Operator semigroups in the mixed topology and the infinitesimal description of Markov processes

We define a class of not necessarily linear $C_0$-semigroups $(P_t)_{t\geq0}$ on $C_b(E)$ (more generally, on $C_\kappa(E):=\frac1\kappa C_b(E)$, for some bounded function $\kappa$, which is the pointwise limit of a decreasing sequence of continuous functions) equipped with the mixed topology $\tau_1^{\mathscr M}$ for a large class of topological state spaces $E$. If these semigroups are linear, classical theory of operator semigroups on locally convex spaces as well as the theory of bicontinuous semigroups apply to them. In particular, they are infinitesimally generated by their generator $(L,D(L))$ and thus reconstructable through an Euler formula from their strong derivative at zero in $(C_b(E),\tau_1^{\mathscr M})$. In the linear case, we characterize such $(P_t)_{t\geq0}$ as integral operators given by measure kernels satisfying certain tightness properties. As a consequence, transition semigroups of Markov processes are $C_0$-semigroups on $(C_b(E),\tau_1^{\mathscr M})$, if they leave $C_b(E)$ invariant and they are jointly weakly continuous in space and time. Hence, they can be reconstructed from their strong derivative at zero and thus have a fully infinitesimal description. Furthermore, we introduce the notion of a Markov core operator $(L_0,D(L_0))$ for the above generators $(L,D(L))$ and prove that uniqueness of the Fokker-Planck-Kolmogorov equations corresponding to $(L_0,D(L_0))$ for all Dirac initial conditions implies that $(L_0,D(L_0))$ is a Markov core operator for $(L,D(L))$. If each $P_t$ is merely convex, we prove that $(P_t)_{t\geq0}$ gives rise to viscosity solutions to the Cauchy problem given by its associated (nonlinear) infinitesimal generator. We also show that value functions of optimal control problems, both, in finite and infinite dimensions are particular instances of convex $C_0$-semigroups on $(C_\kappa(E),\tau_\kappa^{\mathscr M})$.

The literature on Markov processes is huge. We here only refer to a selection from pioneering and/or fundamental books on the subject and to the references therein, as, e.g., [3] [69]. Let us briefly recall the definition of a Markov process: Let (E, B) be a measurable space and, for each x ∈ E, let (Ω, F, (F t ) t≥0 , P x ) be a filtered probability space and X(t) : Ω → E F t /B-measurable maps, t ≥ 0, such that P x [X(0) = x] = 1. Then the tuple M := (Ω, F, (F t ) t≥0 , (X(t)) t≥0 , (P x ) x∈E ) is called a (timehomogeneous) Markov process with state space E, if it satisfies the Markov property, i.e., for all x ∈ E, A ∈ B, t, s ≥ 0, P x [X(s + t) ∈ A|F s ] = P X(s) [X(t) ∈ A] P x -a.s., (1.1) where P x [ · |F s ] denotes the conditional probability of P x given F s . Its corresponding transition semigroup of probability kernels is defined by the time marginal laws of P x under X(t), t ≥ 0, i.e., p t (x, dy) := (P x • X(t) −1 )(dy), x ∈ E, t ≥ 0. (1.2) Usually, one also assumes some path regularity on X(t), t ≥ 0, by considering topological state spaces E together with the corresponding Borel σ-algebra B := B(E) and assuming that, for all x ∈ E, the map [0, ∞) t → X(t) ∈ E is right-continuous P x -a.s.. Define for f : E → R, bounded, B-measurable, P t f (x) := E f (y) p t (x, dy) = E x [f (X(t))], x ∈ E, t ≥ 0, (1.3) where E x denotes the expectation with respect to P x . Then the Markov property (1.1) implies the semigroup property P t+s f (x) = P t (P s f )(x), x ∈ E, t, s ≥ 0. (1.4) A common very natural assumption, which is fulfilled in many situations (in particular, where P x , x ∈ E, are the laws of the solutions of a stochastic differential equation (SDE) with respective initial data x ∈ E and where E is, say a Banach space or just R d ) is the so-called Feller property, i.e., Here C b (E) denotes the set of all bounded real-valued continuous functions on E. Let P(E) denote the set of all probability measures an (E, B(E)). Then (1.5) means: E x → p t (x, dy) ∈ P(E) is continuous in the weak topology (1. 6) on P(E) for all t ≥ 0.
By the assumed right continuity of sample paths and by (1.5) we also have [0, ∞) t → p t (x, dy) ∈ P(E) is right continuous in the weak topology on (1.7) It is well-known that, if we consider C b (E) with its supremum norm · ∞ , then t → P t f is (in general) not continuous at t = 0 for all f ∈ C b (E), i.e., (P t ) t≥0 is not a C 0 -semigroup on (C b (E), · ∞ ).
If E is metric space, then the next natural choice is the space U C b (E) of bounded uniformly continuous functions which, when endowed with the the norm · ∞ , is a closed subspace of C b (E). It turns out that the gain is very limited. It can be shown that if E is a separable Hilberts space and (P t ) t≥0 is a transiton semigroup of an E-valued Wiener process that (P t ) t≥0 is a C 0 -semigroup on U C b (E), see Proposition 3.5.1 in [17]. This result can be easily extended to a general Lévy process. However, the transition semigroup of an Ornstein-Uhlenbeck process in E = R, while it turns out to leave the space U C b (R) invariant, is not strongly continuous there, see Example 6.1 in [14] and Theorem 2.1 in [70]. The latter result also implies that the transition semigroup of a general Ornstein-Uhlenbeck process with nonzero drift is never strongly continuous on C b (E).
Hence the theory of C 0 -semigroups on Banach spaces (see e.g. [56], [25]) does not apply. If it did, P t , t ≥ 0, would be uniquely determined by its derivative at t = 0, i.e., which defines a linear operator L : D(L) ⊂ C b (E) → C b (E) with D(L) being the set of all f ∈ C b (E) for which the limit in (1.8) exists. In this case P t , t ≥ 0, can be recalculated from the operator (L, D(L)), called infinitesimal generator of (P t ) t≥0 , through Euler's formula. But as said, this is in general not possible on (C b (E), · ∞ ).
A way out of this, which only works if E is locally compact (hence excludes, e.g., that E is an infinite dimensional Banach space, which in turn are the typical state spaces for solutions X(t), t ≥ 0, to stochastic partial differential equations (SPDEs) or measure-valued Markov processes) is to replace (1.5) by (1.9) where C ∞ (E) denotes the subset of all elements in C b (E) which vanish at infinity. (P t ) t≥0 , satisfying (1.9) are called Feller semigroups in the literature, which sometimes leads to confusion, since the much weaker property (1.5) is usually called Feller property and the latter makes sense on general topological spaces (see, e.g., [59]). But, if E is locally compact and (1.9) holds, there are a large number of examples, for which (P t ) t≥0 is a C 0 -semigroup on (C ∞ (E), · ∞ ) and thus uniquely determined by and reconstructable from its infinitesimal generator (L, D(L)), i.e., from its strong derivative at zero (see e.g. [25]). This is usually expressed by the symbolic writing P t = e tL , t ≥ 0. On the other hand, condition (1.9) is very strong and in general, of course, not fulfilled, even if E = R d .
Another approach is to avoid the C 0 -(i.e., strong continuity) property and associate to (P t ) t≥0 an operator (L, D(L)), also called generator of (P t ) t≥0 , which is obtained by inverting the resolvent of (P t ) t≥0 , which in turn is given by the Laplace transform of (P t ) t≥0 (see, e.g., [59]). But this definition of generator uses the whole semigroup (P t ) t≥0 and is thus definitely not an infinitesimal generator of (P t ) t≥0 .
