Schauder estimates for elliptic equations in Banach spaces associated with stochastic reaction–diffusion equations

Abstract

We consider some reaction-diffusion equations perturbed by white noise and prove Schauder estimates for the elliptic problem associated with the generator of the corresponding transition semigroup, defined in the Banach space of continuous functions. This requires the proof of some new interpolation result.

Pathwise uniqueness for stochastic reaction-diffusion equations in Banach spaces with an Hölder drift component

“…Next, we recall that in [6,Section 3], by using suitable interpolation estimates for realvalued functions defined in the Banach space E, we have proved the following result.…”

“…A typical tool in Hilbert spaces is the finite dimensional projection or approximation by means of the elements of an orthonormal basis. Here we implement the idea recently developed in [6] of using an orthonormal basis of the Hilbert space L 2 (0, 1) made of elements which belong to E. This method allows to perform certain finite dimensional approximations and in particular to write Itô formulae for certain quantities; the control of many terms is often nontrivial but successful. This paper is, in a sense, the generalization of [8] to Banach spaces (see also [9] on bounded measurable drift and the work in finite dimensions [11] where part of the technique was developed in order to construct stochastic flows).…”

“…These results may have other applications and also an intrinsic interest for infinite dimensional analysis. The Kolmogorov equation associated to reaction-diffusion equation has been investigated in [4], [5], [6] and related works. In our work here we add new informations.…”

Schauder theorems for a class of (pseudo‐)differential operators on finite‐ and infinite‐dimensional state spaces

“…More details are in Sections 5.1 and 5.3. Moreover, Schauder estimates for some nontrivial perturbations of a specific Ornstein-Uhlenbeck operator in the space X = C([0, 1]) were proved in [17].…”

Schauder theorems for Ornstein-Uhlenbeck equations in infinite dimension

“…We would like to remind that there are relevant situations in which Schauder estimates cannot be proved for Hilbert spaces, but only for Banach spaces. This is the case considered in [6], where the transition semigroup T (t) associated with a class of stochastic reaction-diffusion equations defined on a bounded interval [0, 1], with polynomially growing coefficients, is studied in the space X = C([0, 1]). Actually, for that class of equations the analysis of T (t) in X = L 2 (0, 1) is considerably 14 more delicate than in X = C([0, 1]) and it is not possible to prove that when f ∈ C α (L 2 (0, 1)), for some α ∈ (0, 1), the function u defined in (4.2) belongs to C 2 (L 2 (0, 1)).…”