.. _sphx_glr_api_gallery_probability_stochastic_processes:

Random walks and Brownian motion
---------------------------------

Brownian motion paths and Pólya's recurrence theorem for random walks.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Sample paths of standard Brownian motion spread like \sqrt{t}: at time t, W(t) \sim \mathcal N(0, t), so about 95% of paths lie within \pm 2\sqrt t. The quadratic variation \sum (\Delta W)^2 equals t, the signature of a path that is continuous but nowhere differentiable.">

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  .. image:: /api/gallery/probability/stochastic_processes/images/thumb/sphx_glr_plot_01_brownian_motion_thumb.png
    :alt:

  :doc:`/api/gallery/probability/stochastic_processes/plot_01_brownian_motion`

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      <div class="sphx-glr-thumbnail-title">Brownian motion</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="A simple random walk returns to its starting point with probability 1 in one and two dimensions but only about 34% of the time in three. The simulated fraction of walks that have returned by step n keeps climbing toward 1 in 1D (slowly, like 1 - 1/\sqrt{\pi n/2}), crawls upward in 2D, and levels off near 0.3405 in 3D.">

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  .. image:: /api/gallery/probability/stochastic_processes/images/thumb/sphx_glr_plot_02_polya_random_walks_thumb.png
    :alt:

  :doc:`/api/gallery/probability/stochastic_processes/plot_02_polya_random_walks`

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      <div class="sphx-glr-thumbnail-title">Pólya's theorem: a drunk man finds his way home, a drunk bird may not</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/probability/stochastic_processes/plot_01_brownian_motion
   /api/gallery/probability/stochastic_processes/plot_02_polya_random_walks