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第1,081行： − + − 1+ − 953年，杜布出版了《随机过程》一书，这本书对随机过程理论产生了重大影响，并强调了测度理论在概率论中的重要性。<ref name=“Meyer2009”/>+ − + − +
→Stochastic processes after World War II二战后的随机过程
Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of [[potential theory]], with early ideas by [[Shizuo Kakutani]] and then later work by Joseph Doob.<ref name="Meyer2009"/> Further work, considered pioneering, was done by [[Gilbert Hunt]] in the 1950s, connecting Markov processes and potential theory, which had a significant effect on the theory of Lévy processes and led to more interest in studying Markov processes with methods developed by Itô.<ref name="JarrowProtter2004"/><ref name="Bertoin1998pageVIIIandIX">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii and ix}}</ref><ref name="Steele2012page176">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=176}}</ref>
Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of [[potential theory]], with early ideas by [[Shizuo Kakutani]] and then later work by Joseph Doob.<ref name="Meyer2009"/> Further work, considered pioneering, was done by [[Gilbert Hunt]] in the 1950s, connecting Markov processes and potential theory, which had a significant effect on the theory of Lévy processes and led to more interest in studying Markov processes with methods developed by Itô.<ref name="JarrowProtter2004"/><ref name="Bertoin1998pageVIIIandIX">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii and ix}}</ref><ref name="Steele2012page176">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=176}}</ref>
同样从20世纪40年代开始，随机过程（尤其是鞅）与[[势理论]]的数学领域之间建立了联系，[[Shizuo Kakutani]]的早期思想和Joseph Doob后来的工作。<ref name=“Meyer2009”/>在1950年代[[Gilbert Hunt]]完成了被认为是开创性的进一步工作，把马尔可夫过程和势理论联系起来，这对Lévy过程理论产生了重大影响，并使人们对用It开发的方法研究马尔可夫过程产生了更多的兴趣<ref name="Bertoin1998pageVIIIandIX">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books？id=ftcsQgMp5cUC&pg=PR8 | year=1998 | publisher=Cambridge University Press | isbn=978-0-521-64632-1 | page=viii and ix}</ref><ref name=“Steele2012page176”{{引用图书|作者=J.Michael Steele | title=随机微积分和金融应用| url=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer科学与商业媒体| isbn=978-1-4684-9305-4 | page=176}</ref>
同样从20世纪40年代开始，随机过程（尤其是鞅）与[[势理论]]的数学领域之间建立了联系，[[Shizuo Kakutani]]的早期思想和Joseph Doob后来的工作。<ref name=“Meyer2009”/>在1950年代[[Gilbert Hunt]]完成了被认为是开创性的进一步工作，把马尔可夫过程和势理论联系起来，这对Lévy过程理论产生了重大影响，并使人们对用It开发的方法研究马尔可夫过程产生了更多的兴趣<ref name=“JarrowProtter2004”/><ref name=“Bertoin1998pageVIIIandIX”>{cite book | author=Jean Bertoin | title=Lévy Processes |网址=https://books.google.com/books？id=ftcsQgMp5cUC&pg=PR8 | year=1998 | publisher=剑桥大学出版社| isbn=978-0-521-64632-1 | page=viii and ix}}</ref><ref name=“Steele2012page176”{{引用图书|作者=J.Michael Steele | title=随机微积分和金融应用| url=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer科学与商业媒体| isbn=978-1-4684-9305-4 | page=176}</ref>
In 1953 Doob published his book ''Stochastic processes'', which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability.<ref name="Meyer2009"/>
