Stochastic Processes under Parameter Uncertainty

In this paper we study a family of nonlinear (conditional) expectations that can be understood as a stochastic process with uncertain parameters. We develop a general framework which can be seen as a version of the martingale problem method of Stroock and Varadhan with parameter uncertainty. To illustrate our methodology, we explain how it can be used to model nonlinear L\'evy processes in the sense of Neufeld and Nutz, and we introduce the new class of stochastic partial differential equations under parameter uncertainty. Moreover, we study properties of the nonlinear expectations. We prove the dynamic programming principle, i.e., the tower property, and we establish conditions for the (strong) $\textit{USC}_b$-Feller property and a strong Markov selection principle.


Introduction
We study conditional nonlinear expectations on the path space Ω of continuous or càdlàg functions from R + into a Polish space F .More specifically, we are interested in nonlinear expectations of the form E t (ψ)(ω) = sup P ∈C(t,ω) where ψ is a suitable real-valued function on Ω and C(t, ω) is a set of laws of stochastic processes whose paths coincide with ω ∈ Ω till time t ∈ R + .We think of the nonlinear conditional expectation E as a nonlinear stochastic process or a stochastic process under parameter uncertainty.
The systematic study of nonlinear stochastic processes started with the seminal work of Peng [45,46] on the G-Brownian motion.More recently, larger classes of nonlinear semimartingales have been investigated in [8,9,10,20,24,25,32,39].At this point we also highlight the articles [17,38,42] where abstract tools for the study were developed.Next to the measure theoretic approach based on the nonlinear expectation (1.1), nonlinear Markov processes have also been constructed by analytic methods via nonlinear semigroups, see [15,36,37].This construction is based on fundamental ideas of Nisio [41].In the recent paper [33], it was shown that the two approaches are equivalent for the class of nonlinear Lévy processes from [15,39].A key property of the construction via (1.1) is its relation to a class of stochastic processes given through the set C. This connection opens the door to analyze properties of E with powerful tools from probability theory such as stochastic calculus.
In this paper, we propose a new framework for the set C which extends the nonlinear expectation approach to nonlinear stochastic processes beyond semimartingales.The framework is inspired by the martingale problem method of Stroock and Varadhan [53].Consider the family of test processes, where X is the coordinate process on Ω.The set A of test functions is often called a pregenerator.A probability measure P is said to be a solution to the martingale problem associated to A if all test processes in (1.2) are P -martingales.Our idea is to incorporate uncertainty to the pregenerator A.
Let U be a countable index set and let {Y u : u ∈ U } be a family of càdlàg processes on Ω, which serve as test processes for the martingale problem under uncertainty.If P is a probability measure such that Y u is a special P -semimartingale, then there exists a P -unique predictable càdlàg process A P (Y u ) of finite variation, starting in zero, such that Y u − Y u 0 − A P (Y u ) is a P -local martingale.In case the process in (1.2) is a (local) P -martingale, the process f (X) is a special semimartingale and A P (f (X)) = • 0 g(X s )ds.Motivated by this observation, we define C to be the set of all probability measures P under that each test process Y u is a special P -semimartingale with absolutely continuous compensator A P (Y u ) such that (λ \ ⊗ P )-a.e.
dA P (Y u )/dλ \ u∈U ∈ Θ.Here, Θ is a time and path-dependent set-valued mapping on R + × Ω, which captures the uncertainty.
Our framework is tailor made for adding uncertainty to any class of stochastic processes which can be characterized by a martingale problem.For instance, this is the case for semimartingales (with absolutely continuous characteristics), solutions to stochastic partial differential equations and many Markov processes.In particular, it is possible to recover nonlinear semimartingale settings which are built from suitably parameterized absolutely continuous semimartingale characteristics.To illustrate this, we explain how nonlinear Lévy processes, in the sense of [39], can be modeled with our framework.Further, we introduce a novel class of nonlinear Markov processes which are not necessarily semimartingales: the class of nonlinear (in the sense of uncertainty) stochastic partial differential equations (NSPDE).We think this class is of specific interest and deserve further investigation.
