View source for Brownian local time

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b1205001.png" /> be a standard [[Wiener process|Wiener process]] (or, in other words, [[Brownian motion|Brownian motion]]) living on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b1205002.png" /> and started at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b1205003.png" />. The random set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b1205004.png" />, the so-called zero set of the Brownian path, is almost surely perfect (i.e. closed and dense in itself), unbounded and of [[Lebesgue measure|Lebesgue measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b1205005.png" />. The complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b1205006.png" /> is a countable union of open intervals.

A remarkable result of P. Lévy ([[#References|[a6]]], [[#References|[a7]]]) is that there exists a non-decreasing (random) function determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b1205007.png" /> which is constant on the open intervals in the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b1205008.png" /> and which has every point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b1205009.png" /> as a (left and/or right) strict increase point. This function is called the Brownian local time (at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050010.png" />). It is clear that a similar construction can be made at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050011.png" />.

The existence of the local time can be deduced from the fact (also due to Lévy) that the processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050014.png" />, are identical in law. Indeed, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050015.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050016.png" /> has the desired properties of local time; for the proof that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050017.png" />, for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050018.png" />, is determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050019.png" />, see [[#References|[a4]]]. Because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050021.png" /> are identical in law, there exists a function with corresponding properties connected to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050022.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050023.png" /> be the Brownian local time at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050024.png" /> at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050025.png" />. Then almost surely

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050026.png" /></td> </tr></table>

and this leads to the occupation-time formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050027.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050028.png" /> is a Borel-measurable function (cf. also [[Borel function|Borel function]]).

As seen above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050029.png" /> can be viewed as the [[Measure|measure]] of the zero set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050030.png" />. In fact, it has been proved in [[#References|[a11]]] and [[#References|[a9]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050031.png" /> is the random Hausdorff <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050033.png" />-measure (cf. also [[Hausdorff measure|Hausdorff measure]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050034.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050035.png" />.

Introduce for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050036.png" /> the right-continuous inverse of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050037.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050038.png" /></td> </tr></table>

By the strong Markov property (cf. [[Markov property|Markov property]]) and spatial homogeneity of Brownian motion, the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050039.png" /> is increasing and has independent and identically distributed increments, in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050040.png" /> is a subordinator. Because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050042.png" /> are, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050043.png" />, identical in law, also the so-called inverse local time

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050044.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050045.png" /> are identical in law. Hence, the finite-dimensional distributions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050046.png" /> are determined by the Laplace transform

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050047.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050048.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050049.png" /></td> </tr></table>

The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050052.png" />, is continuous. This is due to H.F. Trotter [[#References|[a12]]]; for a proof based on the Itô formula, see, e.g., [[#References|[a3]]].

The behaviour of the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050053.png" /> can be characterized for some stopping times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050054.png" /> (for first hitting times, for instance; cf. also [[Stopping time|Stopping time]]). Results in this direction are called Ray–Knight theorems [[#References|[a10]]], [[#References|[a5]]]; see also [[#References|[a2]]].

The process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050055.png" /> is an example of an additive functional of Brownian motion having support at one point (i.e. at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120500/b12050056.png" />). As such it is unique up to a multiplicative constant. See [[#References|[a1]]].

Brownian local time is an important concept both in the theory and in applications of stochastic processes. It can be used, e.g., to construct diffusions from Brownian motion via random time change and to analyze stochastic differential equations (cf. also [[Stochastic differential equation|Stochastic differential equation]]). There are some natural problems in stochastic optimal control (finite fuel problem) and in financial mathematics (barrier options), for instance, where (Brownian) local time plays a crucial role.

For a survey article, see [[#References|[a8]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.M. Blumenthal,   R.K. Getoor,   "Markov processes and potential theory" , Acad. Press  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.N. Borodin,   P. Salminen,   "Handbook of Brownian motion: Facts and formulae" , Birkhäuser  (1996)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Ikeda,   S. Watanabe,   "Stochastic differential equations and diffusion processes" , North-Holland &amp;Kodansha  (1981)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Itô,   H.P. McKean,   "Diffusion processes and their sample paths" , Springer  (1974)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  F. Knight,   "Random walks and a sojourn density process of Brownian motion"  ''Trans. Amer. Math. Soc.'' , '''109'''  (1963)  pp. 56–86</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Lévy,   "Processus stochastiques et mouvement brownien" , Gauthier-Villars  (1948)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P. Lévy,   "Sur certains processus stochastiques homogénes"  ''Compositio Math.'' , '''7'''  (1939)  pp. 283–339</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  H.P. McKean,   "Brownian local time"  ''Adv. Math.'' , '''15'''  (1975)  pp. 91–111</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E. Perkins,   "The exact Hausdorff measure of the level sets of Brownian motion"  ''ZWvG'' , '''58'''  (1981)  pp. 373–388</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  D.B. Ray,   "Sojourn times of a diffusion process Ill"  ''J. Math.'' , '''7'''  (1963)  pp. 615–630</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  S.J. Taylor,   J.G. Wendel,   "The exact Hausdorff measure of the zero set of a stable process"  ''ZWvG'' , '''6'''  (1966)  pp. 170–180</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  H.F. Trotter,   "A property of Brownian motion paths. Ill"  ''J. Math.'' , '''2'''  (1958)  pp. 425–433</TD></TR></table>