In 1953 Doob published his book ''Stochastic processes'', which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability.<ref name="Meyer2009"/>
1953年，杜布出版了《随机过程》一书，这本书对随机过程理论产生了重大影响，并强调了测度理论在概率论中的重要性。<ref name=“Meyer2009”/>
<ref name="Bingham2005">{{cite journal|last1=Bingham|first1=N. H.|title=Doob: a half-century on|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|pages=257–266|issn=0021-9002|doi=10.1239/jap/1110381385|doi-access=free}}</ref> Doob also chiefly developed the theory of martingales, with later substantial contributions by [[Paul-André Meyer]]. Earlier work had been carried out by [[Sergei Bernstein]], [[Paul Lévy (mathematician)|Paul Lévy]] and [[Jean Ville]], the latter adopting the term martingale for the stochastic process.<ref name="HallHeyde2014page1">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|pages=1, 2}}</ref><ref name="Dynkin1989">{{cite journal|last1=Dynkin|first1=E. B.|title=Kolmogorov and the Theory of Markov Processes|journal=The Annals of Probability|volume=17|issue=3|year=1989|pages=822–832|issn=0091-1798|doi=10.1214/aop/1176991248|doi-access=free}}</ref> Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes.<ref name="Meyer2009"/>
<ref name="Bingham2005">{{cite journal|last1=Bingham|first1=N. H.|title=Doob: a half-century on|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|pages=257–266|issn=0021-9002|doi=10.1239/jap/1110381385|doi-access=free}}</ref> Doob also chiefly developed the theory of martingales, with later substantial contributions by [[Paul-André Meyer]]. Earlier work had been carried out by [[Sergei Bernstein]], [[Paul Lévy (mathematician)|Paul Lévy]] and [[Jean Ville]], the latter adopting the term martingale for the stochastic process.<ref name="HallHeyde2014page1">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|pages=1, 2}}</ref><ref name="Dynkin1989">{{cite journal|last1=Dynkin|first1=E. B.|title=Kolmogorov and the Theory of Markov Processes|journal=The Annals of Probability|volume=17|issue=3|year=1989|pages=822–832|issn=0091-1798|doi=10.1214/aop/1176991248|doi-access=free}}</ref> Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes.<ref name="Meyer2009"/>
{{引用期刊| last1=Bingham | first1=Bingham first1=N.H.；title=Doob:半个世纪on | journal=journal of Appl概率应用概率| volume=42 | issen=1 | year=1 124年=2005年| pages=257–266 | issn=0021-9002 | doi=10.1239/1239/jap/jap/1110381385 |doiaccess=free}}</ref>Doob也主要发展了鞅理论的理论，他主要发展了鞅的理论，他还主要发展在，后来[[Paul AndréMeyer]]提供了大量捐助。早期的工作是由[[谢尔盖.伯恩斯坦]，[[保罗.莱维（数学家）|保罗.莱维]和[[让.维尔]]进行的，后者对随机过程采用了鞅的概念=https://books.google.com/books？id=gqriBQAAQBAJ&pg=PR10 |年份=2014 | publisher=Elsevier Science | isbn=978-1-4832-6322-9 |页数=1，2} </ref><ref name=“dynk198989”>{〈引用期刊| last1=Dynkin | first1=E.B.| title=Kolmogorov和马尔可夫过程理论的马尔可夫过程理论| journal=概率年鉴| volume=17 | issue=3 | year=1989 | pages=822–832 | issn=0091-1798 | doi=10.1214/aop/11766991248 | doi access=free}</ref>方法方法从方法中获取方法从方法中获得方法方法从方法从方法到方法从方法起，方法从方法鞅理论因解决各种问题而流行起来概率问题。研究马尔可夫过程的技术和理论发展到鞅上。相反，鞅理论中的方法被用来处理马尔可夫过程。<ref name=“Meyer2009”/>