Finally, another way out is to replace C b (E) by an L p (E, µ)-space, p ∈ [1, ∞), for some suitable reference measure µ on (E, B(E)) (e.g., an invariant measure for (P t ) t≥0 ). Then (P t ) t≥0 extends to a C 0 -semigroup on L p (E, µ), which has a true infinitesimal generator there (see, e.g., [61] and , [8,Section 4]] and also [35] for symmetrizing measures µ). Clearly, a symmetrizing or invariant measure does not exist in general for (P t ) t≥0 . In [62,Proposition 2.4], however, it was proved that a natural reference measure µ always exists so that the transition semigroup (P t ) t≥0 of a Markov process M as above extends to a C 0 -semigroup on L p (E, µ). But this measure µ again is contructed through the resolvent of (P t ) t≥0 , hence again uses the whole semigroup (P t ) t≥0 . So, the infinitesimal generator of (P t ) t≥0 , extended to a C 0 -semigroup on L p (E, µ), is not really "infinitesimal". In addition, the analysis of this extension of (P t ) t≥0 , depends on the measure µ and statements can always be only made µ-a.e., and the measure µ is in no sense unique.
So, concluding it can be said that it has been an open problem whether the transition semigroup of a general Markov process M as above, which has the Feller property (1.5), is infinitesimally generated by its strong derivative at zero in a "suitable" topology on C b (E).
The first main contribution of this paper concerning the above open problem is to prove that such a "suitable" topology is the well-known mixed topology τ M 1 on C b (E), i.e., the strongest locally convex topology on C b (E) which on · ∞ -bounded subsets of C b (E) coincides with the topology of uniform convergence on compact subsets of E (see Section 2 and Appendix A for details), provided (P t ) t≥0 , satisfies the following very general condition (cf. (1.6) and (1.7) above): [0, ∞) × K (t, x) → p t (x, dy) ∈ P(E) is continuous in the weak topology on (1.10) P(E) for all compact K ⊂ E (see Theorem 3.3 and Proposition 3.6 below). In fact, this is true for very general state spaces E (see Hypothesis 2.1 below). So, in such a very general case the transition semigroup of a Markov process with right continuous sample paths is uniquely determined by its strong derivative at zero with respect to the mixed topology τ M 1 on C b (E) and can be reconstructed through an Euler formula (see Proposition 5.2 (f)).
We would like to mention here that for a special class of stochastic evolution equations on a Hilbert space, similar to those in Section 4.2 below (see the fundamental book [18] for the general theory) the strong continuity of the transition semigroups of their solutions at t = 0 in the mixed topology was first proved in [37] (and the reference therein, in particular [15]). The latter paper was a strong motivation for proving the much more general result above and for developing the corresponding general theory in the present paper. The necessity to relax the norm topology on C b (E) has been well known in the SPDE community and the problem was approached using many ad-hoc constructions, see for example [14,15,17,59]. In the aforementioned works no underlying topology making the semigroups of interest strongly continuous was identified.
Let us summarize the single sections of this paper and at the same time present our further main results. Section 2 contains our setup and necessary definitions, in particular, those concerning the mixed topology. We generalize the situation above by replacing is a continuous function, and consider the mixed topology τ M κ on C κ (E).
In Section 3 we introduce a general class of C 0 -semigroups of operators (P t ) t≥0 on (C κ (E), τ M κ ) (see Definition 3.1). In case these P t are linear, we prove that the semigroup can be represented by a semigroup of measure kernels with certain properties and that any such gives rise to a (linear) C 0 -semigroup on (C κ (E), τ M κ ) (see Theorem 3.3), which is another main result of this paper. The main underlying fact, why this works, is the well-known result that the topological dual of (C b (E), τ M 1 ) coincides with the set M b (E) of all signed Radon measures on (E, B(E)) (see Appendix A for references and a simple proof in Remark A.10 based on the Daniell-Stone Theorem). Section 4 is devoted to examples on finite and infinite dimensional state spaces. We start with transition semigroups coming from a large class of SDEs on Hilbert spaces H (taking the role of E), including, e.g., the 2D-stochastic Navier-Stokes equations as well as stochastic (fast and slow diffusion) porous media equations (see Section 4.1). Here we consider both the norm topology on H and (in Section 4.2) also the bw-topology on H. Furthermore, we look at a class of SPDEs with Levy noise on Banach spaces E, more precisely SDEs of Ornstein-Uhlenbeck (O-U) type, but driven by Levy noise (see equation (4.18)). Their corresponding transition semigroups, called generalized Mehler semigroups, also turn out to be C 0 -semigroups on (C b (E), τ M 1 ) both when E is considered with the norm topology (see Section 4.3) and, provided E is reflexive, also with the bw-topology (see Section 4.4). The interesting feature of the bw-topology is that in this case C b (E) consists of all bounded sequentially weakly continuous functions on E.
In Section 5.1 we define the strong and weak infinitesimal generator (L, D(L)) of a C 0semigroup on (C κ (E), τ M κ ) and prove that they coincide (see Theorem 5.5). Furthermore, we show that the usual "invariance condition" for identifying cores for (L, D(L)) also holds in this case (see Proposition 5.3). We introduce the notions of core operators and Markov core operators for (L, D(L)) (see Definition 5.7). Subsequently, we prove that a sufficient condition for being a Markov core operator (L 0 , D(L 0 )) for (L, D(L)) is, that the Fokker-Planck-Kolmogorov equation for (L 0 , D(L 0 )) has a unique solution for all Dirac measures δ x , x ∈ E (see Theorem 5.9). This is another main result of this paper, which is illustrated by a number of applications, where we identify the Kolmogorov operator of a large class of SDEs on R d (see Section 5.2) or on a Hilbert space H (see Section 5.4) as a Markov core operator for the infinitesimal generator (L, D(L)) of the C 0 -semigroup on (C κ (R d ), τ M κ ) and (C κ (H), τ M κ ), respectively, given by the transition semigroup of the SDE's solutions. Furthermore, in Section 5.3 using results from [47], we identify the Kolmogorov operator of SDE (4.18), i.e., the SDE for the O-U-process with Levy noise on a Hilbert space E, which is a pseudo-differential operator (see equation (5.30)), as a core operator for the generator (L, D(L)) of the corresponding generalized Mehler semigroup on (C b (E), τ M 1 ). These results in Sections 5.1 -5.3 constitute the fourth main contribution of this paper.
In Section 6 we consider the case where the C 0 -semigroup (P t ) t≥0 on (C κ (E), τ M κ ) consists of convex operators. In this case we prove that (P t ) t≥0 , gives rise to viscosity solutions (see Definition 6.1) to the Cauchy problem of its associated infinitesimal generator. Moreover, we show that every convex Markov C 0 -semigroup on (C b (E), τ M κ ) gives rise to a notion of a nonlinear Markov process under a convex expectation. This provides an analytic counterpart to the recent investigations of G-expectations and nonlinear Markov processes, see [57]. The latter appear in the context of financial modeling in terms of a Brownian motion under volatility uncertainty. Generalizations to uncertainty in the generators of Levy processes and a class of Feller processes have been made in [53], [42], [20], [52]. In this context and, more generally, in Mathematical Finance, the so-called continuity from above on C b (E) of related risk measures plays an important role. The main results of this section are formulated in Theorems 6.2 and 6.4. Section 7 contains examples from stochastic optimal control as applications of the result in Section 6, both on finite (Section 7.1) and infinite dimensional (Section 7.2) state spaces.

Basic definitions and setup
In this section we recall basic definitions and some properties of the so called mixed topology on a space of continuous functions ϕ : E → R. A very general definition of this topology was introduced in [72] and, in the special case of the space of bounded continuous functions defined on a completely regular topological space E, it was studied in topological measure theory as one of the strict topologies, see [71]. In this paper we restrict our attention to a special class of completely regular topological spaces E, but many results presented in this section, when appropriately reformulated, hold for larger classes of spaces, or even for every completely regular Hausdorff topological space.
The following hypothesis about the space E is assumed to hold throughout the paper and will not be enunciated again.  (3) are known as k f -or k R -spaces, see [38] or [71]. b) Polish spaces satisfy all three conditions of Hypothesis 2.1. c) Let E = F be the dual of a separable Banach space F endowed with its weak topology. Then, E is a Hausdorff topological vector space and thus completely regular, see [38,Theorem 2.9.2]. We say that a set B ⊂ E is bw-closed if its intersection with every weak -compact set is weak -closed. The corresponding topology is completely regular, and is known as the bw-topology, [22, pages 427-428] or [51,Section 2.7]. Clearly, the bw-topology coincides with the weak -topology on every weak -compact set, and therefore weak -compactness is equivalent to bw-compactness. As a consequence, any function E → R that is continuous on all weak -compacts of E endowed with the weak topology is continuous on E endowed with the bw-topology. Throughout, we consider a continuous weight function κ : E → (0, ∞), and C κ (E) denotes the space of continuous functions ϕ : E → R with If κ ≡ 1, we use the notation C b (E) instead of C 1 (E).
