Related papers: Higher order Lions-Taylor expansions

In this paper we present a new interpretation of the Lions derivative as the Radon-Nikodym derivative of a vector measure, which provides a canonical extension of the Lions derivative for functions taking values in infinite dimensional…

In this paper we investigate the approximation of continuous functions on the Wasserstein space by smooth functions, with smoothness meant in the sense of Lions differentiability. In particular, in the case of a Lipschitz function we are…

In line with the notion of probabilistic rough paths introduced in the previous contribution \cite{salkeld2021Probabilistic}, we address corresponding random controlled rough paths (first introduced in \cite{2019arXiv180205882.2B}), the…

A Bialgebra is a module over a ring that is both an associative algebra and a co-associative coalgebra with the product and coproduct additionally satisfying an appropriate commutative relationship. One application of Bialgebras is in the…

We present several Ito-Wentzell formulae on Wiener spaces for real-valued functional random field of Ito type that depend on measure flows. We distinguish the full- and the marginal-measure flow cases in the spirit of mean-field games.…

Using the recently defined concept of Taylor measures, we propose a generalization of Taylor's theorem to measurable, non-analytic functions, that do not require differentiation. We study consequences of the generalization, including the…

The first objective of the paper is to estimate logarithmic partial derivative for meromorphic functions in several complex variables. Our estimations for logarithmic partial derivatives extend the results of Gundersen \cite{GG2} to the…

High-order Lie derivatives are essential in nonlinear systems analysis. If done symbolically, their evaluation becomes increasingly expensive as the order increases. We present a compact and efficient numerical approach for computing Lie…

Perturbation or error bounds of functions have been of great interest for a long time. If the functions are differentiable, then the mean value theorem and Taylor's theorem come handy for this purpose. While the former is useful in…

We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinite summations…

We present a new methodology, suitable for implementation on computer, to perform the $\epsilon$-expansion of hypergeometric functions with linear $\epsilon$ dependent Pochhammer parameters in any number of variables. Our approach allows…

Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for…

We consider the asymptotic behavior of the multidimensional Laplace-type integral with a perturbed phase function. Under suitable assumptions, we derive a higher-order asymptotic expansion with an error estimate, generalizing some previous…

This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations of a generic autonomous system along a particular solution. The main result of this paper is a compact yet explicit and…

Let $X$ be the number of $k$-term arithmetic progressions contained in the $p$-biased random subset of the first $N$ positive integers. We give asymptotically sharp estimates on the logarithmic upper-tail probability $\log \Pr(X \ge E[X] +…

We define analogues of higher derivatives for $F_q$-linear functions over the field of formal Laurent series with coefficients in $F_q$. This results in a formula for Taylor coefficients of a $F_q$-linear holomorphic function, a definition…

Concentration inequalities are obtained on Poisson space, for random functionals with finite or infinite variance. In particular, dimension free tail estimates and exponential integrability results are given for the Euclidean norm of…

The Representation Theorem of Lions (RTL) is a version of the Lax--Milgram Theorem where completeness of one of the spaces is not complete. In this paper, RTL is deduced from an operator-theoretical version on normed space. The main point…

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are…

We show that descriptive complexity's result extends in High Order Logic to capture the expressivity of Turing Machine which have a finite number of alternation and whose time or space is bounded by a finite tower of exponential. Hence we…