Strongly Continuous Monotone Semigroup

• A strongly continuous monotone semigroup is a family of operators on a Banach lattice that are strongly continuous, monotone, and often convex in nonlinear settings.
• The framework extends classical generator concepts using alternative invariant domains, such as monotone and Lipschitz sets, to ensure solution uniqueness and robustness.
• These semigroups are crucial in modeling nonlinear PDEs, stochastic control, and numerical approximations via stable discretization methods.

A strongly continuous monotone semigroup is a family of operators acting on a Banach lattice of continuous functions, possessing the properties of strong continuity, monotonicity, and—in the convex case—convexity. Such semigroups generalize the classical linear strongly continuous semigroups to nonlinear, order-preserving, and possibly convex settings. The associated generator theory features several nontrivial phenomena, such as the non-invariance of the classical generator’s domain, necessitating alternative domains and extended generators to restore uniqueness and applicability to nonlinear PDEs, stochastic control, and related areas (Denk et al., 2020, Blessing et al., 2023).

1. Fundamental Definitions

Let $X$ denote a real Banach lattice, typically a Riesz subspace of a $\sigma$-Dedekind complete Banach lattice, e.g., $BUC(\mathbb{R})$ or $C_0(\Omega)$. A family $(T(t))_{t \geq 0}$ of (possibly nonlinear) bounded operators $T(t): X \to X$ is a strongly continuous monotone semigroup (or monotone $C_0$-semigroup) if:

• $T(0) = I$ (identity),
• $T(t + s) = T(t) T(s)$ for all $t, s \geq 0$,
• $\|T(t) x - x\| \to 0$ as $t \downarrow 0$ for every $x \in X$ (strong continuity),
• $x \preceq y \implies T(t)x \preceq T(t)y$ for all $t \geq 0$ (monotonicity).

If, additionally, each $T(t)$ is convex, i.e., $$T(t)(\lambda x + (1-\lambda)y) \preceq \lambda T(t)x + (1-\lambda) T(t)y$$ for $\lambda \in [0, 1]$, the semigroup is termed convex monotone (Denk et al., 2020, Blessing et al., 2023).

The classical generator $A$ is defined on $$D(A) = \{ f \in X : \lim_{h \downarrow 0} \frac{T(h)f - f}{h} \text{ exists in } X \}$$ by $Af = \lim_{h \downarrow 0} \frac{T(h)f - f}{h}$.

2. Alternative Domains: Monotone and Lipschitz Sets

A hallmark of the convex-monotone setting is the failure of the invariance of $D(A)$: there exist $f \in D(A)$ such that $T(t)f \notin D(A)$ for some $t > 0$ (Denk et al., 2020). To address this, two broader, flow-invariant domains are introduced:

• The monotone domain $D(A^s)$: $f \in D(A^s)$ if, for any sequence $h_n \downarrow 0$, there exist nets $y_n \in X$ with $\|T(h_n)f - f\| = O(h_n)$ and $y_n \preceq (T(h_n)f - f)/h_n$ converging in order in a Riesz completion $X^s$. For $f \in D(A)$, $A^s f = A f$.
• The symmetric Lipschitz set $D_L$: defined by $$\sup_{0 < h \leq h_0} \frac{\|T(h)f - f\|}{h} < \infty$$ for some $h_0 > 0$, i.e., functions $f$ such that $t \mapsto T(t)f$ is locally Lipschitz in $X$. The intersection $D_I = \{ f \in X : f, -f \in D_L \}$ yields a symmetric, invariant subspace.

Both $D(A^s)$ and $D_L$ are invariant under $T(t)$, a property established using Lipschitz continuity estimates and order-continuity arguments (Denk et al., 2020, Blessing et al., 2023).

3. Extended Generator and Uniqueness Results

The monotone generator $A^s$ (also termed the extended generator) is defined via order-approximate differentiability:

$$D(A^s) = \left\{ f \in X : \text{for any } h_n \downarrow 0, \ \exists y_n \in X \text{ with } T(h_n)f - f = h_n y_n + o(h_n), \ y_n \to y \text{ in order (in } X^s) \right\},$$

setting $A^s f := y$. This generator enjoys invariance of its domain and, crucially, restores the uniqueness property analogous to the linear theory: solutions to the (mild) Cauchy problem on $D(A^s)$ are uniquely generated by $T(t)$. Any other convex-monotone flow with the same monotone generator coincides with $T$ on $D(A^s)$, as rigorously proven in [(Denk et al., 2020), Theorem 4.1].