On C κ (E), we consider various topologies. One of them is the norm topology τ U κ w.r.t. · κ . For any compact set C ⊂ E, we define the seminorm and we denote the locally convex topology on C κ (E) generated by the family of seminorms {p κ,C : C compact} by τ C κ . Note that, by virtue of our assumptions on the weight function κ, the topology τ C κ coincides with the topology τ C 1 of uniform convergence on compact subsets of E, which is generated by the family of seminorms p C (ϕ) := sup x∈C |ϕ(x)|, for ϕ ∈ C κ (E).
We continue with the definition of the mixed topology, which is fundamental for everything that follows. For an arbitrary sequence (C n ) of compact subsets of E and a sequence (a n ) of positive numbers with lim n→∞ a n = 0, we define the seminorm p κ,(Cn),(an) (ϕ) := sup n∈N a n p κ,Cn (ϕ) = sup n∈N sup x∈Cn a n κ(x)|ϕ(x)| . Definition 2.3. The locally convex topology on C κ (E), defined by the family of seminorms p κ,(Cn),(an) : C n ⊂ E compact, 0 < a n → 0 , is called the mixed topology, and is denoted by τ M κ . In the language of topological measure theory, τ M κ belongs to the class of strict topologies, see [71]. By definition, For the reader's convenience, we collect some basic properties of the mixed topology in the Appendices A and B. For a more detailed discussion of mixed (or strict) topologies, we refer to [71] and [72].
We now introduce the dual objects of C κ (E). Let M b (E) denote the space of all signed Radon measures µ : B(E) → R with |µ|(E) < ∞, where |µ| stands for the total variation measure of µ. Recall that, under Hypothesis 2.1, every Baire measure is Borel, and that a Borel measure µ : B(E) → R with |µ|(E) < ∞ is Radon 1 if, for every Borel set B and every ε > 0, there exists a compact set C ε ⊂ B such that We denote the space of all Radon measures µ on (E, B(E)) with be the subset of all nonnegative measures in M κ (E). If µ ∈ M κ (E), then the mapping Throughout, we endow M κ (E) with the narrow topology, i.e., the weakest topology such that, for every ϕ ∈ C κ (E), the mapping By Theorem A.9, the space M κ (E) endowed with the narrow topology is the topological dual of C κ (E), τ M κ . In what follows, we consider (nonlinear) operators on C κ (E), i.e., C κ (E) → C κ (E). We say that an operator T on C κ (E) is norm-bounded if sup ϕ∈B T ϕ κ < ∞ 1 In some papers in topological measure theory this is called a tight measure.
for all norm-bounded sets B ⊂ C κ (E), i.e., sup ϕ∈B ϕ κ < ∞. An operator on C κ (E) is called τ M κ -continuous if it is continuous for the mixed topology τ M κ . In the Appendix B, we characterize norm-bounded linear operators T on C κ (E) that are τ M κ -continuous.

Strongly continuous semigroups on spaces of continuous functions with mixed topology
In this section, we introduce the notion of strongly continuous and locally equicontinuous semigroups on C κ (E), τ M κ , which we will refer to as C 0 -semigroups. The following definition is a straightforward generalisation of the definition of strongly continuous and equicontinuous semigroups of linear operators on locally convex spaces given in [73].
As a consequence for every ϕ ∈ C κ (E) by the continuity of P t ϕ on E we easily obtain that for all compact C ⊂ E the map is continuous.
(ii) Now let us consider the linear case, i.e., each P t of P is a linear operator. Then Indeed, let ϕ ∈ C κ (E). Then by the uniform boundedness principle it suffices to show that sup t≤T P t ϕ κ < ∞.
If this is not the case, there exist t n ∈ [0, T ], n ∈ N, such that We may assume that lim n→∞ t n = t ∈ [0, T ]. Hence by part (i) of this Remark consequently, by Proposition A.4 in the Appendix sup n∈N P tn ϕ κ < ∞, which contradicts (3.2). By the semigroup property (3.1) is equivalent to: There exist M ∈ [1, ∞) and a ∈ R such that P t ϕ κ ≤ M e at ϕ κ for all ϕ ∈ C κ (E) and t ≥ 0.  x ∈ E, t ≥ 0} ⊂ M κ (E) such that: (1) The map E x → µ t (x, B) is measurable for every B ∈ B(E) and t ≥ 0.
(5) For every x ∈ E and any sequence ( where δ x denotes the Dirac measure with barycenter x. Proof. We start with the proof of the implication (b) ⇒ (a). Assume that (b) is satisfied. We have to show (iii),(iv) in Definition 3.1. In order to show that (iii) is satisfied, let T > 0. For n ∈ N let (a n ) ⊂ (0, ∞) and (C n ) be an increasing sequence of compact subsets of E. Let b n := 2 1−n , n ∈ N. By (4), for every l ∈ N, there exists an increasing sequence (K l,n ) n∈N of compacts in E such that Define, for n ∈ N, Then, (K n ) is an increasing sequence of compacts, and, for all n ∈ N, by (3.6). Now, we are going to show (3.4). To that end, let ϕ ∈ C κ (E). By homogeneity, we may assume that p κ,(Kn),(bn) (ϕ) = 1, hence p κ,Kn (ϕ) ≤ 2 n−1 for all n ∈ N. (3.8) which, by (3.7) and (3.8), is dominated by where, by (3), this constant is finite. Hence, by the last part of Remark 3.2 (ii), Property (iii) follows.
We proceed to the proof of (iv). Since by Remark 3.2 (ii) we know that (3.1) holds, by Proposition A.4 we have to show that, for every compact K ⊂ E, Suppose this does not hold. Then, we can find a compact K ⊂ E, ε > 0, t n → 0, and (x n ) ⊂ K such that Since K is compact and metrizable, there exists some x ∈ K such that x n k → x for a subsequence (n k ). Since (3.10) also holds for this subsequence, we get a contradiction to condition (5). It remains to establish the implication (a) ⇒ (b). If (a) holds, then (3.1) holds by Remark 3.2 (ii). Hence by Theorem B.2, there exists a family of measures hence we can use the same arguments as in the proof of Theorem B.2 to prove Property (4). Property (5) is an immediate consequence of the strong continuity of P at zero. Remark 3.4. As just proved above, the dependence of the constant C T in (3.4) on the semi- The following proposition renders a convenient sufficient condition to check conditions (4) and (5) in Theorem 3.3, if E is a so-called Prohorov space, whose definition we recall first (see [6, Definition 4.7.1(i)]). Definition 3.5. Let E be as above (i.e., as in Hypothesis 2.1). Then E is called a Prohorov space, if every compact subset of M + b (E) (equipped with the narrow topology) is tight. (3) in Theorem 3.3 holds and that, for every T ∈ (0, ∞) and every compact C ⊂ E, the map is continuous for every ϕ ∈ C κ (E). Then (4) and (5) in Theorem 3.3 also hold.
Proof. Since the continuous image of a compact set is compact, by the assumptions, it follows Remark 3.7. If E is Polish, then E is Prohorov. Likewise, if E is as in Remark 2.2(3), and equipped with the bounded weak topology τ bw , then [6, Proposition 4.7.6(i)] implies that (E, τ bw ) is Prohorov.

Examples for linear
We are now going to present large classes of examples for C 0 -semigroups on (C κ (E), τ M κ ) given by transition semigroups of solutions to stochastic differential equations (SDEs) on infinite dimensional state spaces, hence including stochastic partial differential equations (SPDEs) as their main examples. The main tool to show that such transition semigroups are indeed C 0 -semigroups on (C κ (E), τ M κ ) will be Proposition 3.6.
