We call φ the root function of Φ. The continuity of Φ implies that (1.2) φ(γ) → |γ| as |γ| → ∞. Moreover the converse is true, if φ is continuous and satisfies this asymptotic formula, then the function Φ formed as in (1.1) is a multiplicative distance function. Theorem 1.1. A continuous function φ : C → [0,∞) is the root function of a multiplicative distance function iff it satisfies (1.2). Ar...

Abstract We consider the action of Mandelbrot multiplicative cascades on probability measures supported a symbolic space. For general measures, we obtain almost sharp criterion non-degeneracy limiting measure; it relies lower and upper Hausdorff dimensions measure entropy random weights. also bounds for packing measure. When original is Gibbs associated with potential certain modulus continuity...

We present a new approach to absolute continuity of laws of Poisson functionals. The theoretical framework is that of local Dirichlet forms as a tool to study probability spaces. The method gives rise to a new explicit calculus that we show first on some simple examples : it consists in adding a particle and taking it back after computing the gradient. Then we apply it to SDE’s driven by Poisso...

We consider a fixed probability μ. For any λ ∈ (−1,∞), we denote by m(λ, μ) the λ-Sugeno measure generated by μ. For a fixed measurable set A, the continuity of the map λ 7−→ ∫ A fdm(λ, μ) is proved, where the integral is either Choquet or Sugeno. The asymptotic (marginal) cases λ = −1 and λ =∞ are studied too. Finally, some concrete computations illustrate the previous theory.

We consider a fixed probability µ. For any () 1, λ ∈ − ∞ , we denote by () , m λ µ the λ-Sugeno measure generated by µ. For a fixed measurable set A, the continuity of the map () , A fdm λ λ µ ∫ ֏ is proved, where the integral is either Choquet or Sugeno. The asymptotic (marginal) cases 1 λ = − and λ = ∞ are studied too. Finally, some concrete computations illustrate the previous theory.

We consider the controllability problem for continuity equation, corresponding to neural ordinary differential equations (ODEs), which describes how a probability measure is pushedforward by flow. show that controlled equation has very strong properties. Particularly, given solution of bounded Lipschitz vector field defines trajectory on set measures. For this trajectory, we there exist piecewi...

We prove that if (X,A, P ) is an arbitrary probability space with countably generated σ-algebra A, (Y,B,Q) is an arbitrary complete probability space with a lifting ρ and R̂ is a complete probability measure on A ⊗̂R B determined by a regular conditional probability {Sy :y ∈ Y } on A with respect to B, then there exist a lifting π on (X × Y,A ⊗̂R B, R̂) and liftings σy on (X, Ây, Ŝy), y ∈ Y , such ...