The first main result in this paper is the dynamic programming principle (DPP) for the nonlinear expectation (1.1), i.e., the tower property In case the uncertainty set Θ has a Markovian structure in the sense that Θ(t, ω) depends on (t, ω) only through the value ω(t−), the DPP induces a nonlinear Markov property given by As in the theory of (linear) Markov processes, there is a strong link to semigroups.Indeed, in this Markovian case, the nonlinear Markov property ensures the semigroup property T t T s = T t+s of the family (T t ) t≥0 defined by where ψ runs through the class of bounded upper semianalytic functions.It is an important question when the semigroup (T t ) t≥0 preserves some regularity.For a continuous path setting, we provide general conditions for the USC b -Feller property of the semigroup (T t ) t≥0 , which means that it is a sublinear Markovian semigroup on the space of bounded upper semicontinuous functions.Further, we establish a strong Markov selection principle, i.e., we prove that, for every bounded upper semicontinuous function ψ : F → R and any time t > 0, there exists a (time inhomogeneous) strong Markov family {P (s,x) : (s, x) ∈ R + × F } such that P 0,x ∈ C(0, x) and T t (ψ)(x) = E P (0,x) ψ(X t ) .
We stress that the nonlinear structure of our setting is reflected by the fact that the strong Markov selection {P (s,x) : (s, x) ∈ R + × F } depends on the input elements ψ and t.
Thereafter, we investigate the Markovian class of NSPDEs in more detail.We establish continuity and linear growth conditions for the USC b -Feller property and the strong Markov selection principle.Moreover, for the subclass of infinite-dimensional nonlinear Cauchy problems with drift uncertainty, we derive conditions for the strong USC b -Feller property, which means that T t (USC b ) ⊂ C b for all t > 0. The strong USC b -Feller property can be seen as a smoothing property of the sublinear semigroup (T t ) t≥0 .To the best of our knowledge, such an effect was first discovered in [8] for nonlinear one-dimensional diffusions.We also emphasis that the strong USC b -Feller property entails the C b -Feller property, i.e., T t (C b ) ⊂ C b for all t ∈ R + .This property is considered to be of fundamental importance for the study of sublinear Markovian semigroups via pointwise generators, see [9,Section 2.4] for some comments.
Let us now comment on some technical aspects of our proofs.To establish the DPP, we invoke a general theorem from [17] which requires that we check a measurable graph condition and stability under conditioning and concatenation.Hereby, we benefit from technical ideas developed in [8,39].In the context of classical optimal control, the relation of a value function and a nonlinear semigroup was discovered in [40].For a nonlinear Markov process framework, the Markov property (1.3) appeared, to the best of our knowledge, for the first time in the thesis [24] for nonlinear Markovian semimartingales.We adapt the proof from [24] to our setting.For the USC b -Feller property, we rely on the theory of correspondences and, for the strong Markov selection principle, we invoke a technique from [16,22] for controlled diffusions, which is based on ideas of Krylov [31] and Stroock and Varadhan [53] about Markovian selection.Again, we also benefit from technical ideas developed in [9] for one-dimensional nonlinear diffusions.The strong USC b -Feller property for nonlinear Cauchy problems is proved by means of a strong Feller selection principle, which adapts ideas from [9,10,53] for finite-dimensional (nonlinear) diffusions to an infinite-dimensional non-semimartingale setting.
We end this section with a summary of the structure of this paper.In Section 2, we introduce our framework and formulate the DPP.Thereafter, in Section 3, we comment on nonlinear Lévy processes and we introduce the new class of NSPDEs.We discuss nonlinear Markov processes in Section 4. In particular, in Section 4.2, we establish the USC b -Feller property and, in Section 4.3, we provide the strong Markov selection principle.In Section 5, these results are tailored to the special case of NSPDEs and the strong USC b -Feller property for nonlinear Cauchy problems with drift uncertainty is established.The remaining sections of this paper are devoted to the proofs of our results.The location of each proof is indicated before the statement of the respective result.

Nonlinear Processes and the Dynamic Programming Principle
2.1.The Ingredients.Let F be a Polish space and define Ω to be either the space of all càdlàg functions R + → F or the space of all continuous functions R + → F endowed with the Skorokhod J 1 topology. 1The canonical process on Ω is denoted by X, i.e., X t (ω) = ω(t) for ω ∈ Ω and t ∈ R + .It is well-known that F B(Ω) = σ(X t , t ≥ 0).We define F (F t ) t≥0 to be the canonical filtration generated by X, i.e., F t σ(X s , s ≤ t) for t ∈ R + .To lighten our notation, for two stopping times S and T , we also define the stochastic interval The set of all probability measures on (Ω, F ) is denoted by P(Ω) and endowed with the usual topology of convergence in distribution, i.e., the weak topology.The shift operator on Ω is denoted by θ = (θ t ) t≥0 , i.e., θ t (ω) = ω( • + t) for all ω ∈ Ω and t ∈ R + .Let K be either the real line R or the complex plane C. Further, let U be a countable index set and, for every u ∈ U , let Y u be a K-valued càdlàg F-adapted process on Ω such that Y u •+t = Y u • θ t for all t ∈ R + .We endow K U with the product Euclidean topology.Finally, let Θ : R + × Ω ։ K U be a correspondence, i.e., a set-valued mapping.