4.1. Transition semigroups of solutions to SDEs on Hilbert spaces of locally monotone type.
The first class of examples come from SDEs in Hilbert spaces of locally monotone type, introduced in [49]. Let us recall the necessary details from [49, Section 5.1]. Let E := H be a separable Hilbert space with inner product , H and H * its dual. Let V be a reflexive Banach space, such that V ⊂ H continuously and densely. Then for its dual space V * it follows that H * ⊂ V * continuously and densely. Identifying H and H * via the Riesz isomorphism we have that continuously and densely and if V * , V denotes the dualization between V * and V (i.e. , be a cylindrical Wiener process in a separable Hilbert space U on a probability space (Ω, F, P) with normal filtration F t , t ∈ [0, ∞). We consider the following stochastic differential equation on H where for some fixed time T > 0 are progressively measurable, where U is another separable Hilbert space and L 2 (U, H) denotes the set of all Hilbert-Schmidt operators from U to H. The coefficients A and B are assumed to satisfy the following conditions: (H2) (Local monotonicity) with α in (H3) and P-a.s.
whereX is any V -valued progressively measurable dt ⊗ P-version ofX.
with some p ≥ β + 2, and there exists a constant C such that Then for every X 0 ∈ L p (Ω, F 0 , P ; H), (4.1) has a unique solution (X(t)) t∈[0,T ] such that For later use we need the following: Consider the situation of Theorem 4.2 and let X(t, x), t ∈ [0, T ], be the unique solution of (4.1) with X(0, x) = x ∈ H. Assume, in addition, that there exists we have by Itô's formula (see e.g. [49,Theorem 4 Hence by (H2), the Burkholder-Davis-Gundy inequality with p = 1 and (4.3) Hence by Gronwall's lemma ∀t ≥ 0 From now on in this section we assume that A and B above do not depend on ω ∈ Ω, t ∈ [0, ∞), and that (H1)-(H4) hold with some constant f ∈ [0, ∞) replacing the function f . Furthermore, we assume that (4.3) holds.
So, let us now consider the transition semigroup of the unique solution from Theorem 4.2, Proof of Claim 1: Clearly, (P t ) t≥0 and µ t (x, dy), t ≥ 0, x ∈ H, satisfy conditions (1), (2), (3) in Theorem 3.3 with E := H (equipped with its norm topology) and κ = 1. To show that also (4) and (5) hold, by Proposition 3.6 we have to show that for every compact C ⊂ H and Clearly, it then follows by Lemma 4.3 that weakly as n → ∞ and (4.6) follows. Therefore, (P t ) defined in (4.4) is a C 0 -semigroup on (C b (E), τ M 1 ) by Theorem 3.3, and Claim 1 is proved.
Proof of Claim 2: Obviously, (P t ) t≥0 satisfies (1), (2) and (for κ as in (4.5)) also (3) in Theorem 3.3, since by (4.2) (applied with p = m) for some C T ∈ (0, ∞) we have As above (4) and (5) in Theorem 3.3 follow from (4.6) above, however, to be proved for all which converges to zero as n → ∞, since we already know from the proof of Claim 1 that X(t n , x n ) → X(t, x) in P-measure and since X(t n , x n ) m H , n ∈ N, are uniformly integrable by (4.2), so that the generalized Lebesgue's dominated convergence theorem applies. Hence Theorem 3.3 implies Claim 2.

Transition semigroups of mild solutions to SDEs on Hilbert spaces with bounded weak topology.
Let H be a separable Hilbert space with inner product ·, · H and norm | · | H . We will denote the space H endowed with the bounded weak topology by H bw . In this section we will consider the following stochastic evolution equation in H: We assume that -W is a cylindrical Wiener process on a separable Hilbert space U , -A generates a C 0 -semigroup T t , t ≥ 0, on H, -F : H → H satisfies the Lipschitz condition with a constant L: -G : E → L(U, E) (:= all continuous linear operators from U to H) is strongly measurable and satisfies the conditions In the above we use the notation B L 2 (U,H) for the Hilbert-Schmidt norm of an operator B : U → H. Under the above assumptions equation (4.8) has a unique mild solution in H given by the formula Moreover, by standard arguments we find, that for all x ∈ H, T > 0 where B r denotes the open centered ball of radius r in H and Following the arguments from [37] we obtain for By an easy modification of the proof in [37] one can prove that under the above assumptions the semigroup κm . The next proposition extends the result in [50].
This is the only part of the proof where compactness of T t is required. We will show that the semigroup (S t ) satisfies conditions (1) We recall here that the Borel σ-algebras of H and H bw coincide and clearly, the mapping is B (H bw )-measurable for every t ≥ 0 and V ∈ B (H bw ), hence condition (1) of Theorem 3.3 holds. By (4.13) condition (2) of Theorem 3.3 is satisfied as well. Invoking (4.9) we obtain for all x ∈ H, T > 0 and condition (3) of Theorem 3.3 follows. Since B r is bw-compact for every r > 0 we can use (4.9) again to show that for every T > 0 and every r > 0 the family of measures is tight, which yields condition (4) of Theorem 3.3. It remains to prove that conditon (5) is satisfied and it is enough to prove this condition for m = 0. Let ϕ ∈ C b (H bw ). Let t n → 0 and x n → x weakly with sup n≥1 |x n | H ≤ r for a certain r > 0. For any ε > 0 and T > 0 we can choose R ≥ r such that sup Let {f k ; |f | H = 1 , k ≥ 1} be a dense set in the sphere {f ∈ H; |f | H = 1}. We recall that the metric defines a Polish topology identical with the weak topology on B R . We have Let ω be the modulus of continuity of the function ϕ on B R . Then .

Generalized Mehler semigroups on Banach spaces with norm topology.
Let E be a separable Banach space and let κ = 1. Let (T t ) t≥0 be C 0 -semigroup of operators on E. Furthermore, let µ t , t ∈ [0, ∞), be probability measures on (E, B(E)) such that: Then (P t ) t≥0 is (by (4.16)) a semigroup of linear operators on C b (E), called a "generalized Mehler semigroup". In this generality such semigroups have been first introduced in [10] and then further analyzed in [34] and many other papers (see e.g. the very recent work [2] and the references therein). They appear as transition semigroups of Ornstein-Uhlenbeck process with Levy noise, i.e. solutions to the following SDEs on E dX(t) = AX(t)dt + dY (t), (4.18) where A is the generator of (T t ) on E and Y (t), t ≥ 0, is the underlying Levy process corresponding to the Levy characteristics appearing in the Levy-Khintchine representation of the exponent of the Fourier transforms of µ t , t ≥ 0. We refer to [34] for details. Obviously, (P t ) has a representation as in (3.5) with is continuous. So, let x n , x ∈ E, t n , t ∈ [0, ∞) such that lim n→∞ t n = t and lim n→∞ x n = x (w.r.t. the norm topology on E). Then we have to show that for all By the Portemanteau theorem we may assume that ϕ is Lipschitz with Lipschitz constant less or equal to one. Then we have which clearly converges to zero as n → ∞ by (4.15) and because (T t ) is a C 0 -semigroup on E.

Generalized Mehler semigroups on Banach spaces with bounded weak topology.
Let κ = 1 and E be a reflexive separable Banach space (in particular, E is as in Remark 2.2 (3)). Let us now consider (E, τ bw ), i.e. E equipped with the bounded weak topology (see Remark 2.2 (3)). Then, since E is separable, we have that B((E, · E )) = B((E, τ bw )). Let µ t , t ∈ [0, ∞), be as in (iii) above, satisfying (4.16), but instead of (4.15), we assume the weaker condition Let (P t ) be defined as in (4.17). We want to show that again by Theorem 3.3 and Proposition 3.6 (P t ) is a C 0 -semigroup on C b ((E, τ bw )), τ M 1 . We recall that C b ((E, τ bw )) are exactly the bounded sequentially weak * -continuous functions on E and that each τ bw -compact C ⊂ E is metrizable (see Remark 2.2 (3)). Obviously (P t ) is a semigroup of linear operators on C b ((E, τ bw )) satisfying conditions (1)-(3) in Theorem 3.3. It remains to prove (4) and (5), which again will follow by Proposition 3.6. So let t n → t in [0, T ], x n → x in (E, τ bw ) and ϕ ∈ C b ((E, τ bw )). We have to show that lim n→∞ P tn ϕ(x n ) = P t ϕ(x). Let us recall the definition of the finitely based C 1 b -functions, i.e.

