Strong laws, random monotone vector fields and gradient flows on metric spaces of nonpositive curvature

@misc{pischke2026strong,
  title = {{Strong laws, random monotone vector fields and gradient flows on metric spaces of nonpositive curvature}},
  author = {Nicholas Pischke},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.16439},
  eprint = {2609.16439},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16439},
}

TY  - UNPB
TI  - Strong laws, random monotone vector fields and gradient flows on metric spaces of nonpositive curvature
AU  - Nicholas Pischke
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.16439
UR  - https://arxiv.org/abs/2609.16439
AB  - We establish a probabilistic Lie-Trotter-Kato formula for the gradient flow generated by an integral function over suitable separable Hadamard spaces. This result leverages Stojković's Lie-Trotter-Kato formula for gradient flows of sums of convex functions and Bačák's result relating resolvent and gradient flow convergence, along with a strong law of large numbers for normal convex integrands. For separable Hilbert spaces and locally compact Hadamard spaces, these strong laws follow from results of King and Wets and Artstein and Wets, respectively, but we crucially go beyond these examples by establishing a new such strong law for normal convex integrands on all Hadamard spaces whose tangent cones are isometric to Hilbert spaces, a class that includes non-Hilbertian, non-locally compact examples such as infinite-dimensional hyperbolic space. This new strong law arises as a combination of a strong law of large numbers for integrable random monotone vector fields on Hilbert-Hadamard spaces, that is, Hadamard spaces whose tangent cones all embed isometrically into Hilbert spaces, extending a result of Salim from Hilbert spaces, together with a new result on the interchangeability of the subdifferential and the integral, which we establish for L^2-Lipschitz integrands. This latter ingredient provides, to our knowledge, the first nonlinear version of (a particular case of) a previous result due variously to Ioffe and Tikhomirov, Levin, Hiriart-Urruty, Thibault as well as Rockafellar and Wets. On the way to this strong law for random monotone vector fields, we also establish a distribution-uniform generalization of Sturm's strong law of large numbers for inductive means of L^1-sequences of i.i.d. random variables on Hadamard spaces via a new effective non-asymptotic concentration inequality. Throughout the paper, we highlight various remaining open problems.
ER  -