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class="infinite-pagination-container"> <div class="grey-box w-100 "> <div class="topic-content pt-sm-15"> <div class="mt-60 mt-sm-0"></div>  <span class="marker before-article"></span> <section id="ref" data-level="1"> <section id="ref32789" data-level="2"> <h1 class="h2"><span id="ref407453"></span><a href="//www.rctutku.com/science/Brownian-motion" class="md-crosslink" data-show-preview="true">布朗运动</一个>过程</h1gydF4y2Ba>  <p class="topic-paragraph">最重要的<一个href="//www.rctutku.com/science/stochastic-process" class="md-crosslink autoxref" data-show-preview="true">随机过程</一个>布朗运动或吗<年代pan id="ref407454"></span><a href="//www.rctutku.com/science/Brownian-motion-process" class="md-crosslink">维纳过程</一个>。这是第一次讨论<年代pan id="ref407455"></span>路易Bachelier(1900),建模感兴趣的金融市场价格的波动,以及<年代pan id="ref407456"></span><a href="//www.rctutku.com/biography/Albert-Einstein" class="md-crosslink" data-show-preview="true">阿尔伯特·爱因斯坦</一个>(1905),给了谁<一个href="//www.rctutku.com/science/mathematical-model" class="md-crosslink autoxref" data-show-preview="true">数学模型</一个>不规则的运动<年代pan id="ref407457"></span><a href="//www.rctutku.com/science/colloid" class="md-crosslink" data-show-preview="true">胶体</一个>粒子由苏格兰植物学家首次发现<一个href="//www.rctutku.com/biography/Robert-Brown-Scottish-botanist" class="md-crosslink" data-show-preview="true">罗伯特•布朗</一个>在1827年。第一个数学上严格的对待这个模型是由维纳(1923)。爱因斯坦的结果导致早期,大量物质的分子理论的确认法国物理学家<年代pan id="ref407458"></span><a href="//www.rctutku.com/biography/Jean-Perrin" class="md-crosslink" data-show-preview="true">琼佩兰</一个>的实验来确定<一个href="//www.rctutku.com/science/Avogadros-law" class="md-crosslink" data-show-preview="true">阿伏伽德罗常数</一个>,佩兰被授予<一个href="//www.rctutku.com/topic/Nobel-Prize" class="md-crosslink autoxref" data-show-preview="true">诺贝尔奖</一个>在1926年。今天有些不同模型对物理布朗运动被认为是更多的<一个class="md-dictionary-link md-dictionary-tt-off eb" data-term="appropriate" href="//www.rctutku.com/dictionary/appropriate" data-type="EB">适当的</一个>比爱因斯坦,但原始数学模型继续发挥核心作用的随机过程的理论和应用。</gydF4y2Bap>  <span class="marker p1"></span>  <span class="marker AM1 am-inline"></span>  <span class="marker MOD1 mod-inline"></span> <p class="topic-paragraph">让<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)表示位移(在一个<一个href="//www.rctutku.com/science/dimension-geometry" class="md-crosslink autoxref" data-show-preview="true">维</一个>为简单起见)的胶体悬浮粒子,由众多的冲击更小分子介质的暂停。这个位移将获得的限制<一个href="//www.rctutku.com/science/random-walk" class="md-crosslink" data-show-preview="true">随机漫步</一个>发生在离散时间步骤变得无限大的数量和每个步骤无限小的大小。假设有时<gydF4y2Baem>k</gydF4y2Baem>δ,<gydF4y2Baem>k</gydF4y2Baem>= 1,2,…,胶体粒子是流离失所的距离<gydF4y2Baem>h</gydF4y2Baem><em>X</gydF4y2Baem><sub><em>k</gydF4y2Baem></sub>,在那里<gydF4y2Baem>X</gydF4y2Baem><sub>1</年代ub>,<gydF4y2Baem>X</gydF4y2Baem><sub>2</年代ub>,…+ 1或−1照扔一个公平的结果<一个href="//www.rctutku.com/topic/coin" class="md-crosslink autoxref" data-show-preview="true">硬币</一个>是正面还是反面。