Strong and Weak infinitesimal generators
5.1. Definitions and (Markov) core operators.
Then, we define its infinitesimal generator L by the formula (5.1) In order to formulate the next result, we first recall that, if X is any sequentially complete locally convex linear space, then a continuous function f : [0, T ] → X is Riemann integrable ,and the function F (t) = t 0 f (s) ds is differentiable with dF dt = f (t) for every t ∈ (0, T ) (see [30] for details). We also recall that, by Theorem A.6, the space C κ (E), τ M κ is complete. In the next propostion we collect some known properties of C 0 -semigroups of operators on C κ (E). Parts (b)-(e) of the Proposition were proved in a more general framework in [40], part (a) in [1].  (f) The Euler formula holds, i.e., for all ϕ ∈ C κ (E) Proof. According to the remarks preceding this theorem, we only have to prove (e). But by Remark 5.6 below this is an immediate consequence of Theorem 5.6 in [11] (attributed there to F. Kühnemund, see [43]) Proof. Since by Proposition 5.2 (c) each P t is continuous in the τ M κ -graph topology of D(L), we have that P t (D L ) ⊂ D L , so we may assume that D = D L . Now let ϕ ∈ D(L). We have to We claim that for every α and t ≥ 0 t 0 P s ϕ α ds ∈ D.
for each ν ∈ M κ (E). In this case, we define the operator L w by the formula We say that L w is the weak generator of the C 0 -semigroup P on C κ (E), τ M κ with domain D (L w ).
Proof. We start with the proof that L = L w , which is a modification of the proof of [56, p. 43, Corollary 1.2] given for C 0 -semigroups in Banach spaces. If ϕ ∈ D(L), then (5.4) follows by Theorem A.9. Hence, L ⊂ L w . To show that L w ⊂ L choose ϕ ∈ D (L w ). Then, by the semigroup property, t → P t ϕ is a weakly continuous differentiable curve in C κ (E), τ M κ . Hence where for the secound equality we used that a continuous linear functional on (C κ (E), τ M κ ) interchanges with the C κ (E)-valued Riemann integral. Taking ν := δ x , for x ∈ E, we get, for all x ∈ E, which shows that ϕ ∈ D(L) and concludes the proof that L = L w . Assume that (5.5) and (5.6) hold. Then, one immediately sees that (5.4) is satisfied for every measure ν ∈ M κ (E), hence ϕ ∈ D(L w ) = D(L) with f = Lϕ by the first part of the proof. Conversely, assume that ϕ ∈ D(L). Then, (5.6) with f = Lϕ is obvious and, by Proposition A.4, (5.5) holds.
Remark 5.6. It is very easy to check that in the linear case our C 0 -semigroups on (C κ (E), τ M κ ) are special cases of the bi-continuous semigroups introduced in [43]. We also refer to [12], [27], [41] and [45] for further developments, and furthermore to [31], where only a sequential C 0 -property is required. In particular, according to the main result in [43] there is a Hille-Yosida-type Theorem for characterizing their infinitesimal generators defined in Definition 5.1. Likewise we have a characterization for the latter through a Hille-Phillips-type Theorem by the very nice recent paper [13] (see Theorems 3.6 and 3.15 therein).
Next we want to discuss examples of infinitesimal generators for C 0 -semigroups on (C κ (E), τ M κ ), which are given by transition semigroups of solutions to S(P)DEs, and the relation to the Kolmogorov operators associated to the latter. In each case we shall proceed in two steps. First, we shall prove that the respective infinitesimal generator is an extension of the Kolmogorov operator associated to the latter. Second, we shall prove "strong uniqueness" or at least "Markov uniqueness" for the respective Kolmogorov operator, which thus uniquely determines the infinitesimal generator of the corresponding C 0 -semigroup on (C κ , τ M κ ). We start with the following definitions.
Proof. Let (L, D(L)) be the infinitesimal generator of a Markov C 0 -semigroup (P t ) t≥0 on x ∈ E, t ≥ 0, be its representing measures from Theorem 3.3. Clearly, (μ t (x, ·)) t≥0 ∈ C([0, ∞), M + κ (E)),μ t (x, E) = 1 for all t ∈ [0, ∞) and (5.9) holds withμ t (x, ·) replacing ν t for all x ∈ E and by Theorem 5.5 (more precisely, (5.7)) it solves (5.8). Hence the assertion follows. Now let us start with an example on a finite-dimensional state space. In fact, the corresponding SDE on R d and the assumptions on the coefficients are the standard ones. So, in this "generic case" our theory of C 0 -semigroups on C κ (E), τ M κ applies and thus identifies the corresponding Kolmogorov operator L 0 with domain D(L 0 ) = C 2 b (R d ) as a Markov core operator for the infinitesimal generator of the C 0 -semigroup on C κ (E), τ M κ given by the transition semigroup of the solutions to the SDE. To the best of our knowledge in this generality this is the first result confirming that the Kolmogorov operator determines the (truly) infinitesimal generator of the said transition semigroup of the Markov process given by the SDE's solution. This appears to have been an open problem for many years.

Applications to SDEs on R d .
Let E := R d and (Ω, F , P) be a complete probability space with normal filtration F t , t ≥ 0, and (W t ) t≥0 be a (standard) (F t )-Wiener process on R d 1 . Let M (d × d 1 , R) denote the set of real d × d 1 -matrices equipped with the Hilbert-Schmidt norm · and let σ : be continuous maps satisfying the following standard assumptions. There exist K ∈ L 1 loc ([0, ∞)) and C ∈ [0, ∞) such that for all R ≥ 0, Here , denotes the Euclideam inner product on R d and | · | the corresponding norm. Then it is well-known (see e.g. [49, Section 3] and the references therein) that the SDE dX(t) = b(X(t))dt + σ(X(t))dW (t), X(0) = x ∈ R d , (5.12) has a unique strong solution X(t, x), t ≥ 0, such that for p ≥ 2 there exists C T,p ∈ [0, ∞) such that |X(t, x)| p ] ≤ C T,p (1 + |x| p ), (5.13) where E denotes expectation w.r.t.P. Indeed, (5.13) is a direct consequence of (5.11) and Itô's formula. Let κ(x) := (1 + |x| m ) −1 (5.14) (4.4)). By [49, Proposition 3.2.1] exactly the same arguments, which prove Claim 2 in Section 4.1, imply that (P t ) t≥0 is a Markov C 0 -semigroup on C κ (R d ), τ M κ . Let (L, D(L)) be its infinitesimal generator and let us consider the Kolmogorov operator (L 0 , D(L 0 )) corresponding to (5.12), defined as we need one more condition on b and σ, namely we additionally assume Here we note that by (5.18) for some c 0 ∈ (0, ∞) So, since µ s (x, dy), x ∈ R d , s ∈ [0, ∞), satisfy (3) in Theorem 3.3 by (5.13), the above use of Fubini's Theorem is justified and L 0 ϕ ∈ C κ (R d ). (5.19) implies that for all We would like to stress at this point that at least if κ = 1, the proof of (5.17) is completely standard (as it is also in Section 5.4 below), as far as the part (5.22) is concerned, while part (5.23) is to be taken care of case by case.
To show that (L 0 , D(L 0 )) is a Markov core operator for (L, D(L)), we furthermore assume that Each σσ T i,j is locally in V M O(R d ). where B r (x) denotes the ball in R d of radius r, centered at x ∈ R d , and |B r (x)| its Lebesgue measure. g belongs locally to the class Under the assumptions (5.10), (5.11), (5.18), (5.24), (5.25) on the continuous maps b and σ above, it now follows by Proposition 5.9 and Theorem 9.3.6 in [9] that the Kolmogorov operator (L 0 , D(L 0 )) in (5.16) corresponding to SDE (5.12) is a Markov core operator for (L, D(L)). In this case one also says that Markov uniqueness holds for (L 0 , D(L 0 )) on Now let us give an example on an infinitely dimensional state space, where we even have that the Kolmogorov operator (L 0 , D(L 0 )) is a core-operator for (L, D(L)).

