Hitting Probabilities for a Class of Gaussian Integrators via Second Quantization

In our paper, we consider Gaussian processes in the form of η(t)=∫01A(1[0,t])(s)dw(s),t∈[0,1], where {w(t);t∈[0,1]} is a standard Wiener process in Rd and A is a continuous linear operator on L2([0,1]) into itself. For the domain of D⊂Rd with a C1 smooth boundary, the following probability is considered for x∈DP{∃τ∈[0,1]:x+η(τ)∈∂D}. It is well known that, for the Wiener process, such hitting probabilities satisfy the parabolic boundary-value problem for heat equations. Note that, in general, the process (η) may be non-Markovian and may fail to be a semimartingale. Despite this, η can be considered an application of a second quantization operator to w. Our main contribution is the derivation of a representation for hitting probabilities as a series of Wiener–Itô multiple integrals with the kernel obtained from the Green function on the domain D and tensor powers of operator A.