In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number p ≥ 1 {\displaystyle p\geq 1} . p-variation is a measure of the regularity or smoothness of a function. Specifically, if f : I → ( M , d ) {\displaystyle f:I\to (M,d)} , where ( M , d ) {\displaystyle (M,d)} is a metric space and I a totally ordered set, its p-variation is:
‖ f ‖ p -var = ( sup D ∑ t k ∈ D d ( f ( t k ) , f ( t k − 1 ) ) p ) 1 / p {\displaystyle \|f\|_{p{\text{-var}}}=\left(\sup _{D}\sum _{t_{k}\in D}d(f(t_{k}),f(t_{k-1}))^{p}\right)^{1/p}}
where D ranges over all finite partitions of the interval I. The p variation of a function decreases with p. If f has finite p-variation and g is an α-Hölder continuous function, then g ∘ f {\displaystyle g\circ f} has finite p α {\displaystyle {\frac {p}{\alpha }}} -variation. The case when p is one is called total variation, and functions with a finite 1-variation are called bounded variation functions. This concept should not be confused with the notion of p-th variation along a sequence of partitions, which is computed as a limit along a given sequence ( D n ) {\displaystyle (D_{n})} of time partitions:
[ f ] p = ( lim n → ∞ ∑ t k n ∈ D n d ( f ( t k n ) , f ( t k − 1 n ) ) p ) {\displaystyle [f]_{p}=\left(\lim _{n\to \infty }\sum _{t_{k}^{n}\in D_{n}}d(f(t_{k}^{n}),f(t_{k-1}^{n}))^{p}\right)}
For example for p=2, this corresponds to the concept of quadratic variation, which is different from 2-variation.
Link with Hölder norm One can interpret the p-variation as a parameter-independent version of the Hölder norm, which also extends to discontinuous functions. If f is α–Hölder continuous (i.e. its α–Hölder norm is finite) then its 1 α {\displaystyle {\frac {1}{\alpha }}} -variation is finite. Specifically, on an interval [a,b], ‖ f ‖ 1 α -var ≤ ‖ f ‖ α ( b − a ) α {\displaystyle \|f\|_{{\frac {1}{\alpha }}{\text{-var}}}\leq \|f\|_{\alpha }(b-a)^{\alpha }} . If p is less than q then the space of functions of finite p-variation on a compact set is continuously embedded with norm 1 into those of finite q-variation. I.e.
‖ f ‖ q -var ≤ ‖ f ‖ p -var {\displaystyle \|f\|_{q{\text{-var}}}\leq \|f\|_{p{\text{-var}}}} . However unlike the analogous situation with Hölder spaces the embedding is not compact. For example, consider the real functions on [0,1] given by f n ( x ) = x n {\displaystyle f_{n}(x)=x^{n}} . They are uniformly bounded in 1-variation and converge pointwise to a discontinuous function f but this not only is not a convergence in p-variation for any p but also is not uniform convergence.
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