arXiv subjects
Publications and source records attributed to Victor Marx.
Introducing an interpolation method we derive lower bounds for the spectral gap for Brownian motion on general domains with sticky-reflecting boundary diffusion associated to the first nontrivial eigenvalue for the Laplace operator with corresponding Wentzell-type boundary condition. In the manifold case our proofs involve novel applications of the celebrated Reilly formula.
We introduce the notion of a conditional distribution to a zero-probability event in a given direction of approximation, and prove that the conditional distribution of a family of independent Brownian particles to the event that their paths coalesce after the meeting coincides with the law of a modified massive Arratia flow, defined in [arXiv:1408.0628].
We introduce in this paper a strategy to prove gradient estimates for some infinite-dimensional diffusions on $L_2$-Wasserstein spaces. For a specific example of a diffusion on the $L_2$-Wasserstein space of the torus, we get a Bismut-Elworthy-Li formula up to a remainder term and deduce a gradient estimate with a rate of blow-up of order $\mathcal{O}(t^{-(2+\epsilon)})$.
Much effort has been spent in recent years on restoring uniqueness of McKean-Vlasov SDEs with non-smooth coefficients. As a typical instance, the velocity field is assumed to be bounded and measurable in its space variable and Lipschitz-continuous with respect to the distance in total variation in its measure variable, see [Jourdain, Mishura-Veretennikov]. In contrast with those works, we consider in this paper a Fokker-Planck equation driven by an infinite-dimensional noise, inspired by the diffusion models on the Wasserstein space studied in [Konarovskyi, Marx]. We prove that well-posedness of that equation holds for a drift function that might be only bounded and measurable in its measure argument, provided that a trade-off is respected between the regularity in the finite-dimensional component and the regularity in the measure argument. In this regard, we show that the higher the regularity of b with respect to its space variable is, the lower regularity we have to assume on b with respect to its measure variable in order to restore uniqueness.
We propose in this paper a construction of a diffusion process on the Wasserstein space P\_2(R) of probability measures with a second-order moment. This process was introduced in several papers by Konarovskyi (see e.g. "A system of coalescing heavy diffusion particles on the real line", 2017) and consists of the limit when N tends to infinity of a system of N coalescing and mass-carrying particles. It has properties analogous to those of a standard Euclidean Brownian motion, in a sense that we will precise in this paper. We also compare it to the Wasserstein diffusion on P\_2(R) constructed by von Renesse and Sturm (see Entropic measure and Wasserstein diffusion). We obtain that process by the construction of a system of particles having short-range interactions and by letting the range of interactions tend to zero. This construction can be seen as an approximation of the singular process of Konarovskyi by a sequence of smoother processes.