Applications to generalized Mehler semigroups (or OU-processes with Levy noise) on Hilbert spaces.
Let E be a separable Hilbert space with inner product · , · and norm · E and let us come back to Section 4.3, i.e., (P t ) t≥0 is the semigroup defined in (4.17), which as shown there, is a C 0 -semigroup on (C b (E), τ M 1 ), (so κ ≡ 1). In order to calculate its infinitesimal generator (L, D(L)) explicitly on a core domain, we need some assumptions. Let λ : E → C satisfy the following hypothesis: (H1) λ is negative definite and Sazonov continuous with λ(0) = 0. We refer e.g. to [34,Section 2] for the corresponding definitions. Then, as is well-known (cf., e.g., [55,Theorem VI. 4.10]), λ posesses a unique Levy-Khintchin representation of the form (5.28) implies thatμ t+s (ξ) =μ s (ξ)μ t (T * s ξ), ξ ∈ E , t, s ≥ 0, which in turn is equivalent to (4.16). Hence for such µ t , P t , t ≥ 0 defined as in (4.17), form indeed a generalized Mehler semigroup, hence by Section 4.3 a Markov C 0 -semigroup on C κ (E), τ M 1 . Now we want to identify its generator (L, D(L)) on a convenient and large enough domain which was suggested in [47]. Let us recall its definition. Let W 0 be the set of functions ϕ that have a representation of the form Let us now recall one of the main results in [47], for which we need to assume the following condition: (H2) There exists an orthonormal basis {ξ n |n ∈ N} of E, consisting of eigenvectors of the adjoint operator A * of A on E.
From this it is easy to see that W is an algebra separating the points of E. Note also that by definition of the Fréchet derivative ϕ of ϕ it follows that ϕ (x) ∈ D(A * ) for all x ∈ E.
From this result if follows easily that for (L 0 , D(L 0 )) we have L 0 ⊂ L. Indeed, by Theorem 5.10 and its consequence that and for all t ∈ [0, 1] (where we recall that κ ≡ 1). Hence Theorem 5.5 implies that D(L 0 ) ⊂ D(L) and L 0 ϕ = Lϕ for all ϕ ∈ D(L 0 ), i.e., Now we shall prove that (L 0 , D(L 0 )) is a core operator for (L, D(L)). For this, according to Proposition 5.3 it suffices to prove Remark 5.11. If λ, restricted to span{ξ 1 , · · · , ξ n }, is infinitely often differentiable for all n ∈ N, then by [47, Theorem 1.3(i)] Hence (5.32) holds. But this is in general not true for general λ as above.
So, to prove (5.32) let us fix t > 0 and ϕ ∈ D(L 0 ). Because P t is linear, we may assume that ϕ is of type (5.29) with f real-valued. It is easily seen that such a ϕ can be approximated in the τ M κ -graph topology of L on D(L) by ϕ n , n ∈ N, of type (5.29) with the corresponding f n ∈ S(R m , C) being Fourier transforms of f n,0 ∈ S(R m , C) with compact supports. Hence we may assume that ϕ is of the form (5.29) with compactly supported f 0 . Consider the following approximation of λ (see (5.26)) for ∈ (0, 1) Obviously, M is again a Lévy measure on (E, B(E)), and λ ε satisfies (H1). Let µ (ε) t , t ≥ 0, be defined analogously to µ t , t ≥ 0, through (5.28) with λ ε replacing λ, and P (ε) . Since by [47,Proposition 3.3], each λ ε fulfills the condition of Remark 5.11, the latter implies (5.35) in the τ M 1 -graph topology of L on D(L), we obtain that P t ϕ ∈ D(L 0 ) L and (5.32) is proved.
(5.35) follows from the following two claims, under the additional condition (5.36) in Claim 1, which is, however, is always fulfilled, if λ is real-valued (see Lemma 5.12 below). Claim 1. Assume that By Claim 1 we have that as ε → 0 the first summand converges to zero in (C b (E), τ M 1 ). For the second summand we have Defining L and hence By [47,Lemma 3.1], λ ε → λ uniformly on bounded subsets of E as ε → 0, so by (5.42) and because f 0 has compact support, we obtain lim ε→0 (L 0 − L (ε) )ϕ 1 = 0, so by (3.1) and Remark 3.2 (ii) the second summand in the r.h.s. of (5.40) also converges to zero in (C b (E), τ M 1 ). Now let us turn to the third summand. So, let x ∈ E, ε ∈ (0, 1). Then by an elementary calculation (see [47, (1.2) By (H2) we know that A * ξ j = α j ξ j , j ∈ N, for some α j ∈ R. Hence by Euler's formula for T * t = e A * t we have We note that and that g ε ∈ S(R m , C) with g ε (−v) = g ε (v), v ∈ R m . Hence P (ε) t ϕ ∈ D(L 0 ), and by (5.41) analogously to (5.42) we obtain Hence by [47, Lemma 3.1], (5.44) and because suppf 0 is compact, we obtain and Claim 2 is proved.
In the above example the Kolmogorov operator (see (5.30)) was a pseudo differential operator on E with symbol λ only dependent on ξ (not on x), i.e., constant diffusion, and linear drift. Therefore, finally we give an example on infinite dimensional state space E, but where the Kolmogorov operator is a partial differential operator with non-constant second order (=diffusion) coefficients and nonlinear first order (=drift) coefficients.

Applications to SDEs on Hilbert spaces of locally monotone type.
Consider the situation of Section 4.1 (so E := H := a separable Hilbert space H which is the pivot space of a Gelfand-triple V ⊂ H ⊂ V * as defined there). Let the coefficients A and B be independent both of ω ∈ Ω and t ∈ [0, T ] and that U = H. Assume that B satisfies (4.3) and we assume that A can be written as a sum of two operators C and F . More precisely, let (C, D(C)) be a self-adjoint operator on H such that −C ≥ θ 0 ∈ (0, ∞). Define V := D((−C) 1 2 ), equipped with the graph norm of (−C) 1 2 , and V * to be its dual. Then it is easy to see that C extends uniquely to a continuous linear operator from V to V * , again denoted by C such that for all u, v ∈ V Define the Kolmogorov operator associated to SDE (4.1) with B as above and A as in (5.47) with domain D(L 0 ) as follows: Here ∂ ∂e i denotes partial derivative in directions e i and we note that all sums in (5.49) are in fact finite sums, since ϕ ∈ D(L 0 ). Now let us prove that For this we need one more condition, i.e. we assume: The eigenbasis of (C, D(C)) above can be chosen in such a way Remark 5.13. (i) A typical example for F : H → V * above is a demicontinuous function (i.e., x → V * F (x), u V is continuous on H for all u ∈ V ) with F (0) ∈ H, which is one sided Lipschitz and of at most polynomial growth. Under condition (5.51) a straightforward application of Itô's formula for Itô-processes in R N , N ∈ N, yields for all ϕ ∈ D(L 0 ) and for the solution X(t, x), t ≥ 0, x ∈ E, to (4.1) with A and B as above: Now exactly the same arguments as in Section 5.2 prove that (5.50) holds.
To prove that (L 0 , D(L 0 )) is a Markov core operator for (L, D(L)) on (C κ (H), τ M κ ) we shall again use Proposition 5.9, i.e. we have to prove uniqueness for the corresponding Fokker-Planck-Kolmogorov equation, which is in general very difficult here, since our state space H is infinite dimensional, and more assumptions are needed. Though there are such results also when B depends on x (see [7]), for simplicity we shall assume that B is constant. More preasely, we additionally assume that: for all x ∈ V with B = B * , B non-negative definite with kerB = {0}, and that for the eigenvalues α k ∈ (0, ∞), k ∈ N, of B. There exists m ∈ [1, ∞) such that 6. Convex C 0 -semigroups on (C κ (E), τ M κ ) We now draw our attention to C 0 -semigroups on (C κ (E), τ M κ ) consisting of convex increasing operators on C κ (E). We show that these lead to viscosity solutions to abstract differential equations that are given in terms of their generator. We start by introducing our notion of a viscosity solution for abstract differential equations of the form (6.1) u (t) = Lu(t), for all t > 0.
