1,721,085 research outputs found
Positivity of solutions in a perturbed age-structured model
No full textFor certain age-structured population models, the cone of positive functions is preserved when the dynamics is perturbed by white noise. Solutions can be forced to assume negative values, even when initial conditions are strictly positive. Necessary and sufficient conditions are expressed under which the solutions are nonnegative
Absolutely continuous solutions for continuity equations in Hilbert spaces
Full text linkDa Prato G, Flandoli F, Röckner M. Absolutely continuous solutions for continuity equations in Hilbert spaces. Journal de Mathématiques Pures et Appliquées. 2019;128:42-86.We prove existence of solutions to continuity equations in a separable Hilbert space. We look for solutions which are absolutely continuous with respect to a reference measure gamma which is Fomin-differentiable with exponentially integrable partial logarithmic derivatives. We describe a class of examples to which our result applies and for which we can prove also uniqueness. Finally, we consider the case where gamma is the invariant measure of a reaction-diffusion equation and prove uniqueness of solutions in this case. We exploit that the gradient operator D-x is closable with respect to L-p(H, gamma) and a recent formula for the commutator DxPt - PtDx where P-t is the transition semigroup corresponding to the reaction-diffusion equation, [10]. We stress that P-t is not necessarily symmetric in this case. This uniqueness result is an extension to such gamma of that in [12] where gamma was the Gaussian invariant measure of a suitable Ornstein-Uhlenbeck process. (C) 2019 Elsevier Masson SAS. All rights reserved
On a class of elliptic and parabolic equations in convex domains without boundary conditions
No full textOn a class of degenerate elliptic operators in L^1 spaces with respect to invariant measures
No full textOrnstein-Uhlenbeck operators with time periodic coefficients
No full textWe study the realization of the differential operator u \mapsto u_t - L(t)u in the space of continuous time periodic functions, and in L^2 with respect to its (unique) invariant measure. Here L(t) is an Ornstein-Uhlenbeck operator in R^n, such that L(t+T) = L(t) for each t in R
Maximal dissipativity of a class of elliptic degenerate operators in weighted L^2 spaces
No full textBV functions in Hilbert spaces
No full textWe study the basic theory of BV functions in a Hilbert space X endowed with a (not necessarily Gaussian) probability measure ν. We present necessary and sufficient conditions in order that a function u∈ Lp(X, ν) is of bounded variation. We also discuss the De Giorgi approach to BV functions through the behavior as t→ 0 of ∫X‖∇T(t)u‖dν, for a smoothing semigroup T(t). Particular attention is devoted to the case where u is the indicator function of a sublevel set {x:g(x
Stochastic viability for regular closed sets in Hilbert spaces
Full text linkWe present necessary and su‰cient conditions to guarantee that at least one solution
of an infinite dimensional stochastic di¤erential equation, which starts from a regular closed
subset K of an Hilbert space, remains in K for all times
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