Solution to viscous Hamilton-Jacobi equation can be written as fixed points

Question

Can anyone provide me some source to read more about the fact that, solution to the viscous (https://arxiv.org/abs/2002.06674) $$ -\Delta u + H(x,Du) = \alpha_0\quad \text{in}\;\mathbb{T}^n$$ can be written as a fixed points, for every $t>0$, of the operator $\mathcal{S}(t): \mathrm{C}(\mathbb{T}^n)\rightarrow \mathrm{C}(\mathbb{T}^n)$, defined on the space $\mathrm{C}(\mathbb{T}^n)$ of continuous $\mathbb{Z}^n$ periodic function on $\mathbb{R}^n$, as follows: $$ (\mathcal{S}(t)u)(x) = \inf_{v}\mathbb{E} \left[u(Y_x(t))+\int_0^t (L(Y_x(s),-v(s))+\alpha_0)\;ds\right]$$ for $x\in \mathbb{T}^n$ and $t>0$. Here $L$ is the Lagrangian associated to $H$ via the Legendre's transform, $v:[0,\infty)\times \Omega \rightarrow \mathbb{R}^n$ is a control process satisfying suitable measurability conditions and $Y_x$ is the solution of the following Stochastic Differential Equation $$ \begin{cases} dY_x(t) = v(t)dt + \sqrt{2}dW_t\\ Y_x(0) = x \end{cases}$$ where $W_t$ denotes a standard Brownian motion on $\mathbb{R}^n$, defined on a probability space $(\Omega, \mathcal{F},\mathbb{P})$.