Existence of solutions of $u_t = Lu$ for pseudo-differential operators $L$?

Question

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.

Answer 1

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.

Answer 2

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 $$h(q)=\int_{\mathbb{R}^{n}}\varphi(x)e^{-iqx}\frac{d^{n}x}{(2\pi)^{n}}$$