按时间<gydF4y2Baem>t</gydF4y2Baem>粒子了<gydF4y2Baem>米</gydF4y2Baem>步骤,<gydF4y2Baem>米</gydF4y2Baem>是最大的整数≤<gydF4y2Baem>t</gydF4y2Baem>/δ,它<一个class="md-dictionary-link md-dictionary-tt-off eb" data-term="displacement" href="//www.rctutku.com/dictionary/displacement" data-type="EB">位移</一个>从原来的位置<gydF4y2Baem>B</gydF4y2Baem><sub><em>米</gydF4y2Baem></sub>(<gydF4y2Baem>t</gydF4y2Baem>)=<gydF4y2Baem>h</gydF4y2Baem>(<gydF4y2Baem>X</gydF4y2Baem><sub>1</年代ub>+⋯+<gydF4y2Baem>X</gydF4y2Baem><sub><em>米</gydF4y2Baem></sub>)。的<一个href="//www.rctutku.com/topic/expected-value" class="md-crosslink autoxref" data-show-preview="true">期望值</一个>的<gydF4y2Baem>B</gydF4y2Baem><sub><em>米</gydF4y2Baem></sub>(<gydF4y2Baem>t</gydF4y2Baem>)= 0,和它的<一个href="//www.rctutku.com/topic/variance" class="md-crosslink autoxref" data-show-preview="true">方差</一个>是<gydF4y2Baem>h</gydF4y2Baem><sup>2</年代up><em>米</gydF4y2Baem>,或者约<gydF4y2Baem>h</gydF4y2Baem><sup>2</年代up><em>t</gydF4y2Baem>/δ。现在假设δ→0,在同一时间<gydF4y2Baem>h</gydF4y2Baem>→0这样的方差<gydF4y2Baem>B</gydF4y2Baem><sub><em>米</gydF4y2Baem></sub>(1)收敛于一些积极的<一个href="//www.rctutku.com/topic/constant" class="md-crosslink autoxref" data-show-preview="true">常数</一个>,σ<年代up>2</年代up>。这意味着<gydF4y2Baem>米</gydF4y2Baem>变得无限大,<gydF4y2Baem>h</gydF4y2Baem>大约是σ(<gydF4y2Baem>t</gydF4y2Baem>/<gydF4y2Baem>米</gydF4y2Baem>)<年代up>1/2</年代up>。它遵循的<一个href="//www.rctutku.com/science/central-limit-theorem" class="md-crosslink autoxref" data-show-preview="true">中心极限定理</一个>(方程<一个href="#ref-14392">12</一个>),林<gydF4y2Baem>P</gydF4y2Baem>{<gydF4y2Baem>B</gydF4y2Baem><sub><em>米</gydF4y2Baem></sub>(<gydF4y2Baem>t</gydF4y2Baem>)≤<gydF4y2Baem>x</gydF4y2Baem>}=<gydF4y2Baem>G</gydF4y2Baem>(<gydF4y2Baem>x</gydF4y2Baem>/σ<gydF4y2Baem>t</gydF4y2Baem><sup>1/2</年代up>),<gydF4y2Baem>G</gydF4y2Baem>(<gydF4y2Baem>x</gydF4y2Baem>)是标准正态<一个href="//www.rctutku.com/science/distribution-function" class="md-crosslink autoxref" data-show-preview="true">累积分布函数</一个>定义下面<一个href="//www.rctutku.com/science/equation" class="md-crosslink autoxref" data-show-preview="true">方程</一个>(12)。布朗运动过程<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)可以被定义为限制在一定技术意义上的<gydF4y2Baem>B</gydF4y2Baem><sub><em>米</gydF4y2Baem></sub>(<gydF4y2Baem>t</gydF4y2Baem>),δ→0<gydF4y2Baem>h</gydF4y2Baem>→0和<gydF4y2Baem>h</gydF4y2Baem><sup>2</年代up>/δ→σ<年代up>2</年代up>。</gydF4y2Bap>  <span class="marker p2"></span>  <span class="marker AM2 am-inline"></span>  <span class="marker MOD2 mod-inline"></span> <p class="topic-paragraph">这个过程<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)有许多其他属性,原则上都是继承了近似随机游走<gydF4y2Baem>B</gydF4y2Baem><sub><em>米</gydF4y2Baem></sub>(<gydF4y2Baem>t</gydF4y2Baem>)。例如,如果(<gydF4y2Baem>年代</gydF4y2Baem><sub>1</年代ub>,<gydF4y2Baem>t</gydF4y2Baem><sub>1</年代ub>)和(<gydF4y2Baem>年代</gydF4y2Baem><sub>2</年代ub>,<gydF4y2Baem>t</gydF4y2Baem><sub>2</年代ub>)不相交的间隔,增量<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem><sub>1</年代ub>)−<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>年代</gydF4y2Baem><sub>1</年代ub>),<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem><sub>2</年代ub>)−<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>年代</gydF4y2Baem><sub>2</年代ub>)是正态分布的独立随机变量的期望0和方差σ<年代up>2</年代up>(<gydF4y2Baem>t</gydF4y2Baem><sub>1</年代ub>−<gydF4y2Baem>年代</gydF4y2Baem><sub>1</年代ub>)和σ<年代up>2</年代up>(<gydF4y2Baem>t</gydF4y2Baem><sub>2</年代ub>−<gydF4y2Baem>年代</gydF4y2Baem><sub>2</年代ub>),分别。