The Kolmogorov-Chapman equation

Kolmogorov - Chapman equation for a one-parameter family of continuous linear operators $\mathbf {P} (t),\;t>0$ ${\ displaystyle \ mathbf {P} (t), \; t> 0}$ ${\ displaystyle \ mathbf {P} (t), \; t> 0}$ in a topological vector space expresses a semigroup property :

\mathbf {P} (t+s)=\mathbf {P} (t)\mathbf {P} (s).

{\ displaystyle \ mathbf {P} (t + s) = \ mathbf {P} (t) \ mathbf {P} (s).}

{\ displaystyle \ mathbf {P} (t + s) = \ mathbf {P} (t) \ mathbf {P} (s).}

Most often, this term is used in the theory of homogeneous Markov random processes , where $\mathbf {P} (t),\;t\geq 0$ ${\ displaystyle \ mathbf {P} (t), \; t \ geq 0}$ ${\ displaystyle \ mathbf {P} (t), \; t \ geq 0}$ - an operator that transforms the probability distribution at the initial moment of time into the probability distribution at the moment of time $t$ ${\ displaystyle t}$ $t$ ( $\mathbf {P} (0)=\mathbf {1}$ ${\ displaystyle \ mathbf {P} (0) = \ mathbf {1}}$ ${\ displaystyle \ mathbf {P} (0) = \ mathbf {1}}$ )

For inhomogeneous processes, two-parameter families of operators are considered $\mathbf {P} (t,h),\;h>t>0$ ${\ displaystyle \ mathbf {P} (t, h), \; h> t> 0}$ ${\ displaystyle \ mathbf {P} (t, h), \; h> t> 0}$ transforming the probability distribution at time $t>0$ ${\ displaystyle t> 0}$ ${\ displaystyle t> 0}$ in the probability distribution at time $h>t>0.$ ${\ displaystyle h> t> 0.}$ ${\ displaystyle h> t> 0.}$ For them, the Kolmogorov – Chapman equation has the form

\mathbf {P} (t,s)=\mathbf {P} (t,h)\mathbf {P} (h,s),\;s>h>t>0.

{\ displaystyle \ mathbf {P} (t, s) = \ mathbf {P} (t, h) \ mathbf {P} (h, s), \; s> h> t> 0.}

{\ displaystyle \ mathbf {P} (t, s) = \ mathbf {P} (t, h) \ mathbf {P} (h, s), \; s> h> t> 0.}

For discrete time systems, the parameters $t, h, s$ ${\ displaystyle t, h, s}$ ${\ displaystyle t, h, s}$ take natural values .

Forward and reverse Kolmogorov equations

Formally differentiating the Kolmogorov – Chapman equation with respect to $s$ ${\ displaystyle s}$ $s$ at $s=0$ ${\ displaystyle s = 0}$ $s=0$ we obtain the direct Kolmogorov equation :

{\frac {d\mathbf {P} (t)}{dt}}=\mathbf {P} (t)\mathbf {Q} ,

{\ displaystyle {\ frac {d \ mathbf {P} (t)} {dt}} = \ mathbf {P} (t) \ mathbf {Q},}

{\frac {d{\mathbf {P}}(t)}{dt}}={\mathbf {P}}(t){\mathbf {Q}},

Where

\mathbf {Q} =\lim _{h\to 0}{\frac {\mathbf {P} (h)-\mathbf {1} }{h}}.

{\ displaystyle \ mathbf {Q} = \ lim _ {h \ to 0} {\ frac {\ mathbf {P} (h) - \ mathbf {1}} {h}}.}

\mathbf {Q} =\lim _{h\to 0}{\frac {\mathbf {P} (h)-\mathbf {1} }{h}}.

Formally differentiating the Kolmogorov-Chapman equation with respect to $t$ ${\ displaystyle t}$ $t$ at $t=0$ ${\ displaystyle t = 0}$ $t=0$ we obtain the inverse Kolmogorov equation

{\frac {d\mathbf {P} (t)}{dt}}=\mathbf {Q} \mathbf {P} (t).

{\ displaystyle {\ frac {d \ mathbf {P} (t)} {dt}} = \ mathbf {Q} \ mathbf {P} (t).}

{\frac {d{\mathbf {P}}(t)}{dt}}={\mathbf {Q}}{\mathbf {P}}(t).

It must be emphasized that for infinite-dimensional spaces the operator $\mathbf {Q}$ ${\ displaystyle \ mathbf {Q}}$ ${\mathbf {Q}}$ is no longer necessarily continuous, and may not be defined everywhere, for example, to be a differential operator in the distribution space.

Examples

We consider homogeneous Markov random processes in $\mathbb {R} ^{n},$ ${\ displaystyle \ mathbb {R} ^ {n},}$ for which the transition probability operator $\mathbf {P} (t)$ ${\ displaystyle \ mathbf {P} (t)}$ defined by transition density $p(t,x,y)$ ${\ displaystyle p (t, x, y)}$ : probability of transition from area $U$ ${\ displaystyle U}$ to the area $W$ ${\ displaystyle W}$ during $t$ ${\ displaystyle t}$ there is $\int \limits _{U}dx\,\int \limits _{V}dy\,p(t,x,y)$ ${\ displaystyle \ int \ limits _ {U} dx \, \ int \ limits _ {V} dy \, p (t, x, y)}$ . The Kolmogorov – Chapman equation for densities has the form:

p(t+s,x,y)=\int \limits _{\mathbb {R} ^{n}}p(t,x,z)p(s,z,y)\,dz.

