What is "your peace" in Matthew 10? In Matthew 10, Jesus is instructing his disciples, whom he is sending out to preach to the children of Israel as follows:


  11 Whatever town or village you enter, search there for some worthy person and stay at their house until you leave. 12 As you enter the home, give it your greeting. 13 If the home is deserving, let your peace rest on it; if it is not, let your peace return to you


What does it mean to "let your peace rest on it", and even more curiously, how would one let his peace return to him?
A similar passage in Luke:


  Luke 10:5-6
  5 Into whatever house you enter, first say, ‘Peace to this household.’
  6 If a peaceful person lives there, your peace will rest on him; but if not, it will return to you.


See also Mark 6:8 . Here is the whole passage:


  Matthew 10:12
  12 As you enter a house, wish it peace.
  13 If the house is worthy, let your peace come upon it; if not, let your peace return to you.*
  14 Whoever will not receive you or listen to your words—go outside that house or town and shake the dust from your feet.


The note says: "The greeting of peace is conceived of not merely as a salutation but as an effective word. If it finds no worthy recipient, it will return to the speaker." Consider what happened after St Paul and St Barnabas were driven out of a city, and shook the dusk from their feet:


  Acts 13:51
  51 So they shook the dust from their feet in protest against them and went to Iconium.
  52 The disciples were filled with joy and the holy Spirit.


See also:


  Luke 2:14
  14 “Glory to God in the highest
  and on earth peace to those on whom his favor rests.”


Peace is effectively a blessing, the favor of the Lord. If the blessing finds no worthy recipient, it will fall back on those who gave it. Those who St Paul and St Barnabas gave peace to would not have that peace, so the peace of the Lord fell back on the two men, and they were "filled with joy and the holy Spirit".