99 research outputs found
BV-regularity for the Malliavin Derivative of the Maximum of the Wiener Process
We prove that, on the classical Wiener space, the random variable admits a measure as second Malliavin derivative, whose total
variation measure is finite and singular w.r.t.\ the Wiener measure
Lagrangian flows driven by fields in Wiener spaces
We establish the renormalization property for essentially bounded solutions
of the continuity equation associated to fields in Wiener spaces, with
values in the associated Cameron-Martin space; thus obtaining, by standard
arguments, new uniqueness and stability results for correspondent Lagrangian
-flows. An example related to Neumann elliptic problems is also
discussed
Zero noise limits using local times
We consider a well-known family of SDEs with irregular drifts and the
correspondent zero noise limits. Using (mollified) local times, we show which
trajectories are selected. The approach is completely probabilistic and relies
on elementary stochastic calculus only
Well-posedness of Multidimensional Diffusion Processes with Weakly Differentiable Coefficients
We investigate well-posedness for martingale solutions of stochastic
differential equations, under low regularity assumptions on their coefficients,
widely extending some results first obtained by A. Figalli. Our main results
are a very general equivalence between different descriptions for
multidimensional diffusion processes, such as Fokker-Planck equations and
martingale problems, under minimal regularity and integrability assumptions,
and new existence and uniqueness results for diffusions having weakly
differentiable coefficients, by means of energy estimates and commutator
inequalities. Our approach relies upon techniques recently developed, jointly
with L. Ambrosio, to address well-posedness for ordinary differential equations
in metric measure spaces: in particular, we employ in a systematic way new
representations and inequalities for commutators between smoothing operators
and diffusion generators.Comment: Added references to further literature on the subjec
A Benamou-Brenier formulation of martingale optimal transport
We introduce a Benamou-Brenier formulation for the continuous-time martingale
optimal transport problem as a weak length relaxation of its discrete-time
counterpart. By the correspondence between classical martingale problems and
Fokker-Planck equations, we obtain an equivalent PDE formulation for which
basic properties such as existence, duality and geodesic equations can be
analytically studied, yielding corresponding results for the stochastic
formulation. In the one dimensional case, sufficient conditions for finiteness
of the cost are also given and a link between geodesics and porous medium
equations is partially investigated
Well posedness of Lagrangian flows and continuity equations in metric measure spaces
We establish, in a rather general setting, an analogue of DiPerna-Lions
theory on well-posedness of flows of ODE's associated to Sobolev vector fields.
Key results are a well-posedness result for the continuity equation associated
to suitably defined Sobolev vector fields, via a commutator estimate, and an
abstract superposition principle in (possibly extended) metric measure spaces,
via an embedding into .
When specialized to the setting of Euclidean or infinite dimensional (e.g.
Gaussian) spaces, large parts of previously known results are recovered at
once. Moreover, the class of metric measure spaces object
of extensive recent research fits into our framework. Therefore we provide, for
the first time, well-posedness results for ODE's under low regularity
assumptions on the velocity and in a nonsmooth context.Comment: Slightly expanded some remarks on the technical assumption (7.11);
Journal reference inserte
Lecture notes on the DiPerna-Lions theory in abstract measure spaces
These notes closely correspond to a series of lectures given by the first
author in Toulouse, on the recent extension of the theory of ODE well-posedness
to abstract spaces, jointly obtained by the two authors. In the last part, we
describe some further developments with respect to the theory of (possibly
degenerate) diffusion processes, in a similar setting, contained in the second
author's PhD thesis
Three superposition principles: currents, continuity equations and curves of measures
We establish a general superposition principle for curves of measures solving
a continuity equation on metric spaces without any smooth structure nor
underlying measure, representing them as marginals of measures concentrated on
the solutions of the associated ODE defined by some algebra of observables. We
relate this result with decomposition of acyclic normal currents in metric
spaces.
As an application, a slightly extended version of a probabilistic
representation for absolutely continuous curves in Kantorovich-Wasserstein
spaces, originally due to S. Lisini, is provided in the metric framework. This
gives a hierarchy of implications between superposition principles for curves
of measures and for metric currents
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