint, Hence for u, v ∈ H equation (16) yields The expression on the right hand side of equality (17) appears in equation (5)  Recall that (g k ) is an orthonormal basis of H such that A Q g k = −α k g k for each k ∈ . Setting u k := 〈u, g k 〉 and v k := 〈v, g k 〉 for k ∈ , we have from equality (17): and the convergence is uniform for u and v in any compact subset of H. The uniform convergence on compact sets is because for any compact set K ⊂ H we have sup{ ∞ j=n 〈u, g j 〉 2 : u ∈ K} → 0 as n goes to infinity. Similarly we have and the convergence is uniform for u in any compact subset of H.
Using the results in (12), (19) and (20), we have for x and y in Q 1 2 (H): hence in this case we have x = y and lim t→0 t ln k(t, x, y) = ∞.