Laplace Asymptotic Expansions for Gaussian Functional Integrals

We obtain a Laplace asymptotic expansion, in orders of $\lambda$, of $$ E^\rho_x \left\{ G(\lambda x) e^{-\lambda ^{-2} F(\lambda x)}\right\}$$ the expectation being with respect to a Gaussian process. We extend a result of Pincus and build upon the previous work of Davies and Truman. Our methods differ from those of Ellis and Rosen in that we use the supremum norm to simplify the application of the result.


Introduction
There is a considerable literature on the development and use of Laplace asymptotic expansions in areas related to mathematical physics. The papers of Schilder [10] and Pincus [9] dealt with Wiener integrals and Gaussian functional integrals respectively. Schilder derived the full structure of the asymptotic expansion and considered examples in the solution of functional equations and the calculus of variations whilst Pincus derived the leading order behaviour for the asymptotic expansion and had applications to Hammerstein integral equations. In the papers of Davies and Truman [1,2,3,4] the Laplace asymptotic expansion of a Conditional Wiener integral ( the underlying process being the Brownian Bridge ) was developed to arbitrarily high orders and applications were made to obtaining generalized Mehler kernel formulae (for Hamiltonians including magnetic fields) and to the Bender-Wu formula (concerned with the behaviour of perturbation series for ground state energies of the anharmonic oscillator). One notable application of this work was in Davies and Truman [5] where the existence of the Meissner-Ochsenfeld effect was proven for an ideal charged Boson gas.
Ellis and Rosen [6,7,8] have developed Laplace asymptotic expansions for Gaussian functional integrals working with the L 2 norm throughout. This gives, perhaps, a cleaner approach to the estimates required in the arguments but in our view the supremum norm is easier to work with especially when one considers applications. It was this approach that was used in the initial extension of Schilder's [10] work and we continued to use it in our subsequent work.
The seminal paper of Schilder has been, and continues to be, of topical interest. In more recent years Azencott and Doss [18] have used asymptotic expansions to study the Schrödinger equation, whilst Azencott [19,20] has used asymptotic expansions to study the density of diffusions for small time and both sequential and parallel annealing. Ben Arous (and co-workers) [21,22,23,24] have developed and utilised Laplace asymptotic techniques to study functional integrals with respect to possibly degenerate diffusions, Strassen's functional law of the iterated logarithm and the asymptotics of solutions to non-homogeneous versions of the KPP equation. Kusuoka and Stroock [25,26] have developed asymptotic expansions of certain Wiener functionals with degenerate extrema for processes on abstract Wiener space whilst Rossignol [27] has used the Newton polyhedron to study the case of Laplace integrals on Wiener space with an isolated degenerate minimum. A generalization of the expansion formula of Ben Arous has been developed by Takanobu and Watanabe [28] in which one can handle Wiener functionals which are smooth in the sense of Malliavin but not necessarily smooth in the sense of Frechet.
Some preliminary notation and the statement of the main result follow in the next section. The subsequent section contains the necessary lemmas (and proofs in some cases) to substantiate the theorem. Those lemmas due to Pincus are included for clarity and his proofs are noted as such. The final section contains the proof of the theorem. The proof is an amalgam and extension of the methods of Pincus [9] and Schilder [10] influenced by our previous work on Conditional Wiener integrals [1,2,3,4].
Given that A has positive eigenvalues let {ρ i } ∞ i=1 be the reciprocal eigenvalues in order of increasing magnitude. We will make use of the notation and Theorem. Let ρ(σ, τ ), 0 ≤ σ, τ ≤ t, be a continuous, symmetric, positivedefinite kernel for which there is a Gaussian process generated by ρ(σ, τ ) having continuous sample paths x(τ ), 0 ≤ τ ≤ t. Let F (x) and G(x) be real valued continuous functionals defined on C ρ [0, t] and suppose that the functional If F (x) and G(x) satisfy the conditions below, then as λ → 0, where the Γ i are integrals dependent only on the functionals F (x), G(x) and their Frechet derivatives evaluated at x * .
We will only prove the theorem in the case of b = 0 since we can deduce the corresponding result for b = 0 by use of the substitution This Theorem is the analogue of Schilder's Theorem C [10] and the proof follows essentially the same route as taken by both Schilder [10] and Pincus [9].
The following three lemmas are concerned with the properties of the operators A and A − 1 2 .
Lemma 2. A − 1 2 is a Hilbert-Schmidt operator with a kernel K(σ, τ ) and is a completely continuous mapping of L 2 [0, t] into C[0, t].
In terms of L 2 A Lemma 2 can be taken to mean that every bounded set in L 2 A is precompact in C. The following lemmas deal with the functional and its properties.
where c 1 < ρ 1 , c 2 ∈ R. It then follows that there exists at least one point Proof. (Pincus) Let B be the set of points at which H(x) attains its global minimum, and let {x n } be a minimising sequence of H(x). We then have Thus we have H(x) bounded below for all x. Without loss of generality we may assume that the global minimum of H(x) is zero. Clearly, when By what immediately follows the proof of Lemma 4 we see that {x n } contains a subsequence {x ni } which forms a Cauchy sequence in We now proceed to show that y ∈ D(A − 1 2 ). {x ni } is a bounded set in L 2 A and so is weakly precompact. Since any Hilbert space is weakly complete we see that there exists a subsequence of {x ni } which converges weakly in L 2 A to a point u ∈ L 2 A . We also denote this subsequence as {x ni } to retain clarity. By the definition of weak convergence we have for all z ∈ D(A −1 ) which implies that u = y ∈ D(A − 1 2 ). In any normed linear space the norm is weakly lower semi-continuous, i.e x n → x weakly implies Applying this to L 2 The first statement of the proof shows us that the above is a contradiction.
Proof. Setting y = Ax, we have for x ∈ L 2 ,

