Path space
Encyclopedia
In mathematics
, the term path space refers to any topological space
of paths
from one specified set into another. In particular, it may refer to
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the term path space refers to any topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
of paths
Path (topology)
In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
from one specified set into another. In particular, it may refer to
- the classical Wiener spaceClassical Wiener spaceIn mathematics, classical Wiener space is the collection of all continuous functions on a given domain , taking values in a metric space . Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions...
of continuousContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
paths; - the Skorokhod spaceCàdlàgIn mathematics, a càdlàg , RCLL , or corlol function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere...
of càdlàgCàdlàgIn mathematics, a càdlàg , RCLL , or corlol function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere...
paths.