Note that the previous definition does, a priori, not require the class of test functions for a viscosity solution to be rich in any sense. Therefore, in order to obtain uniqueness in standard settings, one has to verify on a case by case basis that the operator L is defined on a sufficiently large set D in order to apply standard comparison methods. Concerning the existence of D-viscosity solutions, we have the following theorem.
In order to show that u is a viscosity supersolution, let ψ : (0, ∞) → C κ (E) differentiable with ψ(t) ∈ D(L), ψ(t) (x) = u(t) (x) and ψ(s) ≤ u(s) for all s > 0. Again, using the semigroup property, we find that, for all h ∈ (0, 1) with h < t, where, in the last step, we used the convexity of the map Again, the strong continuity of the semigroup P and ψ t ∈ D(L) imply that as h ↓ 0 in the mixed topology τ M κ . Since u(t) (x) = ψ(t) (x), we find that 0 ≤ − Lψ(t) (x) + ψ (t) (x), and the proof is complete.
Finally, we derive a stochastic representation for P using convex expectations. For a measurable space (Ω, F), we denote the space of all bounded F-measurable functions (random variables) Ω → R by B b (Ω, F). For two bounded random variables X, For a constant m ∈ R, we do not distinguish between m and the constant function taking that value.
. We say that (Ω, F, E) is a convex expectation space if there exists a set of probability measures P on (Ω, F) and a function α : P → [0, ∞) such that where E P (·) denotes the expectation w.r.t. to the probability measure P.
In particular, for all t ≥ 0, x ∈ E, and ϕ ∈ C b (E).
Let E be a Polish space. The quadruple (Ω, F, (E x ) x∈E , (X(t)) t≥0 ) can be seen as a nonlinear version of a Markov process. As an illustration, we consider the case, where the semigroup P and thus E x is linear for all x ∈ E, and choose ψ(x, y) = ϕ(x)1 B (y), for x, y ∈ E, with ϕ ∈ C b (E) and B ∈ B n , where B n denotes the product σ-algebra of the Borel σ-algebra B.
Then, E x = E P x is the expectation w.r.t. a probability measure P x on (Ω, F) for all x ∈ E. Using the continuity from above and Dynkin's lemma, Equation (6.2) reads as E P x ϕ(X(t))1 B (X(t 1 ), . . . , X(t n )) = E P x P t−s ϕ (X(s))1 B (X(t 1 ), . . . , X(t n )) , which is equivalent to the Markov property (6.4) E P x ϕ(X(t))|F s = P t−s u X(s) P x -a.s., where F s := σ {X(u) | 0 ≤ u ≤ s} . On the other hand, if E x = E P x , the Markov property (6.4) implies Property (iii) from Theorem 6.4.

7.
Examples: value functions of optimal control problems 7.1. A finite-dimensional setting. In this section, we show that value functions of a large class of optimal control problems are examples of nonlinear C 0 -semigroups. We illustrate this by means of a simple controlled dynamics in R d , with d ∈ N where the contol acts on the drift of a diffusion process. However, with similar techniques also other classes of controlled diffusions fall into our setup. Throughout, let W = (W (t)) t≥0 be a Brownian Motion on a complete filtered probability space Ω, F, (F t ) t≥0 , P satisfying the usual assumptions and σ > 0. For m ∈ N, we consider a fixed nonempty set A ⊂ R m of controls with 0 ∈ A and define the the set of admissible controls A as the set of all progressively measurable processes For a fixed measurable function b : Ω × R d × A → R d , an admissible control α ∈ A, and an initial value x ∈ R d , we consider the controlled dynamics We assume that the drift term b satisfies the following Lipschitz and growth conditions: there exists a constant C ≥ 0 such that b(x, 0) = 0, P-a.s., for all x ∈ R d , |b(x, a)| ≤ C 1 + |x| + |a| , P-a.s., for all x ∈ R d and a ∈ A.
Under these assumptions, by standard SDE theory, for each initial value x ∈ R d and every admissible control α ∈ A, there exists a unique strong solution (X α (t, x)) t≥0 to the controlled SDE (7.1). We consider the weight function κ ≡ 1 and a running cost function g : A → [0, ∞) with g(0) = 0 and g * (y) := sup a∈A |a|y − g(a) < ∞ for all y ≥ 0. For ϕ ∈ C b (R d ), we consider the value function and we define P t ϕ (x) := V (t, x; ϕ) for all t ≥ 0 and x ∈ R d . We first show that P t : Using the Lipschitz condition of b together with Gronwall's lemma, we obtain the a priori estimate for all t ≥ 0, x 1 , x 2 ∈ R d , and α ∈ A. This shows that the value function V is continuous in the x-variable. Moreover, V (t, · , ϕ) ∞ ≤ ϕ ∞ for all ϕ ∈ C b (R d ),since g(0) = 0. Since the value function V satisfies the dynamic programming principle, cf. Pham [58] or Fabbri et al. [29], the family P = (P t ) t≥0 is a semigroup.
Using the linear growth of b together with Gronwall's lemma,
Hence, for all x ∈ R d with |x| ≤ r, Equation 7.2 implies that On the other hand, for all x ∈ R d with |x| ≤ r, We thus see, that P t ϕ → ϕ uniformly on compact sets. Now, let R ≥ 0, ε > 0, and ϕ 1 , for sufficiently large r > 0.
where, in the last step, we used Equation (7.2) and Equation (7.4). Choosing r > 0 sufficiently large, a symmetry argument yields that With similar arguments together with Itô's formula, one finds that the generator L of P on C 2 b (E) is given by By Theorem 6.2, we thus obtain that (t, x) → V (t, x; ϕ) = P t ϕ (x) is a C 2 b (R d )-viscosity solution to the HJB equation , v(0, x) = ϕ(x).

7.
2. An infinite-dimensional example with linear growth. In this section, we consider a similar setup as in the previous subsection in a separable Hilbert space H with orthonormal base (e k ) k∈N ⊂ H, endowed with the bw-topology. Throughout, let W = (W (t)) t≥0 be a Brownian Motion with trace class covariance operator Σ : H → H on a complete filtered probability space Ω, F, (F t ) t∈[0,T ] , P satisfying the usual assumptions. For p ∈ (1, 2], we define the the set of admissible controls A as the set of all progressively measurable processes For every admissible control α ∈ A and every initial value x ∈ H, we consider the controlled dynamics We For ϕ ∈ C κ (H bw ), we consider the value function and we define P t ϕ (x) := V (t, x; ϕ) for all t ≥ 0 and x ∈ H. We first show that P t : C κ (H bw ) → C κ (H bw ) is well-defined. Let ϕ ∈ C b (H bw ) such that,there exists a constant L ≥ 0 and some n ∈ N with Then, for all t ≥ 0 and x, y ∈ H Moreover, for all q ∈ [1, p], Moreover, for all x ∈ H, ϕ(x) − V (t, x; ϕ) ≤ L Σ tr √ t. In particular, since the bounded finitely based Lipschitz functions are dense in C κ (H bw ), it follows that P t ϕ → ϕ uniformly on compacts for all ϕ ∈ C κ (H bw ).
By Itô's formula and Theorem 6.2, we obtain that (t, x) → V (t, x; ϕ) = P t ϕ (x) is a C 2 b (H bw )-viscosity solution to the HJB equation Appendix A. Some facts on the mixed topology In this section we collect some general properties of the mixed topology that was introduced in Section 2 in a special case suitable for our purposes. For the reader's convenience, we start with some basic facts about topological vector spaces (see e.g. [64]).
