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for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376043.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376045.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376046.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376047.png" /> is called a transition density.
 
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376043.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376045.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376046.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376047.png" /> is called a transition density.
  
Under very general conditions (cf. [[#References|[1]]], [[#References|[2]]]), the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376048.png" /> can be related to a Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376049.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376050.png" /> (in the case of a Markov transition function, this process does not terminate, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376051.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376052.png" />-a.s.). On the other hand, the Markov property for a random process enables one, as a rule, to put the process into correspondence with a transition function [[#References|[3]]].
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Under very general conditions (cf. {{Cite|N}}, {{Cite|GS}}), the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376048.png" /> can be related to a Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376049.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376050.png" /> (in the case of a Markov transition function, this process does not terminate, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376051.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376052.png" />-a.s.). On the other hand, the Markov property for a random process enables one, as a rule, to put the process into correspondence with a transition function {{Cite|K}}.
  
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376053.png" /> be homogeneous in the sense that the set of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376054.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376055.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376056.png" /> forms a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376057.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376058.png" /> under addition (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376061.png" />). If, moreover, the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376062.png" /> depends only on the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376063.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376065.png" /> is a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376068.png" /> satisfying the corresponding form of conditions a)–d), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376069.png" /> is called a homogeneous transition function. The latter name is also given to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376070.png" /> for which (*) takes the form
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376053.png" /> be homogeneous in the sense that the set of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376054.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376055.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376056.png" /> forms a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376057.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376058.png" /> under addition (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376061.png" />). If, moreover, the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376062.png" /> depends only on the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376063.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376065.png" /> is a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376068.png" /> satisfying the corresponding form of conditions a)–d), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376069.png" /> is called a homogeneous transition function. The latter name is also given to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093760/t09376070.png" /> for which (*) takes the form
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970) {{MR|0272004}} {{ZBL|0203.49901}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.E. Kuznetsov, "Any Markov process in a Borel space has a transition function" ''Theory Probab. Appl.'' , '''25''' : 2 (1980) pp. 384–388 ''Teor. Veroyatnost. i ee Primenen.'' , '''25''' : 2 (1980) pp. 389–393 {{MR|0572574}} {{ZBL|0456.60077}} {{ZBL|0431.60071}} </TD></TR></table>
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{|
 
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|valign="top"|{{Ref|N}}|| J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970) {{MR|0272004}} {{ZBL|0203.49901}}
 
+
|-
 +
|valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}}
 +
|-
 +
|valign="top"|{{Ref|K}}|| S.E. Kuznetsov, "Any Markov process in a Borel space has a transition function" ''Theory Probab. Appl.'' , '''25''' : 2 (1980) pp. 384–388 ''Teor. Veroyatnost. i ee Primenen.'' , '''25''' : 2 (1980) pp. 389–393 {{MR|0572574}} {{ZBL|0456.60077}} {{ZBL|0431.60071}}
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|}
  
 
====Comments====
 
====Comments====
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Dellacherie, P.A. Meyer, "Probabilities and potential" , '''1–3''' , North-Holland (1978–1988) pp. Chapts. XII-XVI (Translated from French) {{MR|0939365}} {{MR|0898005}} {{MR|0727641}} {{MR|0745449}} {{MR|0566768}} {{MR|0521810}} {{ZBL|0716.60001}} {{ZBL|0494.60002}} {{ZBL|0494.60001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M.J. Sharpe, "General theory of Markov processes" , Acad. Press (1988) {{MR|0958914}} {{ZBL|0649.60079}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Albeverio, Z.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" ''Forum Math.'' , '''3''' (1991) pp. 389–400</TD></TR></table>
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{|
 +
|valign="top"|{{Ref|DM}}|| C. Dellacherie, P.A. Meyer, "Probabilities and potential" , '''1–3''' , North-Holland (1978–1988) pp. Chapts. XII-XVI (Translated from French) {{MR|0939365}} {{MR|0898005}} {{MR|0727641}} {{MR|0745449}} {{MR|0566768}} {{MR|0521810}} {{ZBL|0716.60001}} {{ZBL|0494.60002}} {{ZBL|0494.60001}}
 +
|-
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|valign="top"|{{Ref|S}}|| M.J. Sharpe, "General theory of Markov processes" , Acad. Press (1988) {{MR|0958914}} {{ZBL|0649.60079}}
 +
|-
 +
|valign="top"|{{Ref|AM}}|| S. Albeverio, Z.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" ''Forum Math.'' , '''3''' (1991) pp. 389–400
 +
|}

Revision as of 14:18, 31 May 2012

transition probability

2020 Mathematics Subject Classification: Primary: 60J35 [MSN][ZBL]

A family of measures used in the theory of Markov processes for determining the distribution at future instants from known states at previous times. Let a measurable space be such that the -algebra contains all one-point subsets from , and let be a subset of the real line . A function given for , , and is called a transition function for if: a) for given , and , it is a measure on , with ; b) for given , and , it is a -measurable function of the point ; c) and for all limit points of from the right in the topology of ,

and d) for all , and from , the Kolmogorov–Chapman equation is fulfilled:

(*)

(in some cases, requirement c) may be omitted or weakened). A transition function is called a Markov transition function if , and a subMarkov transition function otherwise. If is at most countable, then the transition function is specified by means of the matrix of transition probabilities

(see Transition probabilities; Matrix of transition probabilities). It often happens that for any admissible , and the measure has a density with respect to a certain measure. If in this case the following form of equation (*) is satisfied:

for any and from and from , then is called a transition density.

Under very general conditions (cf. [N], [GS]), the transition function can be related to a Markov process for which (in the case of a Markov transition function, this process does not terminate, i.e. -a.s.). On the other hand, the Markov property for a random process enables one, as a rule, to put the process into correspondence with a transition function [K].

Let be homogeneous in the sense that the set of values of for from forms a semi-group in under addition (for example, , , ). If, moreover, the transition function depends only on the difference , i.e. if , where is a function of , , satisfying the corresponding form of conditions a)–d), then is called a homogeneous transition function. The latter name is also given to a function for which (*) takes the form

For certain purposes (such as regularizing transition functions) it is necessary to extend the definition. For example, one takes as given a family of measurable spaces , , while a transition function with respect to this family is defined as a function , where , , , , that satisfies a suitable modification of conditions a)–d).

References

[N] J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970) MR0272004 Zbl 0203.49901
[GS] I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027
[K] S.E. Kuznetsov, "Any Markov process in a Borel space has a transition function" Theory Probab. Appl. , 25 : 2 (1980) pp. 384–388 Teor. Veroyatnost. i ee Primenen. , 25 : 2 (1980) pp. 389–393 MR0572574 Zbl 0456.60077 Zbl 0431.60071

Comments

For additional references see also Markov chain; Markov process.

References

[DM] C. Dellacherie, P.A. Meyer, "Probabilities and potential" , 1–3 , North-Holland (1978–1988) pp. Chapts. XII-XVI (Translated from French) MR0939365 MR0898005 MR0727641 MR0745449 MR0566768 MR0521810 Zbl 0716.60001 Zbl 0494.60002 Zbl 0494.60001
[S] M.J. Sharpe, "General theory of Markov processes" , Acad. Press (1988) MR0958914 Zbl 0649.60079
[AM] S. Albeverio, Z.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" Forum Math. , 3 (1991) pp. 389–400
How to Cite This Entry:
Transition function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transition_function&oldid=26963
This article was adapted from an original article by M.G. Shur (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article