In parallel, on the Lipschitz set $D_L$, a $\Gamma$-generator ($\Gamma f = \limsup_{h \downarrow 0} [T(h)f - f]/h$) can be defined, and a comparison principle ensures uniqueness for semigroups agreeing on a common invariant core in $D_L$ (Blessing et al., 2023).

4. Representative Examples

Two canonical nonlinear $C_0$-semigroups illustrate these phenomena (Denk et al., 2020):

Example	State Space	Semigroup $(T(t))$	Generator on Classical Domain	Invariant Domain	Generator on $D(A^s)$ (a.e.)
Uncertain Shift	$BUC(\mathbb{R})$	$(T(t)f)(x) = \sup_{|y-x|\leq t}f(y)$	$Af = |f'|$ on $BUC^1$	$W^{1,\infty}(\mathbb{R})$	$A^s f = |f'|$
G-Heat Semigroup	$BUC(\mathbb{R})$	$$(T(t)f)(x) = \sup_{\sigma(\cdot)\in\Sigma} \mathbb{E}\left[ f(x + \int_0^t \sigma_s \, dW_s) \right]$$	$Af = \max\{g^2 f'', \bar{\sigma}^2 f''\}$ on $BUC^2$	$W^{2,\infty}(\mathbb{R})$	$A^s f = \max\{g^2 f'', \bar{\sigma}^2 f''\}$

In both examples, the classical domain is not invariant, but the symmetric Lipschitz set is. The monotone generator coincides almost everywhere with the formal nonlinear PDE operator associated with the problem (e.g., a fully nonlinear HJB operator).

5. Stability and Approximation Theory

Semigroup stability under approximation is formulated in terms of convergence of the associated generators in the mixed topology (uniform on compacts plus weighted supremum norm). For strongly continuous convex monotone semigroups $(T_n)$ with generators $(A_n)$ converging on a dense core, the corresponding semigroups converge strongly, and the limit is uniquely characterized by its extended generator (comparison principle) (Blessing et al., 2023).

The Chernoff-type construction yields convergence of iterated one-step convex, monotone, uniformly equi-Lipschitz maps to the flow of the limiting semigroup under suitable consistency conditions. This framework underpins stable discretizations for nonlinear problems, avoiding the reliance on traditional viscosity solution theory.

6. Applications to Nonlinear PDEs and Control

Strongly continuous monotone semigroups provide a unified framework for representing nonlinear evolution equations, especially fully nonlinear Hamilton–Jacobi–Bellman (HJB) and stochastic control problems:

• Nonlinear shift semigroups encode worst-case state propagation and arise in robust control and uncertainty quantification.
• G-heat semigroups represent solutions to nonstandard diffusion equations driven by sublinear expectations, relevant in the theory of $G$-Brownian motion.
• Iterative Markov chain approximations recover robust optimal control semigroups for stochastic processes with uncertain transitions.
• Numerical schemes (Euler, Yosida, finite-difference) for such semigroups guarantee convergence to the continuous flow, provided stability and consistency at the generator level (Blessing et al., 2023).

7. Further Directions

Current research suggests multiple open directions:

• Characterization of which monotone (extended) generators $A^s$ generate a strongly continuous monotone semigroup.
• Invariant regularity theory (Hölder, Sobolev) on the monotone or Lipschitz domains.
• Extension to infinite-dimensional state spaces, such as Banach lattices of path-valued functions.
• Connections to $m$-accretive operators in Banach spaces with order-continuous norms and the broader theory of nonlinear evolution equations (Denk et al., 2020).

The theory thus bridges functional analysis, nonlinear PDEs, stochastic analysis, and optimal control, supplying robust operator-theoretic tools for the study and discretization of nonlinear evolutionary phenomena.