From 61271b92a793dd837d8326c7064cebd0a3fcdb39 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 28 Nov 2021 17:15:39 +0100
Subject: Expand knowledge base

---
 .../know/concept/kolmogorov-equations/index.pdc    | 43 ++++++++++++++++++++++
 1 file changed, 43 insertions(+)

(limited to 'content/know/concept/kolmogorov-equations')

diff --git a/content/know/concept/kolmogorov-equations/index.pdc b/content/know/concept/kolmogorov-equations/index.pdc
index 331d803..a3b11db 100644
--- a/content/know/concept/kolmogorov-equations/index.pdc
+++ b/content/know/concept/kolmogorov-equations/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-11-14
 categories:
 - Mathematics
 - Statistics
+- Stochastic analysis
 
 date: 2021-11-13T21:05:30+01:00
 draft: false
@@ -201,6 +202,48 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
+This can be rewritten in a way
+that highlights the connection between Itō diffusions and physical diffusion,
+if we define the **diffusivity** $D$, **advection** $u$, and **probability flux** $J$:
+
+$$\begin{aligned}
+    D
+    \equiv \frac{1}{2} g^2
+    \qquad \quad
+    u
+    = f - \pdv{D}{x}
+    \qquad \quad
+    J
+    \equiv u \phi - D \pdv{\phi}{x}
+\end{aligned}$$
+
+Such that the forward Kolmogorov equation takes the following **conservative form**,
+so called because it looks like a physical continuity equation:
+
+$$\begin{aligned}
+    \boxed{
+        \pdv{\phi}{t}
+        = - \pdv{J}{x}
+        = - \pdv{x} \Big( u \phi - D \pdv{\phi}{x} \Big)
+    }
+\end{aligned}$$
+
+Note that if $u = 0$, then this reduces to
+[Fick's second law](/know/concept/ficks-laws/).
+The backward Kolmogorov equation can also be rewritten analogously,
+although it is less noteworthy:
+
+$$\begin{aligned}
+    \boxed{
+        - \pdv{\psi}{t}
+        = u \pdv{\psi}{x} + \pdv{x} \Big( D \pdv{\psi}{x} \Big)
+    }
+\end{aligned}$$
+
+Notice that the diffusivity term looks the same
+in both the forward and backward equations;
+we say that diffusion is self-adjoint.
+
 
 
 ## References
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