Feynman–Kac conjecture for the stochastic Bessel operator

Let $\mathcal{B}$ be the stochastic Bessel operator, let $f\in L^2[0,1]$, and let $X$ be the stochastic process formally satisfying

$$dX=\sqrt{2X}\,dB+\left(1+2\sqrt{X/\beta}\,W'(X)\right)\,dt,$$

with functional

$$\Phi(X)=-\frac{a^2}{4}\int_0^t\frac{du}{X_u}-\frac{a}{\sqrt{\beta}}\int_0^1\frac{L_X(x,t)}{\sqrt{x}}\circ dW(x).$$

Let $A$ be the event $\sup_{0\leq u\leq t}X_u\leq 1$. Feynman–Kac conjecture. The stochastic Bessel operator satisfies

$$\exp(-t\mathcal{B})f(x)=\mathbb{E}_x\left[\mathbbm{1}_A\exp\Phi(X)f(X_t)\right].$$

Moreover, if $f_n$ is the orthogonal projection of a fixed $f\in L^2[0,1]$ onto the span of $\{\mathbbm{1}_{[(k-1)/n,k/n)}:k=1,\ldots,n\}$, then $(I-L)^{tn^2}f_n$ should converge in distribution to the right-hand side of this formula, where $L$ is a matrix from the $n\times n$ tridiagonal $\beta$-Laguerre ensemble. This conjecture seeks to identify the stochastic Bessel operator semigroup through a continuum Feynman–Kac representation and to connect it with large powers of the finite-dimensional tridiagonal Laguerre model; the stated convergence is the proposed, rather than established, link.