Standing Assumption 2.1.The correspondence Θ has a measurable graph, i.e., the graph gr Θ = (t, ω, g where M is a local martingale and A is a predictable process of (locally) finite variation, both starting in zero.The decomposition is unique up to indistinguishability.A K-valued càdlàg process Y is called a special semimartingale after a time t * ∈ R + if the process Y •+t * = (Y t+t * ) t≥0 is a special semimartingale for the right-continuous natural filtration of X •+t * = (X t+t * ) t≥0 .For P ∈ P(Ω), the set of special semimartingales after time t * on the probability space (Ω, F , P ) is denoted by S sp (t * , P ).We also write S sp (P ) S sp (0, P ).For a given measure P and a given process Y ∈ S sp (t * , P ), denote by A P (Y •+t * ) the predictable part of (locally) finite variation in the semimartingale decomposition of Y •+t * , and denote by S sp ac (t * , P ) the set of all Y ∈ S sp (t * , P ) such that A P (Y •+t * ) is absolutely continuous w.r.t. to the Lebesgue measure λ \ .Again, we write S sp ac (P ) S sp ac (0, P ).For ω, ω ′ ∈ Ω and t ∈ R + , we define the concatenation ) depends only on the path ω up to time t.
Discussion (Relation to Martingale Problems).The idea behind the set C is inspired by the fact that many classes of stochastic processes can be modeled via martingale problems.Take a set A ⊂ C b (F ; R) × B(F ; R), which is often called pregenerator.Here, B(F ; R) denotes the set of all bounded Borel functions F → R. Following the monograph [19], a probability measure P on (Ω, F ) is said to solve the martingale problem associated to A if all processes of the form are P -martingales.To see the connection to the set C, suppose that P solves the martingale problem associated to A. Then, for every (f, g) ∈ A, the process f (X) is a special Psemimartingale, as it decomposes into the P -martingale given in (2.1) and the continuous (hence, predictable) finite variation process • 0 g(X s )ds, which, in particular, means that dA P (f (X))/dλ \ = g(X).Roughly speaking, the set C mimics the idea of the martingale problem but it allows some part of the pregenerator, namely the density of the compensator, to take values in the set Θ, which incorporates uncertainty.For the important special case where Θ is generated by a controlled coefficient, this interpretation is made explicit by the following proposition.
Proposition 2.4.Let G be a metrizable Souslin space 2 and assume that there exists a predictable Carathéodory function z = (z u ) u∈U : G × [[0, ∞[[ → K U , i.e., z is continuous in the first and predictable in the second variable, such that Here, we still assume that Standing Assumption 2.
By Standing Assumption 2.
For general martingale problems, the pregenerator A is not necessarily a countable set.In the diffusion-type setting of Stroock and Varadhan [53], for example, the pregenertor A is given by where b : R d → R d and σ : R d → R d×r are Borel measurable coefficients.In the definition of the set C we take only countably many test functions into account.In typical situations this is no restriction, as it is often possible to pass to a countable subset of the pregenerator that determines the same martingale problem.For example, in the diffusion-type setting explained above, it is sufficient to consider the set cf. [30,Proposition 5.4.6].We also refer to [19,Proposition 4.3.1]for an abstract result in this direction.In Section 3.1 below, we will also discuss a way to characterize the class of nonlinear Lévy processes as introduced in [39] with a countable pregenerator under parameter uncertainty.Let us also comment on the question to what extend the pregenerator has to determine the martingale problem uniquely.In the classical theory on Feller-Dynkin processes, the (pre)generators characterize the law of the processes in a unique manner, cf., for example, [34,Theorem 3.33].In the classical theory on martingale problems (as in [19,53], for instance), such a uniqueness property is an important question rather than intrinsically given.In particular, there is interest in martingale problems that are not well-posed (i.e., do not have unique solutions for each deterministic initial value).Indeed, by the strong Markov selection principle (see Chapter 12 in [53]), many non-well-posed martingale problems give rise to strong Markov families which can be selected from the set of solutions.Our approach is fully independent of a uniqueness property, i.e., we can also introduce uncertainty to martingale problems that have infinitely many solutions (by convexity, a martingale problem has none, one or infinitely many solutions).In fact, considering martingale problems with infinitely many solutions can lead to interesting effects.We explain an example of such an effect in Remark 4.7 below.
