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: ''For the food sometimes called a Wiener (sausage), see [[hot dog]] or [[Vienna sausage]].''
[[Image:Wiener process 3d.png|thumb|A long, thin Wiener sausage in 3 dimensions]]
[[Image:WienerSausage.jpg|thumb|left|A short, fat Wiener sausage in 2 dimensions]]
In the [[mathematics|mathematical]] field of [[probability]], the '''Wiener sausage''' is a neighborhood of the trace of a [[Brownian motion]] up to a time ''t'', given by taking all points within a fixed distance of  Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion. The Wiener sausage was named after [[Norbert Wiener]]  by {{harvs|txt=yes|authorlink=Monroe D. Donsker |author2-link=S. R. Srinivasa Varadhan|first=M. D.|last= Donsker |first2=S. R. Srinivasa|last2= Varadhan|year=1975}} because of its relation to the [[Wiener process]]; the name is also a pun on [[Vienna sausage]], as "Wiener" means "Viennese" in [[German language|German]].


The Wiener sausage is one of the simplest [[Markov process|non-Markovian]] functionals of Brownian motion. Its applications include [[stochastic]] phenomena including [[heat conduction]]. It was first described  by {{harvs|txt=yes|authorlink=Frank Spitzer|first=Frank|last= Spitzer|year=1964}}, and it was used  by {{harvs|txt=yes|authorlink=Mark Kac |author2-link=Joaquin Mazdak Luttinger |first=Mark|last= Kac |first2=Joaquin Mazdak|last2= Luttinger|year=1973|year2=1974}} to explain results of a [[Bose–Einstein condensate]], with proofs published by {{harvs|txt=yes|authorlink=Monroe D. Donsker |author2-link=S. R. Srinivasa Varadhan|first=M. D.|last= Donsker |first2=S. R. Srinivasa|last2= Varadhan|year=1975}}.


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==Definitions==
The Wiener sausage ''W''<sub>δ</sub>(''t'') of radius δ and length ''t'' is the set-valued [[random variable]] on [[Brownian path]]s '''b''' (in some Euclidean space) defined by
:<math>W_\delta(t)({ b})</math> is the set of points within a distance &delta; of some point '''b'''(''x'') of the path '''b''' with 0&le;''x''&le;''t''.
 
==The volume of the Wiener sausage==
 
There has been a lot of work on the behavior of the volume ([[Lebesgue measure]]) |''W''<sub>δ</sub>(''t'')| of the Wiener sausage as it becomes thin (δ→0); by rescaling, this is essentially equivalent to studying the volume as the sausage becomes long (''t''→∞).
 
{{harvtxt|Spitzer|1964}} showed that in 3 dimensions the expected value of the volume of the sausage is
:<math>E(|W_\delta(t)|) = 2\pi\delta t + 4\delta^2\sqrt{2\pi t} +4\pi\delta^3/3.</math>
In dimension ''d'' at least 3 the volume of the Wiener sausage is asymptotic to  
:<math>\delta^{d-2} \pi^{d/2}2t/\Gamma((d-2)/2)</math>
as ''t'' tends to infinity. In dimensions 1 and 2 this formula gets replaced by <math>\sqrt{8t/\pi}</math> and <math>2{\pi}t/\log(t)</math> respectively. {{harvtxt|Whitman|1964}}, a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compacts sets  than balls.
 
