Nous donnons des conditions permettant de vérifier que l’image d’une mesure cylindrique $\mu$ sur un espace vectoriel topologique $E$, par une application linéaire continue dans un autre espace vectoriel topologique $F$, est une mesure de Randon. Dans une première partie, nous donnons des résultats généraux qui portent, soit sur des propriétés géométriques de l’espace $F$, soit sur la mesure cylindrique $\mu$. Dans une seconde partie, nous donnons des conditions plus précises quand $\mu$ est une mesure cylindrique...

The main concern of this paper is to present some improvements to results on the existence or non-existence of countably additive Borel measures that are not Radon measures on Banach spaces taken with their weak topologies, on the standard axioms (ZFC) of set-theory. However, to put the results in perspective we shall need to say something about consistency results concerning measurable cardinals.

An example of a non-zero non-atomic translation-invariant Borel measure ${\nu }_p$ on the Banach space ${\ell }_p\left(1\le p\le \infty \right)$ is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition "${\nu }_p$-almost every element of ${\ell }_p$ has a property P" implies that “almost every” element of ${\ell }_p$ (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

Let $C[0,t]$ denote a generalized Wiener space, the space of real-valued continuous functions on the interval $[0,t]$, and define a random vector $Z_n:C[0,t]\to {ℝ}^{n+1}$ by $$Z_n\left(x\right)=\left(x\left(0\right)+a\left(0\right),{\int }_0^{t_1}h\left(s\right)\mathrm{d}x\left(s\right)+x\left(0\right)+a\left(t_1\right),\cdots ,{\int }_0^{t_n}h\left(s\right)\mathrm{d}x\left(s\right)+x\left(0\right)+a\left(t_n\right)\right),$$ where $a\in C[0,t]$, $h\in L_2[0,t]$, and $0<t_1<\cdots <t_n\le t$ is a partition of $[0,t]$. Using simple formulas for generalized conditional Wiener integrals, given $Z_n$ we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions $F$ in a Banach algebra which corresponds to Cameron-Storvick’s Banach algebra $𝒮$. Finally, we express the generalized analytic conditional Feynman...

On établit pour le cône $C$ des mesures $\mu$ positives bornées sur $𝒞\left(\mathbf{R}\right)$, quasi-invariantes sous les translations de $𝒟\left(\mathbf{R}\right)$ et vérifiant :$$\mu (f+dw)=\mu \left(dw\right)\exp {\int }_{R}dt\left[\right(w\left(t\right)+\frac{1}{2}f\left(t\right))f^{\text{'}\text{'}}\left(t\right)-P\left(w\right(t)+f\left(t\right)+P\left(w\right(t\left)\right)]$$(avec $P$ polynôme borné inférieurement) les résultats suivants :– Toute mesure de $C$ est intégrale de mesures appartenant aux génératrices extrémales de $C$.– Les génératrices extrémales de $C$ sont composées de mesures markoviennes.

An expression in terms of the Wiener integral for the “scattering length” is given and used to discuss the relation between this quantity and electrostatic capacity.

Following H. Sato - Y. Okazaky we will prove that: if $X$ is a topological vector space, locally convex and reflexive, and $\mu$ is a gaussian measure on $𝐂(X,X^{\prime })$, then $L^2(\mu )$ is separable.

In this paper we obtain several basic formulas for generalized integral transforms, convolution products, first variations and inverse integral transforms of functionals defined on function space.

In this paper we define jump set and approximate limits for BV functions on Wiener spaces and show that the weak gradient admits a decomposition similar to the finite dimensional case. We also define the SBV class of functions of special bounded variation and give a characterisation of SBV via a chain rule and a closure theorem. We also provide a characterisation of BV functions in terms of the short-time behaviour of the Ornstein-Uhlenbeck semigroup following an approach due to Ledoux.