1,721,033 research outputs found
BV-regularity for the Malliavin derivative of the maximum of the Wiener process
No full textWe prove that, on the classical Wiener space, the random variable M = sup_{0≤t≤T} W_t
admits a measure as second Malliavin derivative, whose total variation measure is
finite and singular w.r.t. the Wiener measure
Zero noise limits using local times
No full textWe 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 se-
lected. The approach is completely probabilistic and relies on elementary stochastic
calculus only
Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients
Full text linkWe investigate well-posedness for martingale solutions of stochastic differential equations, under low regularity assumptions on their coefficients, widely extending the results first obtained by A. Figalli in [19]. 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 in [6], to address well-posedness for ordinary differential equations in metric measure spaces: in particular, we employ in a systematic way new representations for commutators between smoothing operators and diffusion generators
Lagrangian flows driven by BV fields in Wiener spaces
No full textWe establish the renormalization property for essentially bounded solutions of the continuity equation associated to bounded variation (BV) 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
Lecture notes on the DiPerna-Lions theory in abstract measure spaces
Full text linkThese notes collect the lectures given by the first author in Toulouse, April 2014, on the well-posedness theory for continuity and transport equation in metric measure spaces, summarizing the joint work appeared in Analysis and PDE. The last part of the notes covers also more recent developments, due to the second author, on diffusion operators on metric measure spaces
Three superposition principles: Currents, continuity equations and curves of measures
Full text linkWe 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
Well-posedness of Lagrangian flows and continuity equations in metric measure spaces
Full text linkWe establish, in a rather general setting, an analogue of DiPerna–Lions theory on well-posedness of flows of ODEs 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 R∞.
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 RCD(K,∞) metric measure spaces, introduced by Ambrosio, Gigli and Savaré [Duke Math. J. 163:7 (2014) 1405–1490] and the object of extensive recent research, fits into our framework. Therefore we provide, for the first time, well-posedness results for ODEs under low regularity assumptions on the velocity and in a nonsmooth context
Wasserstein asymptotics for Brownian motion on the flat torus and Brownian interlacements
No full textWe study the large time behaviour of the optimal transportation cost towards the uniform distribution, for the occupation measure of a stationary Brownian motion on the flat torus in d dimensions, where the cost of transporting a unit of mass is given by a power of the flat distance. We establish a global upper bound, in terms of the limit for the analogue problem concerning the occupation measure of the Brownian interlacement on Rd. We conjecture that our bound is sharp and that our techniques may allow for similar studies on a larger variety of problems, e.g. general diffusion processes on weighted Riemannian manifolds
Quantum Optimal Transport: Quantum Channels and Qubits
No full textThese notes are based on the lectures given by the second author at the
School on Optimal Transport on Quantum Structures at Erd\"os Center in
September 2022. The focus of the exposition is on two recently introduced
approaches on quantum optimal transport: one based on quantum channels as
generalized transport plans, the other based on the notion of
Hamming-Wasserstein distance of order 1 on multiple-qubit systems. The material
is presented in an elementary manner with a focus on the finite-dimensional
setting
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