</gydF4y2Bap>  <span class="marker p3"></span>  <span class="marker AM3 am-inline"></span>  <span class="marker MOD3 mod-inline"></span> <p class="topic-paragraph">爱因斯坦采取了不同的方法和推导过程的各种属性<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>通过显示<一个href="//www.rctutku.com/science/density-function" class="md-crosslink autoxref" data-show-preview="true">概率密度函数</一个>,<gydF4y2Baem>g</gydF4y2Baem>(<gydF4y2Baem>x</gydF4y2Baem>,<gydF4y2Baem>t</gydF4y2Baem>),满足<一个class="md-dictionary-link md-dictionary-tt-off mw" data-term="diffusion" href="https://www.merriam-webster.com/dictionary/diffusion" data-type="MW">扩散</一个>方程∂<gydF4y2Baem>g</gydF4y2Baem>/∂<gydF4y2Baem>t</gydF4y2Baem>=<gydF4y2Baem>D</gydF4y2Baem>∂<年代up>2</年代up><em>g</gydF4y2Baem>/∂<gydF4y2Baem>x</gydF4y2Baem><sup>2</年代up>,在那里<gydF4y2Baem>D</gydF4y2Baem>=σ<年代up>2</年代up>/ 2。重要的<一个class="md-dictionary-link md-dictionary-tt-off mw" data-term="implication" href="https://www.merriam-webster.com/dictionary/implication" data-type="MW">含义</一个>为后续实验研究爱因斯坦的理论是,他确定了扩散常数<gydF4y2Baem>D</gydF4y2Baem>的某些测量粒子的属性(半径)和介质(它的粘度和温度),它允许一个做出预测,因此确认或拒绝的假设存在看不见的分子认为是不规则的布朗运动的原因。因为美丽的混合的数学和物理推理,继任者爱因斯坦模型的简要总结如下所示。</gydF4y2Bap>  <span class="marker p4"></span>  <span class="marker AM4 am-inline"></span>  <span class="marker MOD4 mod-inline"></span> <p class="topic-paragraph">与泊松过程不同,是不可能“画”的照片粒子的路径进行数学布朗运动。<年代pan id="ref407459"></span><a href="//www.rctutku.com/biography/Norbert-Wiener" class="md-crosslink" data-show-preview="true">维纳</一个>(1923)表明,功能<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)<一个class="md-dictionary-link md-dictionary-tt-off eb" data-term="continuous" href="//www.rctutku.com/dictionary/continuous" data-type="EB">连续</一个>作为一个希望,但无处可微。因此,粒子经历数学布朗运动并没有一个明确的速度,和曲线<gydF4y2Baem>y</gydF4y2Baem>=<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)没有一个定义良好的切线的任何值<gydF4y2Baem>t</gydF4y2Baem>。明白为什么会这样,记得<一个href="//www.rctutku.com/science/derivative-mathematics" class="md-crosslink autoxref" data-show-preview="true">导数</一个>的<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>),如果它存在,是极限<gydF4y2Baem>h</gydF4y2Baem>→0的<一个href="//www.rctutku.com/science/ratio" class="md-crosslink autoxref" data-show-preview="true">比</一个>(<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>+<gydF4y2Baem>h</gydF4y2Baem>)−<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)/<gydF4y2Baem>h</gydF4y2Baem>。自<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>+<gydF4y2Baem>h</gydF4y2Baem>)−<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)是正态分布均值为0,<一个href="//www.rctutku.com/topic/standard-deviation" class="md-crosslink autoxref" data-show-preview="true">标准偏差</一个><em>h</gydF4y2Baem><sup>1/2</年代up>σ,在非常粗糙<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>+<gydF4y2Baem>h</gydF4y2Baem>)−<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)可以将等于(正面或负面)的倍数<gydF4y2Baem>h</gydF4y2Baem><sup>1/2</年代up>。