<ref name="Bingham2005">{{引用期刊| last1=Bingham | first1=Bingham first1=N.H.；title=Doob:半个世纪on | journal=journal of Appl概率应用概率| volume=42 | issen=1 | year=1 124年=2005年| pages=257–266 | issn=0021-9002 | doi=10.1239/1239/jap/jap/1110381385 |doiaccess=free}}</ref>Doob还主要发展了鞅理论，后来[[保罗.安德烈.梅耶]也作出了重大贡献。早期的研究是由[[Sergei Bernstein]]、[[Paul Lévy（数学家）| Paul Lévy]]和[[Jean Ville]]进行的，后者采用了随机过程的鞅项。<ref name=“HallHeyde2014page1”>{cite book | author1=P.Hall | author2=C.C.Heyde | title=鞅极限理论及其应用| url=https://books.google.com/books？id=gqriBQAAQBAJ&pg=PR10 |年份=2014 | publisher=Elsevier Science | isbn=978-1-4832-6322-9 | pages=1，2}}</ref><ref name=“dynk198989”>{〈引用期刊| last1=Dynkin | first1=E.B.| title=Kolmogorov和马尔可夫过程理论的马尔可夫过程理论| journal=概率年鉴| volume=17 | issue=3 | year=1989 | pages=822–832 | issn=0091-1798 | doi=10.1214/aop/11766991248 | doi access=free}</ref> <ref name=“Meyer2009”/>鞅理论中的方法已成为解决各种概率问题的常用方法。研究马尔可夫过程的技术和理论发展到鞅上。相反地，从鞅理论中也建立了处理Markov过程的方法。<ref name="Meyer2009"/>
Other fields of probability were developed and used to study stochastic processes, with one main approach being the theory of large deviations.<ref name="Meyer2009"/> The theory has many applications in statistical physics, among other fields, and has core ideas going back to at least the 1930s. Later in the 1960s and 1970s fundamental work was done by Alexander Wentzell in the Soviet Union and [[Monroe D. Donsker]] and [[Srinivasa Varadhan]] in the United States of America,<ref name="Ellis1995page98">{{cite journal|last1=Ellis|first1=Richard S.|title=An overview of the theory of large deviations and applications to statistical mechanics|journal=Scandinavian Actuarial Journal|volume=1995|issue=1|year=1995|page=98|issn=0346-1238|doi=10.1080/03461238.1995.10413952}}</ref> which would later result in Varadhan winning the 2007 Abel Prize.<ref name="RaussenSkau2008">{{cite journal|last1=Raussen|first1=Martin|last2=Skau|first2=Christian|title=Interview with Srinivasa Varadhan|journal=Notices of the AMS|volume=55|issue=2|year=2008|pages=238–246}}</ref> In the 1990s and 2000s the theories of [[Schramm–Loewner evolution]]<ref name="HenkelKarevski2012page113">{{cite book|author1=Malte Henkel|author2=Dragi Karevski|title=Conformal Invariance: an Introduction to Loops, Interfaces and Stochastic Loewner Evolution|url=https://books.google.com/books?id=fnCQWd0GEZ8C&pg=PA113|year=2012|publisher=Springer Science & Business Media|isbn=978-3-642-27933-1|page=113}}</ref> and [[rough paths]]<ref name="FrizVictoir2010page571">{{cite book|author1=Peter K. Friz|author2=Nicolas B. Victoir|title=Multidimensional Stochastic Processes as Rough Paths: Theory and Applications|url=https://books.google.com/books?id=CVgwLatxfGsC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48721-4|page=571}}</ref> were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in [[Fields Medal]]s being awarded to [[Wendelin Werner]]<ref name="Werner2004Fields">{{cite journal|title=2006 Fields Medals Awarded|journal=Notices of the AMS|volume=53|issue=9|year=2015|pages=1041–1044}}</ref> in 2008 and to [[Martin Hairer]] in 2014.<ref name="Hairer2004Fields">{{cite journal|last1=Quastel|first1=Jeremy|title=The Work of the 2014 Fields Medalists|journal=Notices of the AMS|volume=62|issue=11|year=2015|pages=1341–1344}}</ref>
