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This question already has an answer here:

Is it possible to apply a gaussian filter to rasters in ArcMap without applying a calculation or otherwise program it yourself?

marked as duplicate by whuber Dec 13 '16 at 22:11

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You can apply a Gaussian filter using the focal function with the NbrIrregular or NbrWeight arguments to designate an ASCII kernel file representing the desired Gaussian Kernel distribution. You will have to look at the help to see what format the kernel file has to be in as, it is quite specific.

We have a Gaussian Smoothing tool in the Geomorphometry & Gradient Metrics Toolbox that will construct the kernel file using a specified kernel size (window) and sigma (standard deviation). The tool is under "Geomorphometry & Gradient Metrics > Statistics > Gaussian Smoothing"

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ESRI provides two methods in the resample function which apply Gaussian filters to rasters. These are listed as Gauss Blur and Gauss Blur Plus.

See Resample Function in the ESRI help.

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    This is only relevant for resampling and not actual Gaussian decomposition, where the cell size remains the same and a Gaussian distribution is applied to a focal neighborhood. – Jeffrey Evans Dec 13 '16 at 19:55

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