但当<gydF4y2Baem>h</gydF4y2Baem>→0时,<gydF4y2Baem>h</gydF4y2Baem><sup>1/2</年代up>/<gydF4y2Baem>h</gydF4y2Baem>= 1 /<gydF4y2Baem>h</gydF4y2Baem><sup>1/2</年代up>是<一个class="md-dictionary-link md-dictionary-tt-off mw" data-term="infinite" href="https://www.merriam-webster.com/dictionary/infinite" data-type="MW">无限</一个>。一个相关的事实说明了极端的不规则性<gydF4y2Baem>B</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>),在每一个区间的时候,无论多小,粒子经历数学布朗运动传播无限的距离。尽管这些属性与常识的功能,明确写下来确实是很困难的一个例子连续,nowhere-differentiable函数是典型的大型一类随机过程,调用<年代pan id="ref407460"></span>布朗运动的扩散过程中,最突出的成员。尤其值得注意的贡献的数学理论是由布朗运动和扩散过程<一个href="//www.rctutku.com/biography/Paul-Levy" class="md-crosslink" data-show-preview="true">保罗利维</一个>在1930年- 60和威廉樵夫。</gydF4y2Bap>  <span class="marker p5"></span>  <span class="marker AM5 am-inline"></span>  <span class="marker MOD5 mod-inline"></span> <p class="topic-paragraph">更复杂的物理布朗运动的描述可以建立在一个简单的应用程序<年代pan id="ref407461"></span><a href="//www.rctutku.com/science/Newtons-laws-of-motion" class="md-crosslink" data-show-preview="true">牛顿第二定律</一个>:<gydF4y2Baem>F</gydF4y2Baem>=<gydF4y2Baem>米</gydF4y2Baem><em>一个</gydF4y2Baem>。让<gydF4y2Baem>V</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)表示质量胶体粒子的速度<gydF4y2Baem>米</gydF4y2Baem>。假设<年代pan class="md-formula"><span id="ref-14388"></span><img src="https://cdn.britannica.com/88/14388-004-4B538268/Equation.jpg" alt="方程。gydF4y2Ba" class="inline-image-baseline"></span></p>  <span class="marker p6"></span>  <span class="marker AM6 am-inline"></span>  <span class="marker MOD6 mod-inline"></span> <p class="topic-paragraph">的数量<gydF4y2Baem>f</gydF4y2Baem>阻碍粒子的运动是由于<一个href="//www.rctutku.com/science/friction" class="md-crosslink autoxref" data-show-preview="true">摩擦</一个>造成周围的介质。这个词<gydF4y2Baem>d</gydF4y2Baem><em>一个</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)的贡献是非常频繁的碰撞的粒子与看不见的分子介质。假设<gydF4y2Baem>f</gydF4y2Baem>由经典吗<一个href="//www.rctutku.com/science/fluid-mechanics" class="md-crosslink autoxref" data-show-preview="true">流体力学</一个>,在周围介质中分子编造很多所以中可以被认为是光滑的和小的<一个class="md-dictionary-link md-dictionary-tt-off mw" data-term="homogeneous" href="https://www.merriam-webster.com/dictionary/homogeneous" data-type="MW">均匀</一个>。然后<年代pan id="ref407462"></span><a href="//www.rctutku.com/science/Stokess-law" class="md-crosslink" data-show-preview="true">斯托克斯定律</一个>球形颗粒的气体,<gydF4y2Baem>f</gydF4y2Baem>= 6π<gydF4y2Baem>一个</gydF4y2Baem>η,<gydF4y2Baem>一个</gydF4y2Baem>是粒子的半径和η介质的粘度系数。<一个class="md-dictionary-link md-dictionary-tt-off mw" data-term="Hypotheses" href="https://www.merriam-webster.com/dictionary/Hypotheses" data-type="MW">假设</一个>有关<gydF4y2Baem>一个</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)不太具体,因为分子周围的介质不能直接观测到。