Adapted Wasserstein distance for SDEs with irregular coefficients
Dienstag, 7.11.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We solve an optimal transport problem under probabilistic constraints, where the marginals are laws of solutions of stochastic differential equations with irregular, that is non-globally Lipschitz continuous coefficients. Numerical methods are employed as a theoretical tool\nto bound the adapted Wasserstein distance. This opens the door for\ncomputing the adapted Wasserstein distance in a simple way.\n\n\nJoint work with B. Robinson (University of Vienna).
1D approximation in Wasserstein spaces
Dienstag, 14.11.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Given a Borel probability measure, we seek to approximate it with a measure uniformly\ndistributed over a 1-dimensional set. With this end, we minimize the Wasserstein distance of this fixed measure to all probability measures uniformly distributed to connected 1 dimensional sets and a regularization term given by their length. To show existence of solution to this problem, one cannot easily resort to the direct method in the calculus of variations due to concentration of mass effects. Therefore, we propose a relaxed problem in the space of probability measures which always admits a solution. In the sequel, we show that whenever the initial measureis absolutely continuous w.r.t. the 1-Hausdorff measure (in particular for absolutely continuous measures w.r.t. Lebesgue measure in R^d) then the solution will be a rectiable measure. This allows us to perform a blow-up argument that, in dimension 2, shows that the solution has a uniform density, being therefore a solution to the original problem. Finally, we prove a phase-field approximation for this problem in the form of a Gamma-convergence result of a functional reminiscent of the Ambrosio-Tortorelli approximation for the Mumford-Shah problem, with the additional property of enforcing connectivity of the 1-dimensional sets that emerges from the approximation. This last feature is achieved with the connectivity functional introduced by Dondl and Wojtowytsch.
Resolvent estimates for one-dimensional Schroedinger operators with complex potentials
Dienstag, 28.11.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We study one-dimensional Schroedinger operators with unbounded complex potentials of various growths (from iterated logs to super-exponentials). We derive asymptotic formulas for the norm of the resolvent as the spectral parameter diverges along the imaginary and real axes. In each case, our analysis yields an explicit leading order term as well as an optimal estimate of the remainder. We also discuss several extensions of the main results, their interrelation with the complementary estimates based on non-semiclassical pseudomode construction in [KS-19] and several examples.\n\nThe talk is based on the joint work [AS-23] with A. Arnal.\n\nReferences:\n\n[AS-23] A. Arnal and P. Siegl: Resolvent estimates for one-dimensional Schroedinger operators with complex potentials, 2023, J. Funct. Anal. 284, 109856\n\n[KS-19] D. Krejcirik and P. Siegl: Pseudomodes for Schroedinger operators with complex potentials, 2019, J. Funct. Anal. 276, 2856-2900