Other fields of probability were developed and used to study stochastic processes, with one main approach being the theory of large deviations.<ref name="Meyer2009"/> The theory has many applications in statistical physics, among other fields, and has core ideas going back to at least the 1930s. Later in the 1960s and 1970s fundamental work was done by Alexander Wentzell in the Soviet Union and [[Monroe D. Donsker]] and [[Srinivasa Varadhan]] in the United States of America,<ref name="Ellis1995page98">{{cite journal|last1=Ellis|first1=Richard S.|title=An overview of the theory of large deviations and applications to statistical mechanics|journal=Scandinavian Actuarial Journal|volume=1995|issue=1|year=1995|page=98|issn=0346-1238|doi=10.1080/03461238.1995.10413952}}</ref> which would later result in Varadhan winning the 2007 Abel Prize.<ref name="RaussenSkau2008">{{cite journal|last1=Raussen|first1=Martin|last2=Skau|first2=Christian|title=Interview with Srinivasa Varadhan|journal=Notices of the AMS|volume=55|issue=2|year=2008|pages=238–246}}</ref> In the 1990s and 2000s the theories of [[Schramm–Loewner evolution]]<ref name="HenkelKarevski2012page113">{{cite book|author1=Malte Henkel|author2=Dragi Karevski|title=Conformal Invariance: an Introduction to Loops, Interfaces and Stochastic Loewner Evolution|url=https://books.google.com/books?id=fnCQWd0GEZ8C&pg=PA113|year=2012|publisher=Springer Science & Business Media|isbn=978-3-642-27933-1|page=113}}</ref> and [[rough paths]]<ref name="FrizVictoir2010page571">{{cite book|author1=Peter K. Friz|author2=Nicolas B. Victoir|title=Multidimensional Stochastic Processes as Rough Paths: Theory and Applications|url=https://books.google.com/books?id=CVgwLatxfGsC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48721-4|page=571}}</ref> were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in [[Fields Medal]]s being awarded to [[Wendelin Werner]]<ref name="Werner2004Fields">{{cite journal|title=2006 Fields Medals Awarded|journal=Notices of the AMS|volume=53|issue=9|year=2015|pages=1041–1044}}</ref> in 2008 and to [[Martin Hairer]] in 2014.<ref name="Hairer2004Fields">{{cite journal|last1=Quastel|first1=Jeremy|title=The Work of the 2014 Fields Medalists|journal=Notices of the AMS|volume=62|issue=11|year=2015|pages=1341–1344}}</ref>