例如,假设<gydF4y2Baem>t</gydF4y2Baem>≠<gydF4y2Baem>年代</gydF4y2Baem>,<一个href="//www.rctutku.com/science/infinitesimal" class="md-crosslink autoxref" data-show-preview="true">无穷小</一个>随机增量<gydF4y2Baem>d</gydF4y2Baem><em>一个</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)=<gydF4y2Baem>一个</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>+<gydF4y2Baem>d</gydF4y2Baem><em>t</gydF4y2Baem>)−<gydF4y2Baem>一个</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>),<gydF4y2Baem>一个</gydF4y2Baem>(<gydF4y2Baem>年代</gydF4y2Baem>+<gydF4y2Baem>d</gydF4y2Baem><em>年代</gydF4y2Baem>)−<gydF4y2Baem>一个</gydF4y2Baem>(<gydF4y2Baem>年代</gydF4y2Baem>)碰撞引起的粒子与周围介质的分子是独立随机变量分布均值为0,方差σ不明<年代up>2</年代up><em>d</gydF4y2Baem><em>t</gydF4y2Baem>和σ<年代up>2</年代up><em>d</gydF4y2Baem><em>年代</gydF4y2Baem>这<gydF4y2Baem>d</gydF4y2Baem><em>一个</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)是独立的<gydF4y2Baem>d</gydF4y2Baem><em>V</gydF4y2Baem>(<gydF4y2Baem>年代</gydF4y2Baem>)<gydF4y2Baem>年代</gydF4y2Baem><<gydF4y2Baem>t</gydF4y2Baem>。</gydF4y2Bap>  <span class="marker p7"></span>  <span class="marker AM7 am-inline"></span>  <span class="marker MOD7 mod-inline"></span> <p class="topic-paragraph">的<一个href="//www.rctutku.com/science/differential-equation" class="md-crosslink autoxref" data-show-preview="true">微分方程</一个>(<一个href="#ref-14388">18</一个>)解决方案<年代pan class="md-formula"><span id="ref-14387"></span><img src="https://cdn.britannica.com/87/14387-004-89079E7F/Equation.jpg" alt="方程。gydF4y2Ba" class="inline-image-baseline"></span>β=<gydF4y2Baem>f</gydF4y2Baem>/<gydF4y2Baem>米</gydF4y2Baem>。从这个方程和假设的属性<gydF4y2Baem>一个</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>),它遵循<gydF4y2Baem>E</gydF4y2Baem>(<gydF4y2Baem>V</gydF4y2Baem><sup>2</年代up>(<gydF4y2Baem>t</gydF4y2Baem>)→σ<年代up>2</年代up>/ (2<gydF4y2Baem>米</gydF4y2Baem><em>f</gydF4y2Baem>),<gydF4y2Baem>t</gydF4y2Baem>→∞。现在,假设按照的原则<一个href="//www.rctutku.com/science/equipartition-of-energy" class="md-crosslink autoxref" data-show-preview="true">能量均分</一个>稳态平均<一个href="//www.rctutku.com/science/kinetic-energy" class="md-crosslink autoxref" data-show-preview="true">动能</一个>的粒子,<gydF4y2Baem>米</gydF4y2Baem>lim<年代ub><em>t</gydF4y2Baem>→∞</年代ub><em>E</gydF4y2Baem>(<gydF4y2Baem>V</gydF4y2Baem><sup>2</年代up>(<gydF4y2Baem>t</gydF4y2Baem>)/ 2,等于平均水平<一个class="md-dictionary-link md-dictionary-tt-off eb" data-term="kinetic" href="//www.rctutku.com/dictionary/kinetic" data-type="EB">动能</一个>分子的能量的媒介。根据<年代pan id="ref407463"></span><a href="//www.rctutku.com/science/kinetic-theory-of-gases" class="md-crosslink" data-show-preview="true">理想气体分子运动论</一个>,这是<gydF4y2Baem>R</gydF4y2Baem><em>T</gydF4y2Baem>/ 2<gydF4y2Baem>N</gydF4y2Baem>,在那里<gydF4y2Baem>R</gydF4y2Baem>是<一个href="//www.rctutku.com/science/ideal-gas" class="md-crosslink autoxref" data-show-preview="true">理想气体</一个>常数,<gydF4y2Baem>T</gydF4y2Baem>在开尔文气体的温度,然后呢<gydF4y2Baem>N</gydF4y2Baem>是<一个href="//www.rctutku.com/science/Avogadros-number" class="md-crosslink autoxref" data-show-preview="true">阿伏伽德罗常数</一个>,分子的数目<一个href="//www.rctutku.com/science/mole-chemistry" class="md-crosslink autoxref" data-show-preview="true">克分子量</一个>的气体。由此可见,未知的σ的价值<年代up>2</年代up>可以确定:σ<年代up>2</年代up>= 2<gydF4y2Baem>R</gydF4y2Baem><em>T</gydF4y2Baem><em>f</gydF4y2Baem>/<gydF4y2Baem>N</gydF4y2Baem>。</gydF4y2Bap>  <span class="marker p8"></span>  <span class="marker AM8 am-inline"></span>  <span class="marker MOD8 mod-inline"></span> <p class="topic-paragraph">如果一个还假定函数<gydF4y2Baem>V</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)是连续的,这当然是合理的从物理因素,它遵循数学<一个href="//www.rctutku.com/science/analysis-mathematics" class="md-crosslink autoxref" data-show-preview="true">分析</一个>那<gydF4y2Baem>一个</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)是一个布朗运动过程如上定义。