In this section, we finally prove Theorem 1.28. One part of this theorem is the construction and commutativity of the following diagram (1.28).

This last section is devoted to the proof of the asymptotic result as ε↘0 contained in Theorem 2.4. Throughout the section, (xε,yε,ξε,ξΓ,ε) and (x,y,ξ,ξΓ) are the unique solutions to (2.1) with additive noise B in the cases ε>0 and ε=0, respectively…

This section is devoted to the proof of Theorem 3.2. In what follows the letter C will denote a constant in R that may change from line to line; if the constant C will depend on some parameters α1,…,αk we will write Cα1,…,αk. Moreover, for any ρ>0, …

In the final section, we want to establish further properties of arbitrary minimizing curves and of solutions of the evolution equation. We start showing that any minimizing curve has a similar structure as the one constructed in the previous sectio…

By a Lie pair (L,A), we mean an ordinary (non-graded) Lie algebroid (L,[⋅,⋅]L,ρL) over an ordinary (non-graded) smooth manifold M, together with a Lie subalgebroid (A,[⋅,⋅]A,ρA) of L over the same base M.

This paper aims at building a unified framework to deal with a wide class of local and nonlocal translation-invariant geometric flows. First, we introduce a class of generalized curvatures, and prove the existence and uniqueness for the level set fo…