{\ displaystyle p (t + s, x, y) = \ int \ limits _ {\ mathbb {R} ^ {n}} p (t, x, z) p (s, z, y) \, dz. }

At $t>0,\,t\to 0$ ${\ displaystyle t> 0, \, t \ to 0}$ transition density $p(t,x,y)$ ${\ displaystyle p (t, x, y)}$ tends to a δ-function (in the sense of the weak limit of generalized functions ): $\lim _{t\to 0}p(t,x,y)=\delta (x-y)$ ${\ displaystyle \ lim _ {t \ to 0} p (t, x, y) = \ delta (xy)}$ . It means that $\lim _{t\to 0}\mathbf {P} (t)=\mathbf {1} .$ ${\ displaystyle \ lim _ {t \ to 0} \ mathbf {P} (t) = \ mathbf {1}.}$ Suppose there is a limit (also a generalized function)

q(x,y)=\lim _{h\to 0}{\frac {p(h,x,y)-\delta (x-y)}{h}}.

{\ displaystyle q (x, y) = \ lim _ {h \ to 0} {\ frac {p (h, x, y) - \ delta (xy)} {h}}.}

Then the operator $\mathbf {Q}$ ${\ displaystyle \ mathbf {Q}}$ acts on functions $f(x)$ ${\ displaystyle f (x)}$ defined on $\mathbb {R} ^{n},$ ${\ displaystyle \ mathbb {R} ^ {n},}$ as $(\mathbf {Q} f)(x)=\int \limits _{\mathbb {R} ^{n}}q(x,y)f(y)\,dy,$ ${\ displaystyle (\ mathbf {Q} f) (x) = \ int \ limits _ {\ mathbb {R} ^ {n}} q (x, y) f (y) \, dy,}$ and the direct Kolmogorov equation takes the form

{\frac {\partial p(t,x,y)}{\partial t}}=\int \limits _{\mathbb {R} ^{n}}p(t,x,z)q(z,y)\,dz,

{\ displaystyle {\ frac {\ partial p (t, x, y)} {\ partial t}} = \ int \ limits _ {\ mathbb {R} ^ {n}} p (t, x, z) q (z, y) \, dz,}

and the inverse Kolmogorov equation

{\frac {\partial p(t,x,y)}{\partial t}}=\int \limits _{\mathbb {R} ^{n}}q(x,z)p(t,z,y)\,dz.

{\ displaystyle {\ frac {\ partial p (t, x, y)} {\ partial t}} = \ int \ limits _ {\ mathbb {R} ^ {n}} q (x, z) p (t , z, y) \, dz.}

Let the operator $\mathbf {Q}$ ${\ displaystyle \ mathbf {Q}}$ - second-order differential operator with continuous coefficients:

(\mathbf {Q} f)={\frac {1}{2}}\sum _{i,j}a^{ij}(x){\frac {\partial ^{2}f}{\partial x^{i}\partial x^{j}}}+\sum _{j}b^{j}(x){\frac {\partial f}{\partial x^{j}}}.

{\ displaystyle (\ mathbf {Q} f) = {\ frac {1} {2}} \ sum _ {i, j} a ^ {ij} (x) {\ frac {\ partial ^ {2} f} {\ partial x ^ {i} \ partial x ^ {j}}} + \ sum _ {j} b ^ {j} (x) {\ frac {\ partial f} {\ partial x ^ {j}}} .}

(it means that $q(x,y)$ ${\ displaystyle q (x, y)}$ there is a linear combination of the first and second derivatives $\delta (x-y)$ ${\ displaystyle \ delta (xy)}$ with continuous coefficients). Matrix $a^{ij}$ ${\ displaystyle a ^ {ij}}$ symmetrical. Let it be positive definite at each point ( diffusion ). The direct Kolmogorov equation has the form

{\frac {\partial p(t,x,y)}{\partial t}}={\frac {1}{2}}\sum _{i,j}{\frac {\partial ^{2}}{\partial y^{i}\partial y^{j}}}(a^{ij}(y)p(t,x,y))-\sum _{j}{\frac {\partial }{\partial y^{j}}}(b^{j}(y)p(t,x,y)).

{\ displaystyle {\ frac {\ partial p (t, x, y)} {\ partial t}} = {\ frac {1} {2}} \ sum _ {i, j} {\ frac {\ partial ^ {2}} {\ partial y ^ {i} \ partial y ^ {j}}} (a ^ {ij} (y) p (t, x, y)) - \ sum _ {j} {\ frac { \ partial} {\ partial y ^ {j}}} (b ^ {j} (y) p (t, x, y)).}

This equation is called the Fokker-Planck equation . Vector $b^{j}$ ${\ displaystyle b ^ {j}}$ in the physical literature is called the drift vector, and the matrix $a^{ij}$ ${\ displaystyle a ^ {ij}}$ - diffusion tensor The inverse Kolmogorov equation in this case

{\frac {\partial p(t,x,y)}{\partial t}}={\frac {1}{2}}\sum _{i,j}a^{ij}(x){\frac {\partial ^{2}}{\partial x^{i}\partial x^{j}}}p(t,x,y)+\sum _{j}b^{j}(x){\frac {\partial }{\partial x^{j}}}p(t,x,y).

{\ displaystyle {\ frac {\ partial p (t, x, y)} {\ partial t}} = {\ frac {1} {2}} \ sum _ {i, j} a ^ {ij} (x ) {\ frac {\ partial ^ {2}} {\ partial x ^ {i} \ partial x ^ {j}}} p (t, x, y) + \ sum _ {j} b ^ {j} ( x) {\ frac {\ partial} {\ partial x ^ {j}}} p (t, x, y).}

Literature

Wentzel A.D. , A course in the theory of random processes. - M .: Nauka, 1996 .-- 400 p.

The Kolmogorov-Chapman equation

Forward and reverse Kolmogorov equations

Examples

See also

Literature

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