Kolmogorov backward equations (diffusion)

The Kolmogorov backward equation (KBE) (diffusion) and its <a href="/facts/Adjoint_of_an_operator/gN6RHphd">adjoint</a> sometimes known as the Kolmogorov forward equation (diffusion) are <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a> (PDE) that arise in the theory of continuous-time continuous-state <a href="/facts/Markov_process/5oP5hntk">Markov processes</a>. Both were published by <a href="/facts/Andrey_Kolmogorov/bEKwtrs6">Andrey Kolmogorov</a> in 1931. Later it was realized that the forward equation was already known to physicists under the name <a href="/facts/Fokker%E2%80%93Planck_equation/974pPTer">Fokker–Planck equation</a>; the KBE on the other hand was new.
Informally, the Kolmogorov forward equation addresses the following problem. We have information about the state x of the system at time t (namely a <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> 
 
 
 
 
 p
 
 t
 
 
 (
 x
 )
 
 
 {\displaystyle p_{t}(x)}
 
); we want to know the probability distribution of the state at a later time 
 
 
 
 s
 >
 t
 
 
 {\displaystyle s>t}
 
. The adjective 'forward' refers to the fact that 
 
 
 
 
 p
 
 t
 
 
 (
 x
 )
 
 
 {\displaystyle p_{t}(x)}
 
 serves as the initial condition and the PDE is integrated forward in time (in the common case where the initial state is known exactly, 
 
 
 
 
 p
 
 t
 
 
 (
 x
 )
 
 
 {\displaystyle p_{t}(x)}
 
 is a <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a> centered on the known initial state).
The Kolmogorov backward equation on the other hand is useful when we are interested at time t in whether at a future time s the system will be in a given subset of states B, sometimes called the target set. The target is described by a given function 
 
 
 
 
 u
 
 s
 
 
 (
 x
 )
 
 
 {\displaystyle u_{s}(x)}
 
 which is equal to 1 if state x is in the target set at time s, and zero otherwise. In other words, 
 
 
 
 
 u
 
 s
 
 
 (
 x
 )
 =
 
 1
 
 B
 
 
 
 
 {\displaystyle u_{s}(x)=1_{B}}
 
, the indicator function for the set B. We want to know for every state x at time 
 
 
 
 t
 ,
  
 (
 t
 <
 s
 )
 
 
 {\displaystyle t,\ (t<s)}
 
 what is the probability of ending up in the target set at time s (sometimes called the hit probability). In this case 
 
 
 
 
 u
 
 s
 
 
 (
 x
 )
 
 
 {\displaystyle u_{s}(x)}
 
 serves as the final condition of the PDE, which is integrated backward in time, from s to t.

Kolmogorov backward equations (diffusion) open-in-new

Kolmogorov backward equations (diffusion)