Classical Wiener space#Tightness in classical Wiener space

Consider $E\backslash subseteq\; \backslash mathbb\{R\}^n$ and a metric space $(M,d)$. The classical Wiener space $C(E,M)$ is the space of all continuous functions $f:E\backslash to\; M.$ That is, for every fixed $t\backslash in\; E,$

:$d(f(s),\; f(t))\; \backslash to\; 0$ as $|\; s\; -\; t\; |\; \backslash to\; 0.$

In almost all applications, one takes $E=[0,T]$ or $E=\backslash R\_+=[0,\; +\backslash infty)$ and $M=\backslash mathbb\{R\}^n$ for some $n\backslash in\backslash mathbb\{N\}.$ For brevity, write $C$ for $C([0,T]);$ this is a vector space. Write $C\_0$ for the linear subspace consisting only of those functions that take the value zero at the infimum of the set $E.$ Many authors refer to $C\_0$ as "classical Wiener space".

The vector space $C$ can be equipped with the uniform norm

:$\backslash |\; f\; \backslash |\; :=\; \backslash sup\_\{t\; \backslash in\; [0,\backslash ,T]\}\; |f(t)|$

turning it into a normed vector space (in fact a Banach space since $[0,T]$ is compact). This norm induces a metric on $C$ in the usual way: $d\; (f,\; g)\; :=\; \backslash |\; f-g\; \backslash |$. The topology generated by the open sets in this metric is the topology of uniform convergence on $[0,T],$ or the uniform topology.

Thinking of the domain $[0,T]$ as "time" and the range $\backslash R^n$ as "space", an intuitive view of the uniform topology is that two functions are "close" if we can "wiggle space slightly" and get the graph of $f$ to lie on top of the graph of $g$, while leaving time fixed. Contrast this with the Skorokhod topology, which allows us to "wiggle" both space and time.

If one looks at the more general domain $\backslash R\_\{+\}$ with

:$\backslash |\; f\; \backslash |\; :=\; \backslash sup\_\{t\; \backslash geq\; 0\}\; |f(t)|,$

then the Wiener space is no longer a Banach space, however it can be made into one if the Wiener space is defined under the additional constraint

:$\backslash lim\backslash limits\_\{s\backslash to\backslash infty\}s^\{-1\}|f(s)|=0.$

With respect to the uniform metric, $C$ is both a separable and a complete space:

• Separability is a consequence of the Stone–Weierstrass theorem;
• Completeness is a consequence of the fact that the uniform limit of a sequence of continuous functions is itself continuous.

Since it is both separable and complete, $C$ is a Polish space.

Recall that the modulus of continuity for a function $f:[0,T]\backslash to\backslash R^n$ is defined by

:$\backslash omega\_\{f\}\; (\backslash delta)\; :=\; \backslash sup\; \backslash left\backslash \{\; |f(s)\; -\; f(t)|\; :\; s,\; t\; \backslash in\; [0,\; T],\backslash ,\; |s\; -\; t|\; \backslash leq\; \backslash delta\; \backslash right\backslash \}.$

This definition makes sense even if $f$ is not continuous, and it can be shown that $f$ is continuous if and only if its modulus of continuity tends to zero as $\backslash delta\backslash to\; 0:$

:$f\; \backslash in\; C\; \backslash iff\; \backslash omega\_\{f\}\; (\backslash delta)\; \backslash to\; 0\; \backslash text\{\; as\; \}\; \backslash delta\; \backslash to\; 0$.

By an application of the Arzelà-Ascoli theorem, one can show that a sequence $(\backslash mu\_\{n\})\_\{n\; =\; 1\}^\{\backslash infty\}$ of probability measures on classical Wiener space $C$ is tight if and only if both the following conditions are met:

:$\backslash lim\_\{a\; \backslash to\; \backslash infty\}\; \backslash limsup\_\{n\; \backslash to\; \backslash infty\}\; \backslash mu\_\{n\}\; \backslash \{\; f\; \backslash in\; C\; :\; |\; f(0)\; |\; \backslash geq\; a\; \backslash \}\; =\; 0,$ and

