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Although not perfect, using a Paint Bucket tool with the maximum Tolerance of 80 to flood the backgroud area can also achieve the effect of thinning the black area. If the edges of the original letterings are feathered first this method would probably work better.

It can be done without a specific blurring brush. 1. Create a duplicate image and add it to the second layer. 2. Blur the first image using things like Gaussian Blur. 3. Use the eraser with suitable size and hardness to erase the unblurred top layer image. The erased areas will appear blurred.