On a Theorem by A.S. Cherny for Semilinear Stochastic Partial Differential Equations

We consider analytically weak solutions to semilinear stochastic partial differential equations with non-anticipating coefficients driven by a cylindrical Brownian motion. The solutions are allowed to take values in Banach spaces. We show that weak uniqueness is equivalent to weak joint uniqueness, and thereby generalize a theorem by A.S. Cherny to an infinite dimensional setting. Our proof for the technical key step is different from Cherny’s and uses cylindrical martingale problems. As an application, we deduce a dual version of the Yamada–Watanabe theorem, i.e. we show that strong existence and weak uniqueness imply weak existence and strong uniqueness.

Brownian motion on Perelman’s almost Ricci-flat manifold

We study Brownian motion and stochastic parallel transport on Perelman’s almost Ricci flat manifold {\mathcal{M}=M\times\mathbb{S}^{N}\times I}, whose dimension depends on a parameter N unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for {N\to\infty} converge to the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on {\mathcal{M}} and for the horizontal Laplacian on the orthonormal frame bundle {\mathcal{OM}}. As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman’s manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and the second author.

Path-dependent martingale problems and additive functionals

The paper introduces and investigates the natural extension to the path-dependent setup of the usual concept of canonical Markov class introduced by Dynkin and which is at the basis of the theory of Markov processes. That extension, indexed by starting paths rather than starting points, will be called path-dependent canonical class. Associated with this is the generalization of the notions of semi-group and of additive functionals to the path-dependent framework. A typical example of such family is constituted by the laws [Formula: see text], where for fixed time [Formula: see text] and fixed path [Formula: see text] defined on [Formula: see text], [Formula: see text] is the (unique) solution of a path-dependent martingale problem or more specifically the weak solution of a path-dependent SDE with jumps, with initial path [Formula: see text]. In a companion paper we apply those results to study path-dependent analysis problems associated with BSDEs.