==References==
*{{citation|first1= M. D.|last1= Donsker|author1-link=Monroe D. Donsker |first2=S. R. S. |last2=Varadhan|author2-link=S. R. Srinivasa Varadhan|title=Asymptotics for the Wiener sausage|journal=Communications in Pure and Applied Mathematics| volume =28 |year=1975|pages= 525&ndash;565
|doi=10.1002/cpa.3160280406|issue= 4}}
*{{springer
| title=Wiener sausage
| id=W/w130100
| first=F. den |last=Hollander}}
*{{citation|mr=0342114|last= Kac|first= M.|author1-link=Mark Kac|last2= Luttinger|first2= J. M. |title=Bose-Einstein condensation in the presence of impurities  |journal=J. Mathematical Phys.|volume=  14  |year=1973|pages= 1626–1628|doi=10.1063/1.1666234|issue=11}}
*{{citation|mr=0342115|last= Kac|first= M.|author1-link=Mark Kac|last2= Luttinger|first2= J. M. |title=Bose-Einstein condensation in the presence of impurities. II |journal=J. Mathematical Phys.|volume=  15  |year=1974|pages= 183–186
|doi=10.1063/1.1666617|issue=2}}
*{{citation|mr=2105995|last= Simon|first= Barry|authorlink=Barry Simon|title= Functional integration and quantum physics|publisher= AMS Chelsea Publishing|place= Providence, RI|year= 2005|isbn= 0-8218-3582-3|unused_data=isbn status=May be invalid - please double check}} Especially chapter 22.
*{{citation|first=F.|last= Spitzer|authorlink=Frank Spitzer|title=Electrostatic capacity, heat flow and Brownian motion|journal= Probability Theory and Related Fields  |volume= 3  |year=1964|pages= 110–121|doi=10.1007/BF00535970|issue=2}}
*{{citation|last=Spitzer|title=  Principles of random walks|mr=0171290
|first= Frank|authorlink=Frank Spitzer
|series=Graduate Texts in Mathematics|volume= 34|publisher= Springer-Verlag|place= New York-Heidelberg, |year=1976|page=40}} (Reprint of 1964 edition)
*{{citation|mr=1717054
|last=Sznitman|first= Alain-Sol
|title=Brownian motion, obstacles and random media
|series=Springer Monographs in Mathematics|publisher= Springer-Verlag|place= Berlin|year= 1998|isbn= 3-540-64554-3|unused_data=isbn status=May be invalid - please double check}} An advanced monograph covering the Wiener sausage.
*{{citation|last=Whitman|first=Walter William|title=Some Strong Laws for Random Walks and Brownian Motion|series= PhD Thesis|year=1964|publisher=Cornell U.}}
 
{{Stochastic processes}}
 
[[Category:Mathematical physics]]
[[Category:Statistical mechanics]]
[[Category:Stochastic processes]]

Latest revision as of 11:35, 28 January 2013

For the food sometimes called a Wiener (sausage), see hot dog or Vienna sausage.
A long, thin Wiener sausage in 3 dimensions
A short, fat Wiener sausage in 2 dimensions

In the mathematical field of probability, the Wiener sausage is a neighborhood of the trace of a Brownian motion up to a time t, given by taking all points within a fixed distance of Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion. The Wiener sausage was named after Norbert Wiener by Template:Harvs because of its relation to the Wiener process; the name is also a pun on Vienna sausage, as "Wiener" means "Viennese" in German.

The Wiener sausage is one of the simplest non-Markovian functionals of Brownian motion. Its applications include stochastic phenomena including heat conduction. It was first described by Template:Harvs, and it was used by Template:Harvs to explain results of a Bose–Einstein condensate, with proofs published by Template:Harvs.

Definitions

The Wiener sausage Wδ(t) of radius δ and length t is the set-valued random variable on Brownian paths b (in some Euclidean space) defined by

Wδ(t)(b) is the set of points within a distance δ of some point b(x) of the path b with 0≤xt.

The volume of the Wiener sausage

There has been a lot of work on the behavior of the volume (Lebesgue measure) |Wδ(t)| of the Wiener sausage as it becomes thin (δ→0); by rescaling, this is essentially equivalent to studying the volume as the sausage becomes long (t→∞).

Template:Harvtxt showed that in 3 dimensions the expected value of the volume of the sausage is

E(|Wδ(t)|)=2πδt+4δ22πt+4πδ3/3.

In dimension d at least 3 the volume of the Wiener sausage is asymptotic to

δd2πd/22t/Γ((d2)/2)

as t tends to infinity. In dimensions 1 and 2 this formula gets replaced by 8t/π and 2πt/log(t) respectively. Template:Harvtxt, a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compacts sets than balls.

References

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    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 Especially chapter 22.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (Reprint of 1964 edition)
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 An advanced monograph covering the Wiener sausage.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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