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The space‐time distribution,Q  A    (  x  ,  d  t    d  ξ  )say, of Brownian hitting of a bounded Borel set   A   ofR   d    is studied. We derive the asymptotic form of the leading term of the time‐derivativeQ  A    (  x  ,  d  t    d  ξ  )   /  d  t    for each    d  =  2  ,  3  ,  …   , valid uniformly with respect to the starting point   x   of the Brownian motion, which result significantly extends the classical ones forQ  A    (  x  ,  d  t    d  ξ  )itself by Hunt (   d  =  2  )   , Joffe and Spitzer (   d  ⩾  3   ). The results obtained are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joining the origin to a distant point.

The Brownian hitting distributions in space‐time of bounded sets and the expected volume of the Wiener sausage for a Brownian bridge