 Probabilistic representation of the heat equation with time-dependent Dirichlet data

ID: probabilistic-representation-of-the-heat-equation-with-time-dependent-dirichlet-data

For

u_{t} = \frac{1}{2} u_{xx}

(0, 1)

with initial data

g

f_{1}, f_{2}

, stop Brownian motion at its first hit of

0

1

. Applying Itô formula to

u (t - s, B_{s})

yields

u (t, x) = E_{x} [g (B_{t}) 1_{{t < τ_{0} \land τ_{1}}} + f_{1} (t - τ_{0}) 1_{{τ_{0} < t \land τ_{1}}} + f_{2} (t - τ_{1}) 1_{{τ_{1} < t \land τ_{0}}}] .

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