这一结论提出了质疑的意思初始方程(<一个href="#ref-14388">18</一个>),因为数学布朗运动<gydF4y2Baem>d</gydF4y2Baem><em>一个</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)不存在通常意义上的导数。一些额外的数学分析表明,随机微分方程(<一个href="#ref-14388">18</一个>)及其解决方案方程(<一个href="#ref-14387">19</一个>)有一个精确的数学解释。这个过程<gydF4y2Baem>V</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)称为<年代pan id="ref407464"></span>Ornstein-Uhlenbeck过程,在物理学家伦纳德·所罗门奥恩斯坦和之后<一个href="//www.rctutku.com/biography/George-Eugene-Uhlenbeck" class="md-crosslink" data-show-preview="true">乔治·尤金·乌伦贝克</一个>。这些尝试的逻辑结果<一个class="md-dictionary-link md-dictionary-tt-off mw" data-term="differentiate" href="https://www.merriam-webster.com/dictionary/differentiate" data-type="MW">区分</一个>和<一个class="md-dictionary-link md-dictionary-tt-off mw" data-term="integrate" href="https://www.merriam-webster.com/dictionary/integrate" data-type="MW">集成</一个>对布朗运动过程<年代pan id="ref407465"></span>伊藤(以日本数学家ItōKiyosi)随机<一个href="//www.rctutku.com/science/calculus-mathematics" class="md-crosslink autoxref" data-show-preview="true">微积分</一个>,它扮演着一个重要的角色在现代随机过程的理论。</gydF4y2Bap>  <span class="marker p9"></span>  <span class="marker AM9 am-inline"></span>  <span class="marker MOD9 mod-inline"></span> <p class="topic-paragraph">位移时<gydF4y2Baem>t</gydF4y2Baem>粒子的速度是由方程(<一个href="#ref-14387">19</一个>)是<年代pan class="md-formula"><span id="ref-14365"></span><img src="https://cdn.britannica.com/65/14365-004-C97F59BD/Equation.jpg" alt="方程。gydF4y2Ba" class="inline-image-baseline"></span></p>  <span class="marker p10"></span>  <span class="marker AM10 am-inline"></span>  <span class="marker MOD10 mod-inline"></span> <p class="topic-paragraph">为<gydF4y2Baem>t</gydF4y2Baem>大与β相比,第一和第三项与第二个表达式相比非常小。因此,<gydF4y2Baem>X</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)−<gydF4y2Baem>X</gydF4y2Baem>(0)约等于<gydF4y2Baem>一个</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)/<gydF4y2Baem>f</gydF4y2Baem>平均平方位移,<gydF4y2Baem>E</gydF4y2Baem>{(<gydF4y2Baem>X</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)−<gydF4y2Baem>X</gydF4y2Baem>(0))<年代up>2</年代up>},大约是σ<年代up>2</年代up>/<gydF4y2Baem>f</gydF4y2Baem><sup>2</年代up>=<gydF4y2Baem>R</gydF4y2Baem><em>T</gydF4y2Baem>/(3π<gydF4y2Baem>一个</gydF4y2Baem>η<gydF4y2Baem>N</gydF4y2Baem>)。这些最终结论符合爱因斯坦的模型,虽然他们在这里出现作为一个近似模型得到方程(<一个href="#ref-14387">19</一个>)。因为它主要是观测结果的结论,基本上是没有新的实验<一个class="md-dictionary-link md-dictionary-tt-off mw" data-term="implications" href="https://www.merriam-webster.com/dictionary/implications" data-type="MW">影响</一个>。然而,分析直接从牛顿第二定律,收益率过程有一个明确的速度在每一个点,似乎更令人满意的理论比爱因斯坦的原始模型。