On the differentiability of solutions of stochastic evolution equations with respect to their initial values

Abstract

In this article we study the differentiability of solutions of parabolic semilinear stochastic evolution equations (SEEs) with respect to their initial values. We prove that if the nonlinear drift coefficients and the nonlinear diffusion coefficients of the considered SEEs are n-times continuously Fréchet differentiable, then the solutions of the considered SEEs are also ntimes continuously Fréchet differentiable with respect to their initial values. Moreover, a key contribution of this work is to establish suitable enhanced regularity properties of the derivative processes of the considered SEE in the sense that the dominating linear operator appearing in the SEE smoothes the higher order derivative processes.

Fréchet differentiable drift dependence of Perron–Frobenius and Koopman operators for non-deterministic dynamics

“…The same dependencies also arise for non-deterministic systems, e.g. [FGP10,AJKW17] for Fréchet-type dependencies on the initial data, and [FLL + 99, GM05, Mon13,DG14] for the dependence of path functionals or their expectations with respect to changes in the drift. This paper uses the approach of [FLL + 99], which establishes a Gâteaux-type dependence on the data by establishing the existence of directional derivatives with respect to the drift, in order to establish the Fréchet-type dependence of the solution operator with respect to an additive change of drift in a sufficiently smooth setting: for a suitable observable g, we provide in Theorem 3.1 the Fréchet derivative at γ = 0 of the non-linear functional u x g (γ) := E g(X γ ) X γ 0 = x with respect to additive perturbations in γ; above, X γ denotes the solution of the perturbed stochastic differential equation (2.7) below.…”

Regularity Properties for Solutions of Infinite Dimensional Kolmogorov Equations in Hilbert Spaces

“…It thus remains to prove items (iv)-(viii). To prove item (iv) we first note that (51) and item (ii) of Theorem 2.1 in [2] (with T = T , η = η, H = H, U = U, W = W , A = A, n = n, F = F , B = B, α = α, β = β, k = k, p = p, δ = δ for δ ∈ D k , p ∈ [2, ∞), k ∈ {1, 2, . .…”

“…. , n} : |F | Lip l (H,H −α ) + |B| Lip l (H,HS(U,H −β )) < ∞} in the notation of Theorem 2.1 in [2]) ensure that for all k ∈ {1, 2, . .…”

“…Moreover, Theorem 3.3 below provides explicit bounds for the left hand sides of (2) and (3) (see items (vii) and (x) in Theorem 3.3 below). Next we would like to emphasize that Theorem 1.1 and Theorem 3.3, respectively, prove finiteness of (2) and (3) even though the denominators in (2) and (3) contain rather weak norms from negative Sobolev-type spaces for the multilinear arguments of the derivatives of the generalized solution. In particular, item (iv) in Theorem 1.1 above and item (vii) in Theorem 3.3 below, respectively, reveal for every p ∈ [1, ∞), k ∈ {1, 2, .…”

Weak convergence rates for stochastic evolution equations and applications to nonlinear stochastic wave, HJMM, stochastic Schrödinger and linearized stochastic Korteweg–de Vries equations

“…N denotes the natural numbers without zero. Function spaces are denoted as follows: B denotes bounded functions with the supremum norm, C denotes continuous functions, C b denotes continuous bounded functions with the supremum norm, C k denotes continuous functions which are k-times Fréchet differentiable on the interior of the domain and whose derivatives up to order k extend to continuous functions on the domain, C k b denotes the subset of C k whose derivatives of orders 1 to k belong to C b with norm f C k b = f (0) + f ′ C b + · · · + f (k) C b , L denotes linear operators with the operator norm, Lip denotes Lipschitz functions with norm f Lip = f (0) +sup x =y f (x)−f (y) / x−y , L p denotes strongly measurable p-integrable functions, L p denotes the corresponding equivalence classes modulo equality almost surely, L (2) denotes bilinear operators, L 2 denotes Hilbert-Schmidt operators, W α,p denotes the Sobolev-Slobodeckij space with smoothness parameter α and integrability parameter p, H α = W α,2 denotes the Bessel potential space, and H α 0 denotes the closure of the compactly supported smooth functions in H α . For any Hilbert space H, B(H) denotes the Borel σ-algebra on H, [X] P,B(H) ∈ L 0 (Ω; H) denotes the P-equivalence class of X ∈ L 0 (Ω; H), and σ P (A) denotes the point spectrum of a linear operator A : D(A) ⊆ H → H. Note that we do not require functions in C k b to be bounded.…”