A numerical scheme for a stochastic partial differential equation of heat equation type is considered where the drift is locally bounded and the dispersion may be state dependent. Uniform convergence in probability is obtained. (2000): Mathematics Subject Classifications

Abstract. We investigate mapping properties for the Bargmann transform and prove that this transform is isometric and bijective from modulation spaces to convenient Banach spaces of analytic functions. IntroductionIn [1], V. Bargmannn introduce a transform V which is bijective and isometric from L 2 (R d ) into the Hilbert space A 2 (C d ) of all entire analytic functions F on C d such that F · e −| · | 2 /2 ∈ L 2 (C d ). (We use the usual notations for the usual function and distribution spaces. See e. g. [21], and refer to Section 1 for specific definitions and other notations.) Furthermore, several important properties for V were established. For example:• the Hermite functions are mapped into the normalized analytical monomials. Furthermore, the latter set forms an orthonormal basis for A 2 (C d ); • the creation and annihilation operators, and harmonic oscillator on appropriate elements in L 2 , are transformed by V into simple operators; • there is a convenient reproducing formula for elements in A 2 . In [2], Bargmann continued his work and discussed mapping properties for V on more general spaces. For example, he proves that V(S ′ ), the image of S ′ under the Bargmann transform is given by the formulaHere A p,q (ω) (C d ) is the set of all entire functions F on C d such that F · e −| · | 2 /2 · ω 0 belongs to the mixed Lebesgue space L p,q (C d ), for some appropriate modification ω 0 of the weight function ω.The Bargmann transform can easily be reformulated in terms of the short-time Fourier transform, with a particular Gauss function as window function. In this context we remark that the (classical) modulation spaces M p,q , p, q ∈ [1, ∞], as introduced by Feichtinger in [5], consist of all tempered distributions whose shorttime Fourier transforms (STFT) have finite mixed L p,q norm. It follows that the parameters p and q to some extent quantify the degrees of asymptotic decay and singularity of the distributions in M p,q . The theory of modulation spaces was developed further and generalized in [7, 9, 10, 14], where Feichtinger and Gröchenig established the theory of coorbit spaces. In particular, the modulation space M p,q (ω) , where ω denotes a weight function on phase (or time-frequency shift) space, appears 1 as the set of tempered (ultra-) distributions whose STFT belong to the weighted and mixed Lebesgue space L p,q (ω) . Here the weight ω quantifies the degree of asymptotic decay and singularity of the distribution in M p,q (ω) . A major idea behind the design of these spaces was to find useful Banach spaces, which are defined in a way similar to Besov spaces, in the sense of replacing the dyadic decomposition on the Fourier transform side, characteristic to Besov spaces, with a uniform decomposition. From the construction of these spaces, it turns out that modulation spaces and Besov spaces in some sense are rather similar, and sharp embeddings between these spaces can be found in [26,27], which are improvements of certain embeddings in [13]. (See also [25] for verification of the sharpne...

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We prove that any linear operator with kernel in a Pilipović or GelfandShilov space can be factorized by two operators in the same class. We also give links on numerical approximations for such compositions. We apply these composition rules to deduce estimates of singular values and establish Schatten-von Neumann properties for such operators.