Existence of solutions of $u_t = Lu$ for pseudo-differential operators $L$? I'm wondering if it is known if existence results for the equation $$\left\{\begin{array}{ll} 
\partial_t u = Lu & \text{on }(0,T)\times\mathbb{R}^n\\
u(0,-) = \varphi & \text{on }\mathbb{R}^n\end{array}\right.$$ for $\varphi\in C^\infty_c(\mathbb{R}^n)$ are known. In particular, I'm interested in the case where $L$ is of the form $$Lf(x) = \frac{1}{2}\sum_{i,j} A_{i,j} \partial_i\partial_j f(x) + \sum_j b_j\partial_j f(x)+\int_{\mathbb{R}^n}f(x+y)-f(x)-\frac{(y,\nabla f(x))}{1+|y|^2}\,M(dy),$$ where $M$ is a Levy measure on $\mathbb{R}^n$.
The hypothesis that a solution exists for all $T>0$ and $\varphi\in C^\infty_c$ is part of Theorem 2.2.6 in Stroock's Markov Processes from K. Ito's Perspective, and as far as I can tell it is part of hypotheses of subsequent results. 
Any references for general or specific cases would be greatly appreciated.
 A: In the Raymond's book, Here, you will find a clear demonstration of solutions to abstract problems $u_t-iLu=0$. His demonstration was taken from the Hörmander's book, Chapter XXIII and is based transposition method. In the Joshi's notes there is a proof too, in the page 38.
A: The solution to this equation is rather simple in spirit as the operator is linear. It is
$$u(t, x)=\exp\Big(tL\Big)\varphi(x)$$
The hard bit is to compute the exponential of $L$)) 
To simplify the calculation we consider a Fourier image of $u(t, x)$ 
$$u(t, x)=\int_{\mathbb{R}^{n}}\hat{u}(t, q)e^{iqx}\frac{d^{n}q}{(2\pi)^{n}}$$
Then
$$Lu(t, x)=\int_{\mathbb{R}^{n}}\Big(-\frac{1}{2}\sum_{i,j} A_{i,j}q_{i}q_{j}  + i\sum_j b_jq_{j}\Big) \hat{u}(t, q) e^{iqx}\frac{d^{n}q}{(2\pi)^{n}}+$$
$$+\int_{\mathbb{R}^n}\Big[\int_{\mathbb{R}^{n}}\Big(e^{iqy}-1-i\frac{(y,q)}{1+|y|^2}\Big)M(dy)\Big]\hat{u}(t, q) e^{iqx}\frac{d^{n}q}{(2\pi)^{n}}$$
We let 
$$g(q)=\int_{\mathbb{R}^{n}}\Big(e^{iqy}-1-i\frac{(y,q)}{1+|y|^2}\Big)M(dy)$$
So the equation becomes
$$\partial_{t}\hat{u}(t, q)=\Big(-\frac{1}{2}\sum_{i,j} A_{i,j}q_{i}q_{j}  + i\sum_j b_jq_{j}+g(q)\Big)\hat{u}(t, q)$$
The solution is thus
$$\hat{u}(t, q)=\exp\Big[\Big(-\frac{1}{2}\sum_{i,j} A_{i,j}q_{i}q_{j}  + i\sum_j b_jq_{j}+g(q)\Big)t\Big]h(q)$$
For some function $h(q)$ to be determined.So
$$u(t, x)=\int_{\mathbb{R}^{n}}\exp\Big[\Big(-\frac{1}{2}\sum_{i,j} A_{i,j}q_{i}q_{j}  + i\sum_j b_jq_{j}+g(q)\Big)t\Big]h(q)e^{iqx}\frac{d^{n}q}{(2\pi)^{n}}$$
Using
$$u(0, x)=\varphi(x)$$
We find out
$$h(q)=\int_{\mathbb{R}^{n}}\varphi(x)e^{-iqx}\frac{d^{n}x}{(2\pi)^{n}}$$
