http://debianws.lexgopc.com/wiki143/index.php?action=history&feed=atom&title=Reversible_diffusion 2026-05-06T14:51:50Z Revision history for this page on the wiki MediaWiki 1.43.1 http://debianws.lexgopc.com/wiki143/index.php?title=Reversible_diffusion&diff=5486660&oldid=prev 2025-04-14T06:35:12Z

<p>Moving <a href="/wiki143/index.php?title=Category:Probability_theorems&amp;action=edit&amp;redlink=1" class="new" title="Category:Probability theorems (page does not exist)">Category:Probability theorems</a> to <a href="/wiki143/index.php?title=Category:Theorems_in_probability_theory&amp;action=edit&amp;redlink=1" class="new" title="Category:Theorems in probability theory (page does not exist)">Category:Theorems in probability theory</a> per <a href="https://en.wikipedia.org/wiki/Categories_for_discussion/Speedy" class="extiw" title="wikipedia:Categories for discussion/Speedy">Wikipedia:Categories for discussion/Speedy</a></p> <p><b>New page</b></p><div>In [[mathematics]], a &#039;&#039;&#039;reversible diffusion&#039;&#039;&#039; is a specific example of a [[reversible dynamics|reversible]] [[stochastic process]]. Reversible diffusions have an elegant [[characterization (mathematics)|characterization]] due to the [[Russia]]n [[mathematician]] [[Andrey Nikolaevich Kolmogorov]].<br /> <br /> ==Kolmogorov&#039;s characterization of reversible diffusions==<br /> <br /> Let &#039;&#039;B&#039;&#039; denote a &#039;&#039;d&#039;&#039;-[[dimension]]al standard [[Brownian motion]]; let &#039;&#039;b&#039;&#039;&amp;nbsp;:&amp;nbsp;&#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;&#039;&#039;d&#039;&#039;&lt;/sup&gt;&amp;nbsp;→&amp;nbsp;&#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;&#039;&#039;d&#039;&#039;&lt;/sup&gt; be a [[Lipschitz continuous]] [[vector field]]. Let &#039;&#039;X&#039;&#039;&amp;nbsp;:&amp;nbsp;[0,&amp;nbsp;+∞)&amp;nbsp;&amp;times;&amp;nbsp;Ω&amp;nbsp;→&amp;nbsp;&#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;&#039;&#039;d&#039;&#039;&lt;/sup&gt; be an [[Itō diffusion]] defined on a [[probability space]] (Ω,&amp;nbsp;Σ,&amp;nbsp;&#039;&#039;&#039;P&#039;&#039;&#039;) and solving the Itō [[stochastic differential equation]]<br /> &lt;math display=&quot;block&quot;&gt;\mathrm{d} X_{t} = b(X_{t}) \, \mathrm{d} t + \mathrm{d} B_{t}&lt;/math&gt;<br /> with square-integrable initial condition, i.e. &#039;&#039;X&#039;&#039;&lt;sub&gt;0&lt;/sub&gt;&amp;nbsp;∈&amp;nbsp;&#039;&#039;L&#039;&#039;&lt;sup&gt;2&lt;/sup&gt;(Ω,&amp;nbsp;Σ,&amp;nbsp;&#039;&#039;&#039;P&#039;&#039;&#039;;&amp;nbsp;&#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;&#039;&#039;d&#039;&#039;&lt;/sup&gt;). Then the following are equivalent:<br /> <br /> * The process &#039;&#039;X&#039;&#039; is reversible with [[stationary distribution]] &#039;&#039;μ&#039;&#039; on &#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;&#039;&#039;d&#039;&#039;&lt;/sup&gt;.<br /> * There exists a [[scalar potential]] Φ&amp;nbsp;:&amp;nbsp;&#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;&#039;&#039;d&#039;&#039;&lt;/sup&gt;&amp;nbsp;→&amp;nbsp;&#039;&#039;&#039;R&#039;&#039;&#039; such that &#039;&#039;b&#039;&#039;&amp;nbsp;=&amp;nbsp;&amp;minus;∇Φ, &#039;&#039;μ&#039;&#039; has [[Radon–Nikodym derivative]] &lt;math display=&quot;block&quot;&gt;\frac{\mathrm{d} \mu (x)}{\mathrm{d} x} = \exp \left( - 2 \Phi (x) \right)&lt;/math&gt; and &lt;math display=&quot;block&quot;&gt;\int_{\mathbf{R}^{d}} \exp \left( - 2 \Phi (x) \right) \, \mathrm{d} x = 1.&lt;/math&gt;<br /> <br /> (Of course, the condition that &#039;&#039;b&#039;&#039; be the negative of the [[gradient]] of Φ only determines Φ [[up to]] an additive constant; this constant may be chosen so that exp(&amp;minus;2Φ(·)) is a [[probability density function]] with integral 1.)<br /> <br /> ==References==<br /> <br /> * {{cite thesis<br /> | last = Voß<br /> | first = Jochen<br /> | title = Some large deviation results for diffusion processes<br /> | year = 2004<br /> | publisher = PhD thesis<br /> | location = Universität Kaiserslautern<br /> | url = https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1559<br /> }} (See theorem 1.4)<br /> <br /> [[Category:Stochastic differential equations]]<br /> [[Category:Theorems in probability theory]]</div>

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