</gydF4y2Bap>  <span class="marker p11"></span>  <span class="marker AM11 am-inline"></span>  <span class="marker MOD11 mod-inline"></span> </section> </section>  <span class="marker h10"></span> <section id="ref32790" data-level="1"> <h2 class="h1">随机过程</h2gydF4y2Ba> <p class="topic-paragraph">随机过程是一个随机变量的家庭<gydF4y2Baem>X</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)被<一个class="md-dictionary-link md-dictionary-tt-off mw" data-term="parameter" href="https://www.merriam-webster.com/dictionary/parameter" data-type="MW">参数</一个><em>t</gydF4y2Baem>,这通常需要在离散值<一个href="//www.rctutku.com/topic/set-mathematics-and-logic" class="md-crosslink autoxref" data-show-preview="true">集</一个>Τ={0,1,2,…}或连续组Τ= [0,+∞)。在许多情况下<gydF4y2Baem>t</gydF4y2Baem>表示时间,<gydF4y2Baem>X</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)是一个<一个href="//www.rctutku.com/topic/random-variable" class="md-crosslink autoxref" data-show-preview="true">随机变量</一个>观察到时间<gydF4y2Baem>t</gydF4y2Baem>。的例子是泊松过程、布朗运动过程,Ornstein-Uhlenbeck前一节中描述的过程。视为一个整体,随机变量的家庭{<gydF4y2Baem>X</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>),<gydF4y2Baem>t</gydF4y2Baem>∊Τ}<一个class="md-dictionary-link md-dictionary-tt-off mw" data-term="constitutes" href="https://www.merriam-webster.com/dictionary/constitutes" data-type="MW">构成</一个>一个“随机函数”。</gydF4y2Bap>  <span class="marker p12"></span>  <span class="marker AM12 am-inline"></span>  <span class="marker MOD12 mod-inline"></span> <section id="ref32791" data-level="2"> <h2 class="h2"><span id="ref407466"></span>固定的流程</h2gydF4y2Ba> <p class="topic-paragraph">随机过程的数学理论试图定义类的流程可以开发一个统一的理论。最重要的类是固定流程和马尔可夫过程。一个随机过程是平稳,如果呼吁所有<gydF4y2Baem>n</gydF4y2Baem>,<gydF4y2Baem>t</gydF4y2Baem><sub>1</年代ub><<gydF4y2Baem>t</gydF4y2Baem><sub>2</年代ub><⋯<<gydF4y2Baem>t</gydF4y2Baem><sub><em>n</gydF4y2Baem></sub>,<gydF4y2Baem>h</gydF4y2Baem>> 0,的联合分布<gydF4y2Baem>X</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem><sub>1</年代ub>+<gydF4y2Baem>h</gydF4y2Baem>),…<gydF4y2Baem>X</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem><sub><em>n</gydF4y2Baem></sub>+<gydF4y2Baem>h</gydF4y2Baem>)不依赖<gydF4y2Baem>h</gydF4y2Baem>。这意味着,实际上没有时间轴上的起源;平稳过程的随机行为是相同的,无论过程是观察。一个独立同分布的随机变量序列是平稳过程的一个例子。定义一个不同的例子如下:<gydF4y2Baem>U</gydF4y2Baem>(0)[0,1]上均匀分布;为每一个<gydF4y2Baem>t</gydF4y2Baem>= 1,2,…<gydF4y2Baem>U</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)= 2<gydF4y2Baem>U</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>−1)如果<gydF4y2Baem>U</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>−1)≤1/2,<gydF4y2Baem>U</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)= 2<gydF4y2Baem>U</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>−1)−1<gydF4y2Baem>U</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>−1)> 1/2。的<一个class="md-dictionary-link md-dictionary-tt-off eb" data-term="marginal" href="//www.rctutku.com/dictionary/marginal" data-type="EB">边际</一个>分布的<gydF4y2Baem>U</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>),<gydF4y2Baem>t</gydF4y2Baem>= 0,1,…是均匀分布在[0,1],但是,与独立同分布的随机变量的情况下,可以从知识预测整个序列<gydF4y2Baem>U</gydF4y2Baem>(0),平稳过程的第三个例子<年代pan class="md-formula"><span id="ref-14362"></span><img src="https://cdn.britannica.com/62/14362-004-E51EF9D1/Equation.jpg" alt="方程。gydF4y2Ba" class="inline-image-baseline"></span>在哪里<gydF4y2Baem>Y</gydF4y2Baem>年代和<gydF4y2Baem>Z</gydF4y2Baem>年代是独立正态分布随机变量平均值为0和单位方差,和<gydF4y2Baem>c</gydF4y2Baem>年代和θs是常数。这种可用于建模的过程大约季节性或周期性现象。