Asymptotic expansions for ornstein-uhlenbeck semigroups perturbed by potentials over banach spaces

Asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths

“…First, we devote ourselves to give the proof of Theorem 3.2 based on the argument of Albeverio, Röckle and Steblovskaya [3]. We consider the precise asymptotic behavior of the following integral as ε 0:…”

“…The contribution for I (2) 0 (ε) is treated in the same way as above and we can easily see the estimate I (2) 0 (ε) c 2 (n)ε n+1 . Next we proceed to estimate the term I (3) 0 (ε). We use an elementary inequality…”

“…In this case we set, for y, z ∈ Y and w ∈ Y ⊗ X, f 3 (y, z, w) := ∇ 2 σ (y) z, w . Then, it is easy to see that f 3 …”

Borel summation of the small time expansion of some SDE’s driven by Gaussian white noise

“…(ii) One has k(t, x, y) = ψ −1 (x) u(t, x, y) ψ(y), with u(t, x, y) as in Theorem 2.5 (with ω > 0). u(t, x, y) satisfies the asymptotic expansion given by (14) and (15), and consequently k(t, x, y) has a corresponding asymptotic expansion, given by (14), (15) multiplied…”

Theory and Applications of Infinite Dimensional Oscillatory Integrals

“…In the latter setting the integral I Φ (f ) is realized as the pairing T Φ , f with respect to the standard Gaussian measure N (0, I L 2 (R n ) ) of a white noise distribution T φ ∈ (S ) (which, heuristically, can be interpreted as e i 2 Φ(γ)+ 1 2 γ,γ) ) and a regular f ∈ (S) , where (S), (S ) are elements of the Gelfand triple (S) ⊂ L 2 (N (0, I L 2 (R n ) )) ⊂ (S ) (see [47] for details). It is interesting to note that formula (26) shows a deep connection between infinite dimensional oscillatory integrals and probabilistic Gaussian integrals. Indeed, under suitable assumptions on the function f that is integrated and on the phase function Φ, the oscillatory integral of f with respect to Φ is equal to a Gaussian integral.…”