Let (X, τ ) be a topological vector space. Then, there exists a basis U (τ ) of neighbourhoods of zero such that: (i) if U ∈ U (τ ) and λ ∈ R, then λU ∈ U (τ ), (ii) if U ∈ U (τ ) and |λ| ≤ 1, then λU ∈ U (τ ), (iii) if U ∈ U (τ ) and x ∈ X, then there exists some λ ∈ R such that x ∈ λU , The topological vector space (X, τ ) is Hausdorff if, for every x ∈ X, there exists a neighbourhood U ∈ U (τ ) of zero such that x / ∈ U . A topological vector space (X, τ ) is locally convex if there exists a basis U (τ ) of neighbourhoods of zero consisting of convex sets satisfying the properties (i) -(v) above. In what follows, all topological vector spaces are Hausdorff and locally convex. We say that a subset B ⊂ X is bounded if, for every U ∈ U (τ ), there exists some r > 0 such that rB ⊂ U .
We recall the definition of the mixed topology τ M . We closely follow [72], where a more general situation is studied. Let X be a linear space, endowed with two topologies τ 1 and τ 2 , with corresponding bases U (τ 1 ) and U (τ 2 ) of neighbourhoods of zero satisfying (i) -(v), respectively. We assume that X, τ 1 and X, τ 2 are Hausdorff topological vector spaces with τ 1 ⊂ τ 2 . For a sequence γ = U 1 n ⊂ U (τ 1 ), and any U 2 ∈ U (τ 2 ), we define a set Then, the family U γ, U 2 : γ = U 1 n ⊂ U τ 1 , U 2 ∈ U τ 2 satisifies the conditions (i) -(v), and therefore defines a basis of neighbourhoods of zero for a topology τ M = τ M τ 1 , τ 2 making (X, τ M ) Hausdorff topological vector space. The topology τ M is known as the mixed topology In the present paper, we use this definition only in the case, where X = C κ (E) with E a completely regular topological Hausdorff space, τ 1 = τ C κ and τ 2 = τ U κ , cf. Section 2 for the notations. We list some basic properties of the mixed topology in this case. is identical with the topology τ M κ defined in Section 2 via the family of seminorms p κ,(Cn),(an) . Proof. For κ ≡ 1, this lemma has been proved in [72]. The case of arbitrary κ is an easy modification of the proof in [72].
Let us recall that a subset of a locally convex space is bounded if it is absorbed by every neighbourhood of zero. The topology τ M κ can be defined as the weakest topology τ on C κ (E) such that for every locally convex space F and every linear operator T : C κ (E) → F , T is τ -continuous if and only if T is τ C κ -continuous on τ U κ -bounded sets. Another equivalent definition of the mixed topology τ M κ is given by the following construction. Let W 0 (E) denote the family of weights consisting of all bounded functions w : E → [0, ∞) such that, for every ε > 0, the set {x ∈ E : κ(x)w(x) ≥ ε} is compact. For every weight w ∈ W 0 (E) we define the seminorm The locally convex topology on C κ (E) defined by the family of seminorms {p κ,w : w ∈ W 0 (E)} is identical with the topology τ M κ . The following result is a special case of Theorem 10.6 in [72]. Since in our case the proof is simple, we include it fot the reader's convenience.
Lemma A.5. For each weight κ, the mapping for all three topologies τ U 1 , τ C 1 , τ M 1 and τ U κ , τ C κ , τ M κ , respectively. Proof. The proof is obvious for τ C κ and τ U κ . By Proposition A.2(c), it is enough to check continuity of I κ and I −1 κ on balls of C b (E) and C κ (E) respectively. But this follows from Proposition A.2(a). (1) sup ϕ∈B ϕ κ < ∞ , (2) B is equicontinuous on every compact subset of E.
Theorem A.8. ( [33,Theorem 11]) Suppose that A ⊂ C κ (E) is an algebra that separates the points of E, and that, for each x ∈ E, there exists a ∈ A with a(x) = 0. Then A is τ M κ -dense in C κ (E). Proof. For κ = 1, the theorem was proved in [33]. For general κ, it follows from Lemma A.5.
Theorem A.9. The space M κ (E) is the topological dual space of C κ (E), τ M κ . Moreover, if M ⊂ M κ (E) such that the set of measures κ −1 µ; µ ∈ M is tight and bounded in variation norm, then M considered as a set of functions on C κ (E) is τ M κ -equicontinuous. Proof. If κ = 1 then by [46] (see also [33]) the dual space of C 1 (E), τ M 1 is identified as the space of Baire measures of finite variation on E for any completely regular space E. In this paper Borel and Baire σ-algebras on E coincide, hence the first claim follows. For the proof of the second part of the assertion see [16,p. 136,Proposition 3.6] . For arbitrary κ the proof follows from Lemma A.5.
Remark A. 10. We note that if, in addition, to Hypothesis 2.1 we assume that our underlying space E is Radon, i.e. every finite measure on (E, B(E)) is tight, which is the case in all examples in this paper, then the first assertion of Theorem A.9 is a trivial consequence of the Daniell-Stone Theorem (see e.g. [21]). Indeed, again by Lemma A.5 we may assume that κ = 1. Obviously, each µ ∈ M b (E) is in the topological dual C b (E), τ M To prove the converse we first note that it is well-known that every element of the latter can be written as a difference = + − − , with + , − ∈ C b (E), τ M 1 and both are nonnegative on nonnegative elements in C b (E) (see e.g. [44]). Hence we may assume that itself has this property. Since C b (E) is a Stone vector lattice, which by assumption (2) in Hypothesis 2.1 generates B(E), we only have to show the Daniell continuity, because then is represented by a unique finite nonnegative measure µ, which, since E is a Radon space, is in M b (E). But if ϕ n ∈ C b (E), ϕ n ≥ 0, n ∈ N, such that ϕ n ↓ 0 pointwise on E, then by Proposition A.4 and Dini's Theorem, we conclude that |µ|(x, dy) κ(y) < ε Proof. We first prove the theorem for the case κ ≡ 1.
(i) ⇒ (ii): Assume (i). We start by showing (a). By Proposition B.1, for every x ∈ E, the functional l x (ϕ) = T ϕ(x) is continuous in the topology τ M 1 . Therefore, by Theorem A.9, there exists a measure µ(x, · ) ∈ M 1 (E) such that l x (ϕ) = T ϕ(x) = E ϕ(y)µ(x, dy), which proves (a). In order to prove (b), let U ⊂ E be open. Let R ∞ denote the Polish space of infinite sequences of real numbers. Since the Baire σ-algebra Ba(E) is identical with the Borel σ-algebra B(E), by [5,Lemma 6.3.3], there exists an open set V ⊂ R ∞ and a continuous function f : E → R ∞ such that U = f −1 (V ). Without loss of generality, we may assume that the measures µ(x, · ) are non-negative. Since R ∞ is a Polish space, there exists a sequence (ϕ n ) ⊂ C b (R ∞ ) such that 0 ≤ ϕ n ≤ 1 and lim n→∞ ϕ n (z) = I V (z). By the dominated convergence theorem, µ(x, U ) = Hence, the function x → µ(x, U ) is Borel measurable as a pointwise limit of continuous functions. Finally, the measurability of the function µ( · , B) for any Borel set B ⊂ E follows from Dynkin's lemma. Next, we prove (c). Invoking the lattice properties of C b (E) and M b (E), we have which shows that (B.1) holds. Let K 1 ⊂ E be compact. Since T is τ M 1 -continuous, for every ε > 0 there exists a τ M 1neighbourhood of zero such that, for ϕ ∈ U , we have p K 1 (T ϕ) < ε. For x ∈ E, we have T ϕ(x) = l x (ϕ) and p K 1 (T ϕ) = sup  This shows that T ϕ ∈ U , for every ϕ ∈ U 1 , and concludes the proof for κ ≡ 1. For general κ, observe that, if (ii) holds, then is τ M 1 -τ M 1 -continuous on C b (E) by the first part of the proof. Hence, by Lemma A.5, T is τ M κ -τ M κ -continuous on C κ (E). The converse implication follows by a similar argument.
Corollary B.3. Assume that a linear operator T : C κ (E) → C κ (E) is τ M κ -τ M κ -continuous. Then T is positive if and only its representing measures µ(x, · ) are a non-negative for every x ∈ E.
It follows from Theorem B.2 that every norm-bounded τ M κ -τ M κ -continuous linear operator on C κ (E) can be extended to a linear operator from κ −1 B b (E) to κ −1 B b (E), where B b (E) refers to the space of all bounded Borel measurable functions.