I am reading the book "stochastic differential equations and diffusion processes" written by Ikeda and Watanabe. In the chapter IV about uniqueness of stochastic differential equation, there is a theorem:
Consider the equation of the time homogeneous Markovian case. If $a(x)=\sigma(x)\sigma(x)^{*}$ is uniformly positive definite, bounded and continuous and $b(x)$ is bounded and Borel measurable, then the uniqueness of solutions holds.
In the proof, it comes out that $$\mu_{\lambda}(\lambda f - \frac{1}{2}\Delta f) = f(x) + \frac{1}{2}\mu_{\lambda}(\sum_{i,j=1}^d c^{ij}(.)\frac{\partial^2 f}{\partial x^i\partial x^j}(.))$$ Letting $f\in C^2_{b}(R^d)=V_{\lambda}h$ for $h \in C_K^{\infty}(R^d)$, it gives $$\mu_{\lambda}(h) = V_{\lambda}h(x) + \frac{1}{2}\mu_{\lambda}(\sum_{i,j=1}^d c^{ij}(.)\frac{\partial^2 V_{\lambda}h}{\partial x^i\partial x^j}(.))$$ My question is how to show that $\mu_{\lambda}(h)=\mu_{\lambda}(\lambda f - \frac{1}{2}\Delta f)$? I hope someone who are specialist in stochastic differential equation here can give me some hints/show me how to arrive the conclusion if anyone reads the book or came across the equation before.
There are some terms which are needed for the above the definition: $$g_t(x) = (2\pi t)^{-d/2} \exp(-\frac{|x|^2}{2t}), t >0, x\in R^{d}$$ $$\nu_{\lambda}(x) = \int_0^{\infty} e^{-\lambda t} g_t(x) dt, \lambda > 0, x\in R^{d}$$ $$V_{\lambda}(x) = \int_{R^d} \nu_{\lambda}(x-y)f(y)dy$$ $$\mu_{\lambda}(h) = E_{x}[\int_0^{\infty} e^{-\lambda t}h(w(t))dt]$$