概率的其他领域也被发展和用于研究随机过程，其中一个主要方法是大偏差理论。<ref name=“Meyer2009”/>该理论在统计物理等领域有许多应用，其核心思想至少可以追溯到20世纪30年代。20世纪60年代和70年代后期，苏联的亚历山大·温策尔和美利坚合众国的[[Monroe D.Donsker]]和[[Srinivasa Varadhan]]完成了基础工作，<ref name=“Ellis1995page98”>{cite journal | last1=Ellis | first1=Richard S.| title=大理论概述统计力学的偏差与应用| journal=斯堪的纳维亚精算杂志| volume=1995 | issue=1 | year=1995 | page=98 | issn=0346-1238 | doi=10.1080/03461238.1995.10413952}</ref>，这将使瓦拉丹获得2007年阿贝尔奖。<ref name=“RaussenSkau2008”>{citejournal | last 1=Raussen | first1=Martin | last2=Skau | first2=Christian | title=专访Srinivasa Varadhan | journal=注意AMS | volume=55 | issue=2 | year=2008 | pages=238–246}</ref>上世纪90年代和2000年代的理论[[施拉姆–Loewner演化]]]<ref name=“HenkelkeKarevskI2012Page113”>{引用书〈引书| AuthorAuthorAuthorAuthorAuthorAuthorAuthorAuth1=马尔特-汉克尔| author2=德拉吉Karevski | title=共形不变性：循环、接口和随机Loewner演化简介| url=https://books.google.com/books？id=fnCQWd0GEZ8C&pg=PA113 | year=2012 | publisher=Springer Science&Business Media | isbn=978-3-642-27933-1 | page=113}</ref>和[[粗略路径]]<ref name=“frizvictoir201page571”>{cite book | author1=Peter K.Friz | author2=Nicolas B.Victoir | title=多维随机过程作为粗糙路径：理论和应用程序| url=https://books.google.com/books？id=CVgwLatxfGsC | year=2010 | publisher=Cambridge University Press | isbn=978-1-139-48721-4 | page=571}</ref>被引入和发展来研究概率论中的随机过程和其他数学对象，分别在2008年和2014年分别授予[[Wendelin Werner]]<ref name=“Werner2004Fields”>{cite journal | title=2006菲尔兹勋章| journal=AMS通知|卷=53 |问题=9 |年=2015 |页=1041-1044}</ref>和2014年授予[[Martin Haier]]journal | last1=Quastel | first1=Jeremy | title=2014年菲尔兹奖获得者的作品| journal=AMS的通知|卷=62 |问题=11 |年=2015 |页=1341-1344}</ref>
概率的其他领域也被发展和用于研究随机过程，其中一个主要方法是大偏差理论。<ref name=“Meyer2009”/>该理论在统计物理等领域有许多应用，其核心思想至少可以追溯到20世纪30年代。20世纪60年代和70年代后期，苏联的亚历山大·温策尔和美利坚合众国的[[Monroe D.Donsker]]和[[Srinivasa Varadhan]]完成了基础工作，<ref name=“Ellis1995page98”>{cite journal | last1=Ellis | first1=Richard S.| title=大理论概述统计力学的偏差与应用| journal=斯堪的纳维亚精算杂志| volume=1995 | issue=1 | year=1995 | page=98 | issn=0346-1238 | doi=10.1080/03461238.1995.10413952}</ref>，这将使瓦拉丹获得2007年阿贝尔奖。<ref name=“RaussenSkau2008”>{cite journal | last1=Raussen | first1=Martin | last2=Skau | title=采访Srinivasa Varadhan | journal=AMS通知|卷=55 |问题=2 |年=2008 |页=238–246}</ref>上世纪90年代和2000年代的理论[[施拉姆–Loewner演化]]]<ref name=“HenkelkeKarevskI2012Page113”>{引用书〈引书| AuthorAuthorAuthorAuthorAuthorAuthorAuthorAuth1=马尔特-汉克尔| author2=德拉吉Karevski | title=共形不变性：循环、接口和随机Loewner演化简介| url=https://books.google.com/books？id=fnCQWd0GEZ8C&pg=PA113 | year=2012 | publisher=Springer Science&Business Media | isbn=978-3-642-27933-1 | page=113}</ref>和[[粗略路径]]<ref name=“frizvictoir201page571”>{cite book | author1=Peter K.Friz | author2=Nicolas B.Victoir | title=多维随机过程作为粗糙路径：理论和应用程序| url=https://books.google.com/books？id=CVgwLatxfGsC | year=2010 | publisher=Cambridge University Press | isbn=978-1-139-48721-4 | page=571}</ref>被引入和发展来研究概率论中的随机过程和其他数学对象，分别在2008年和2014年分别授予[[Wendelin Werner]]<ref name=“Werner2004Fields”>{cite journal | title=2006菲尔兹勋章| journal=AMS通知|卷=53 |问题=9 |年=2015 |页=1041-1044}</ref>和2014年授予[[Martin Haier]]journal | last1=Quastel | first1=Jeremy | title=2014年菲尔兹奖获得者的作品| journal=AMS的通知|卷=62 |问题=11 |年=2015 |页=1341-1344}</ref>
The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.<ref name="BlathImkeller2011"/><ref name="Applebaum2004page1336"/>
The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.<ref name="BlathImkeller2011"/><ref name="Applebaum2004page1336"/>