Past exam of the stochastic analysis and partial differential equations course of the University of Oxford / 2025 / 3

Let $E$ be the underlying Banach space. By the Laplace-transform formula for a semigroup resolvent,These are Bochner integrals, defined for every $f\in E$ and $\lambda >0$. The contraction semigroup bound and semigroup property giveInterchanging and regrouping these Bochner integrals is justified by the integrable bound $e^{-\lambda s-\mu t}\Vert f\Vert$, whose integral is $\Vert f\Vert /(\lambda \mu )$. For $\lambda \ne \mu$, set $v=s+t$ and evaluate the inner scalar integral:Since $f$ was arbitrary, the resolvent identity follows:For $\lambda =\mu$, both sides vanish, so that case holds as well.

For $\lambda \ne \mu$, apply the resolvent identity from part (i) in both orders:The two quotients are equal. For $\lambda =\mu$ the assertion is immediate. HenceThus resolvent operators from the same strongly continuous semigroup commute. This conclusion does not require commuting arbitrary bounded linear operators; it uses the fact that both arise from the same semigroup generator.

A second-order nondivergence-form elliptic operator on a domain $D\subseteq {\mathbb{R}}^d$ has the formOnly the symmetric part of $a$ affects the Hessian matrix term, so one may take $a(x)$ symmetric. The operator is a uniformly elliptic operator if there is a constant $\kappa >0$, independent of $x$, such thatA common bounded-coefficient convention additionally requires an upper bound $\sum a_{jk}(x){\xi }_j{\xi }_k\le \Lambda |\xi {|}^2$ with $\Lambda <\infty$. The lower bound says that the principal quadratic form cannot degenerate in any direction or at any location.

For an explicit stochastic example, standard $d$-dimensional Brownian motion has infinitesimal generatorIts principal quadratic form is $|\xi {|}^2/2$, so it is uniformly elliptic with $\kappa =\Lambda =1/2$. More generally, a sufficiently regular stochastic differential equation $d{X}_t=b(X_t)dt+\Sigma (X_t)d{B}_t$ has diffusion generator $b\cdot \nabla +\frac{1}{2}(\Sigma {\Sigma }^{\mathsf{T}}):D^2$; it is uniformly elliptic when $\Sigma {\Sigma }^{\mathsf{T}}$ has a uniformly positive lower bound. A conservative diffusion process has no zeroth-order term, since its infinitesimal generator annihilates constants.

The convergence is in the strong operator topology: for each fixed $f\in E$, one must prove $\Vert \lambda R_{\lambda }f-f\Vert \to 0$. First, the contraction semigroup bound givesThus $\lambda R_{\lambda }$ is a contraction. This is the useful scaling of the contraction estimate in the source's hint. Since ${\int }_0^{\infty }\lambda e^{-\lambda t}dt=1$,Fix $\epsilon >0$. By strong continuity, choose $\delta >0$ so that $\Vert P_{t}f-f\Vert <\epsilon$ for $0\le t\le \delta$. Split the integral there and use $\Vert P_{t}f-f\Vert \le 2\Vert f\Vert$ for the tail:Let $\lambda \to \infty$ and then $\epsilon \downarrow 0$. ThereforeThe distinction from operator norm convergence is essential. For example, the multiplication semigroup on ${\ell }^2(\mathbb{N})$ defined by $(P_{t}f{)}_n=e^{-nt}{f}_n$ is a strongly continuous semigroup of contractions: coordinatewise convergence and dominated convergence applied to ${\sum }_n|f_n{|}^2$ prove strong continuity. Its resolvent operator is $(R_{\lambda }f{)}_n=f_n/(\lambda +n)$, andThus the claimed convergence cannot in general be read as convergence in operator norm.

Take $f$ in the generator domain $D(L)$. The semigroup commutation with its unbounded generator implies $P_{t}f\in D(L)$ andIndeed, applying the bounded operator $P_t$ to the difference quotient at zero gives the right derivative $P_{t}Lf$; strong continuity gives the full orbit derivative for $t>0$. Now integration by parts in the Bochner integral yieldsThe boundary term at infinity vanishes because $\Vert e^{-\lambda t}{P}_{t}f\Vert \le e^{-\lambda t}\Vert f\Vert$. The derivative integral exists as a Bochner integral, since its integrand is bounded in norm by $e^{-\lambda t}\Vert Lf\Vert$.

Multiplying this identity by $\lambda$ givesPart (b) applies to the fixed vector $Lf\in E$, soThe bounded operators $L_{\lambda }={\lambda }^2{R}_{\lambda }-\lambda I$ are the Yosida approximation of the semigroup generator. The proof only applies the unbounded operator $L$ to $f\in D(L)$; the contraction and limit arguments then act on the ordinary vector $Lf$.