:$\backslash lim\_\{\backslash delta\; \backslash to\; 0\}\; \backslash limsup\_\{n\; \backslash to\; \backslash infty\}\; \backslash mu\_\{n\}\; \backslash \{\; f\; \backslash in\; C\; :\; \backslash omega\_\{f\}\; (\backslash delta)\; \backslash geq\; \backslash varepsilon\; \backslash \}\; =\; 0$ for all $\backslash varepsilon\; >0.$

There is a "standard" measure on $C\_0,$ known as classical Wiener measure (or simply Wiener measure). Wiener measure has (at least) two equivalent characterizations:

If one defines Brownian motion to be a Markov stochastic process $B:[0,T]\backslash times\backslash Omega\backslash to\backslash R^n,$ starting at the origin, with almost surely continuous paths and independent increments

:$B\_\{t\}\; -\; B\_\{s\}\; \backslash sim\backslash ,\; \backslash mathrm\{Normal\}\; \backslash left(\; 0,\; |t\; -\; s|\; \backslash right),$

then classical Wiener measure $\backslash gamma$ is the law of the process $B.$

Alternatively, one may use the abstract Wiener space construction, in which classical Wiener measure $\backslash gamma$ is the radonification of the canonical Gaussian cylinder set measure on the Cameron-Martin Hilbert space corresponding to $C\_0.$

Classical Wiener measure is a Gaussian measure: in particular, it is a strictly positive probability measure.

Given classical Wiener measure $\backslash gamma$ on $C\_0,$ the product measure $\backslash gamma^n\backslash times\backslash gamma$ is a probability measure on $C$, where $\backslash gamma^n$ denotes the standard Gaussian measure on $\backslash R^n.$

For a stochastic process $\backslash \{X\_t,t\backslash in\; [0,T]\backslash \}:(\backslash Omega,\backslash mathcal\{F\},P)\backslash to\; (M,\backslash mathcal\{B\})$ and the function space $M^E\backslash equiv\backslash \{E\backslash to\; M\backslash \}$ of all functions from $E$ to $M$, one looks at the map $\backslash varphi:\backslash Omega\backslash to\; M^E$. One can then define the coordinate maps or canonical versions $Y\_t:M^E\backslash to\; M$ defined by $Y\_t(\backslash omega)=\backslash omega(t)$. The $\backslash \{Y\_t,t\backslash in\; E\backslash \}$ form another process. For $M=\backslash mathbb\{R\}$ and $E=\backslash R\_\{+\}$, the Wiener measure is then the unique measure on $C\_0(\backslash R\_\{+\},\backslash R)$ such that the coordinate process is a Brownian motion.{{cite book|title=Continuous Martingales and Brownian Motion|first1=Daniel|last1=Revuz|first2=Marc|last2=Yor|date=1999|series=Grundlehren der mathematischen Wissenschaften|volume=293|publisher=Springer|pages=33–37}}

Let $H\backslash subset\; C\_0([0,R])$ be a Hilbert space that is continuously embbeded and let $\backslash gamma$ be the Wiener measure then $\backslash gamma(H)=0$. This was proven in 1973 by Smolyanov and Uglanov and in the same year independently by Guerquin.{{cite journal |first1=Oleg G. |last1=Smolyanov |first2=Alexei V. |last2=Uglanov |title=Every Hilbert subspace of a Wiener space has measure zero |journal=Mathematical Notes |volume=14 |number=3 | date=1973| pages=772–774 |doi=10.1007/BF01147453}}{{cite journal|first=Małgorzata |last=Guerquin |title=Non-hilbertian structure of the Wiener measure |journal=Colloq. Math. |volume=28 |pages=145–146 |date=1973|doi=10.4064/cm-28-1-145-146 }} However, there exists a Hilbert space $H\backslash subset\; C\_0([0,R])$ with weaker topology such that $\backslash gamma(H)=1$ which was proven in 1993 by Uglanov.{{cite journal|first1=Alexei V. |last1=Uglanov |title=Hilbert supports of Wiener measure |journal=Math Notes |volume=51 |number=6 |date=1992 |pages=589–592 |doi=10.1007/BF01263304}}

• Abstract Wiener space
• Gaussian probability space
• Malliavin calculus
• Malliavin derivative
• Skorokhod space, a generalization of classical Wiener space, which allows functions to be discontinuous
• Wiener process

{{Measure theory}}

{{Analysis in topological vector spaces}}

Category:Measure theory

Category:Metric geometry

Category:Stochastic processes