</gydF4y2Bap>  <span class="marker p13"></span>  <span class="marker AM13 am-inline"></span>  <span class="marker MOD13 mod-inline"></span> <p class="topic-paragraph">一个了不起的泛化的强劲<一个href="//www.rctutku.com/science/law-of-large-numbers" class="md-crosslink autoxref" data-show-preview="true">大数定律</一个>是<年代pan id="ref407467"></span><a href="//www.rctutku.com/topic/ergodic-theorem" class="md-crosslink">遍历性定理</一个>:如果<gydF4y2Baem>X</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>),<gydF4y2Baem>t</gydF4y2Baem>= 0,1,…<一个class="md-dictionary-link md-dictionary-tt-off eb" data-term="discrete" href="//www.rctutku.com/dictionary/discrete" data-type="EB">离散</一个>案例或0≤<gydF4y2Baem>t</gydF4y2Baem><∞连续情况下,是一个固定的过程,这样<gydF4y2Baem>E</gydF4y2Baem>(<gydF4y2Baem>X</gydF4y2Baem>(0))是有限的,那么<一个href="//www.rctutku.com/science/probability" class="md-crosslink autoxref" data-show-preview="true">概率</一个>1的平均<年代pan class="md-formula"><span id="ref-14361"></span><img src="https://cdn.britannica.com/61/14361-004-E75A732F/Elements-theorem.jpg" alt="的元素遍历定理。gydF4y2Ba" class="inline-image-baseline"></span>如果<gydF4y2Baem>t</gydF4y2Baem>是连续的,收敛到一个极限<gydF4y2Baem>年代</gydF4y2Baem>→∞。在特殊的情况下<gydF4y2Baem>t</gydF4y2Baem>是离散的,<gydF4y2Baem>X</gydF4y2Baem>年代是独立同分布,强大数定律也适用,表明限制必须相等<gydF4y2Baem>E</gydF4y2Baem>{<gydF4y2Baem>X</gydF4y2Baem>(0)}。然而,这个例子<gydF4y2Baem>X</gydF4y2Baem>(0)是一个任意随机变量<gydF4y2Baem>X</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)≡<gydF4y2Baem>X</gydF4y2Baem>(0)<gydF4y2Baem>t</gydF4y2Baem>> 0显示,一般不能如此。极限是平等的<gydF4y2Baem>E</gydF4y2Baem>{<gydF4y2Baem>X</gydF4y2Baem>(0)}一个额外的技术性假设下的效果没有状态空间的子集,在概率严格在0和1之间,这个过程可以卡住,从不逃避。这种假设是不满足的例子<gydF4y2Baem>X</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>)≡<gydF4y2Baem>X</gydF4y2Baem>(0)<gydF4y2Baem>t</gydF4y2Baem>卡住,立刻在其初始值。它是满意的序列<gydF4y2Baem>U</gydF4y2Baem>(<gydF4y2Baem>t</gydF4y2Baem>由遍历定理)上面定义的,所以这些变量的平均收敛于1/2的概率1。遍历定理首次推测由美国化学家<年代pan id="ref407468"></span><a href="//www.rctutku.com/biography/J-Willard-Gibbs" class="md-crosslink" data-show-preview="true">威拉德·j·吉布斯</一个>在1900年代早期<一个class="md-dictionary-link md-dictionary-tt-off mw" data-term="context" href="https://www.merriam-webster.com/dictionary/context" data-type="MW">上下文</一个>的<一个href="//www.rctutku.com/science/statistical-mechanics" class="md-crosslink autoxref" data-show-preview="true">统计力学</一个>并证明了纠正,抽象的配方由美国数学家<一个href="//www.rctutku.com/biography/George-David-Birkhoff" class="md-crosslink" data-show-preview="true">乔治·大卫·比尔科夫</一个>在1931年。</gydF4y2Bap>  <span class="marker p14"></span>  <span class="marker AM14 am-inline"></span>  <span class="marker MOD14 mod-inline"></span> </section> </section>  <span class="marker end-of-content"></span>  <span class="marker after-article"></span> </div> </div> </div> </div> <aside class="col-md-da-320"></aside> </div> </div> </div> </div> </article> </div> </div> </div> </div> </main> <div id="md-footer"></div> <noscript> <iframe src="https://www.rctutku.com/ns.html?id=GTM-5W6NC8" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript> <div 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