and convergence in L 2
A implies uniform convergence, we have the lemma. We now state and prove six lemmas which give us two transformations for the functional integral and specific bounds to ensure its existence.
where if one side of the equality exists then so does the other and they are equal.
Lemma 9. Given ρ(σ, τ ) continuous, G(x) an integrable functional and d > 0, then where D ρ ( ) is the Fredholm determinant of ρ(σ, τ ). As in Lemma 8, if one side of the inequality exists then so does the other and they are equal.
Proof. Refer to Varberg [17] for the proof.
Proof. From Lemma 9 we have, Lemma 3, (b), enables us to write being the characteristic function of the set J. Thus, Note that K β is a monotonic decreasing function of β bounded below at infinity by 1.
Lemma 11. Suppose F (x) and G(x) are real valued measurable functions defined on C[0, t] satisfying Proof. Lemma 9 with d = λ −1 gives the equality of the two functional integrals and using the given conditions on F (x) and G(x) we may write By Lemma 3, we have Using the above and Lemma 3 again we have Setting f(u) = Prob{ x ∞ < u} the integral above may be written as which is finite by virtue of Lemma 1. We have the existence of D ρ (−λ −1 ) by Lemma 10 and so Lemma 12. If the covariance function ρ(σ, τ ) is continuous with 0 < α ≤ 1, K > 0 and 0 < λ ≤ 1 then Proof. (Pincus) Let g(u) = Prob{ x 2 < u}, then since exp {Ku α /λ 2−α − u 2 /λ} ≥ 1 for u in the latter range of integration and ∞ 0 dg(u) = 1. The supremum of the exponent above will be less than Ku α /λ 2−α evaluated at the largest possible value of u, and so Since (4−3α)/(2−α) ≤ 2−α/2 and the fact that the exponent is always greater than 1, the lemma is proven.
Lemma 13. Let F (x) and G(x) be real valued, continuous functionals on C[0, t] such that the following conditions are satisfied.

There exists an
Furthermore, suppose that x * is the only point in the sphere {x ∈ C[0, t] : x − x * ∞ ≤ R} at which H(x) attains its global minimum of zero.
5. Both F (x) and G(x) are measurable.
Then for δ > 0 sufficiently small and the set J 1 , defined by ξ > 0, for sufficiently small λ, J c 1 being the complement of J 1 .
This Lemma lies at the heart of the proof of the Theorem. It highlights the connection with the standard large deviation estimates (Stroock [12]) but our interest in a fully constructive result leads to the proof being somewhat involved.
Proof. Choose a δ such that δ < min {1, R} and θ(δ) < 1 where θ is as defined in Lemma 6 and choose λ > 0 such that λ < min{1, (1 − c 1 /ρ 1 )/(c 1 + 2Mc 3 /ρ 1 ), (1/cv 1 − 1/ρ 1 ), (γ/4c 3 ) Let J 1 be as defined in the hypothesis of the lemma and split J c 1 into the four sets From Lemma 10 we have that and we may vary β > 0 as we desire. We will consider the integral as given in the hypothesis over the above sets. Let E 2 be given by Now recall from Lemma 7 that inf x∈L 2 {(Ax/2, x) + F (Ax)} = 0 and so By condition (2) Given that x ∈ J 2 we have λx ∞ and Ax ∞ both bounded and so we may choose K 2 ∈ R + , an absolute constant, such that Also for x ∈ J 2 we have Ax and so we obtain by use of the Cauchy-Schwarz inequality. Now apply the result of Lemma 1 and Lemma 12 to get Now consider the integral in equation (1) over the set J 3 . Let E 3 be given as using two of our previous arguments. Letting y = Ax we have (Ax/2, x) + F (Ax) = H(y) and x ∈ J 3 implies δ/2 < y − x * ∞ ≤ R. Thus by Lemma 6, Therefore, by lemma 12.
Next define E 4 by By condition (2) of the hypothesis of the lemma By condition (1) of our hypothesis and Lemma 3 Thus, by the Cauchy-Schwarz inequality. Using Lemma 1 and the bound of λ we may now write We finally consider E 5 , From the proof of Lemma 11 we have and so we have If x ∈ J 5 then Ax − x * ∞ > R, and from Lemma 3 we have From the given conditions on H(x) we have and also We now have (Ax, x) for some choice of K 5 ∈ R + by choice of λ we finally obtain Recalling equation (1) we see that We have the following equalities to consider.
It can be seen from the Hölder inequality, Lemma 1 and condition (5) of the theorem that where P is a polynomial and η is a positive constant. Replacing X by [1−(1−X)] finally gives [λf 3 (0, x) + · · · + λ n−3 f n−1 (0, x)] i + O(λ n−2 ), so that where the Γ i are only dependent on the Frechet derivatives of F and G at x * for i = 1, 2, 3, . . ., n − 3. This completes the proof of the theorem.