Strong Feller solutions to SPDE's are strong Feller in the weak topology

Bohdan Maslowski; Jan Seidler

doi:10.4064/sm148-2-2

Strong Feller solutions to SPDE's are strong Feller in the weak topology

Studia Mathematica ◽

10.4064/sm148-2-2 ◽

2001 ◽

Vol 148 (2) ◽

pp. 111-129 ◽

Cited By ~ 3

Author(s):

Bohdan Maslowski ◽

Jan Seidler

Keyword(s):

Weak Topology ◽

Strong Feller

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Existence of Solutions for an integral equation of Chandrasekhar type in Banach algebras with respect to the weak topology

Operator Theory ◽

10.1515/9783110598193-013 ◽

2021 ◽

pp. 177-182

Author(s):

Amor Fahem ◽

Aref Jeribi ◽

Najib Kaddachi

Keyword(s):

Integral Equation ◽

Weak Topology ◽

Existence Of Solutions ◽

Banach Algebras

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All adapted topologies are equal

Probability Theory and Related Fields ◽

10.1007/s00440-020-00993-8 ◽

2020 ◽

Vol 178 (3-4) ◽

pp. 1125-1172

Author(s):

Julio Backhoff-Veraguas ◽

Daniel Bartl ◽

Mathias Beiglböck ◽

Manu Eder

Keyword(s):

Stochastic Processes ◽

Weak Topology ◽

Equilibrium Problems ◽

Diffusion Processes ◽

Temporal Structure ◽

Wasserstein Distance ◽

Optimal Stopping Problems ◽

Reward Functions ◽

Nested Distance ◽

The Stability

Abstract A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion–Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle’s causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this ‘weak adapted topology’ is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions.

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