The purpose of this paper is to investigate the problem of autocorrelated moving boundaries for a one-dimensional heat equation, where the moving boundary is a fractional Brownian motion \(B^H\) with Hurst parameter \(H >1/2\) and the dependent Volterra-type Dirichlet boundary term \(\xi \) is verified to lie in the exponential Orlicz space \( \mathcal {S}_{\Psi }^{(v)}(\Omega \times (0,T])\) with \(v\in (0,1]\) . This problem is interpreted as a single-layer potential involving stochastic heat kernel \(\begin{aligned} G\left( t-s,x-B^H(s)\right) = \frac{1}{\sqrt{4\pi (t-s)}} \exp \left( -\frac{\left( x-B^H(s)\right) ^2}{4(t-s)}\right) , \end{aligned}\) with \(0<s< t\le T\) , \(x\in D_{B^{H}}^{-} (t) \) and \(D_{B^{H}}^{-} (t) = \{x\in \mathbb {R}: {x<B^H(t)} \}\) . To derive regularity estimates and asymptotic behavior, our approach relies on Dudley’s entropy integral method and a universal bound for the excursion probability \(\begin{aligned} P \left( \sup _{(s,t)\in \mathcal {Q}} \frac{ B^H(t)-B^H(s) }{(t-s)^{\frac{1}{2}+\gamma }}>z \right) ,\quad z> E \left[ \sup _{(s,t)\in \mathcal {Q}} \frac{ B^H(t)-B^H(s)}{(t-s)^{\frac{1}{2}+\gamma }} \right] , \end{aligned}\) where \(\mathcal {Q}=\{(s,t): 0\le s \le t\le T \}\) and \(\gamma \in [0,H-1/2)\) . By potential theoretic arguments, we establish the unique weak solution to the fractional Brownian moving boundary problem in \( L^p(\Omega ; C_{(v)}((0,T];C_b(-\infty ,B^{H}(\cdot ))))\) with \(p>1\) and \(v\in (0,1]\) .