2.2.Dynamic Programming Principle.We now define a nonlinear conditional expectation and we provide the corresponding dynamic programming principle (DPP), i.e., the tower property of the nonlinear conditional expectation.Suppose that ψ : Ω → [−∞, ∞] is an upper semianalytic function, i.e., the set {ω ∈ Ω : ψ(ω) > c} is analytic for every c ∈ R, and define the value function v by v(t, ω) sup P ∈C(t,ω) The following theorem is proved in Section 6. Theorem The value function v induces a nonlinear expectation E via the formula and the DPP provides its tower property, i.e., (2.3) means that By the pathwise structure of this property, it follows also immediately that for all finite stopping times 0 ≤ ρ ≤ τ < ∞.For x ∈ F , we define R(x) P ∈ P(Ω) : P (X 0 = x) = 1, ∀ u∈U Y u ∈ S sp ac (P ), (λ \ ⊗ P )-a.e.(dA P (Y u )/dλ \ ) u∈U ∈ Θ , and Thinking of classical martingale problems, we interpret {E x : x ∈ F } as a nonlinear stochastic process.In Section 4, we specify our setting to a Markovian situation and we establish more properties of {E x : x ∈ F } to justify our interpretation.Before we start this program, let us discuss some important examples.

Examples
In this section we explain how our framework can be used to model nonlinear Lévy processes in the sense of [39] and we introduce some new classes of nonlinear processes, namely the class of nonlinear Markov chains and the class of nonlinear (in the sense of uncertainty) stochastic partial differential equations (NSPDEs).
3.1.Nonlinear Lévy processes.Nonlinear Lévy processes have been introduced in [39] via an uncertain set of Lévy-Khinchine triplets.We now explain how such nonlinear processes can be constructed in our framework.Let us emphasis that the discussion can be extended to more general classes of semimartingales under parameter uncertainty.
To fix the basic setting, we consider F R d , K C, U Q d and we take Ω to be the space of all càdlàg functions R + → F = R d .Let S d be the space of symmetric non-negative definite real-valued d × d matrices.Further, let L be the space of all Lévy measure on R d , i.e., the space of all measures K on (R d , B(R d )) such that As in [38,39], we endow L with the weakest topology under which all maps are continuous.By Lemma A.1 below, the space L is Polish.Let Θ ⊂ R d × S d × L be an analytic subset, where the product space is endowed with the product topology (and S d and R d are endowed with their usual topologies), such that sup For every u ∈ Q d , we set Y u e i u,X and we define where h : R d → R d is a fixed continuous truncation function and i denotes the imaginary number.We also assume that Standing Assumption 2.1 holds, which is for instance the case when Θ is compact (by [8, Lemma 2.12]).We now relate this setting to those from [39].First, for a given P ∈ R(x 0 ), we show that the coordinate process X is an R d -valued P -semimartingale whose characteristics (B P , C P , ν P ) are absolutely continuous with densities (b P , c P , K P ) such that (λ \ ⊗ P )-a.e.(b P , c P , K P ) ∈ Θ.Take P ∈ R(x 0 ).For every k ∈ {1, . . ., d} and n ∈ N, the process sin(X (k) /n) is a P -semimartingale.Here, X (k) denotes the k-th coordinate of X.There exists a function f ∈ C 2 (R; R) such that f (sin(x)) = x for all |x| ≤ 1/2.Hence, for [28,Proposition I.4.25], it follows that X (k) is a P -semimartingale.Of course, this means that X is an R d -valued P -semimartingale.For every u ∈ R d , by Lemma A.2 below, the map is continuous.Using this observation, we deduce from Proposition 2.4 that there exists a predictable function g = g(P By virtue of [28, Theorem II.2.42], we get that P -a.s. It is clear that P -a.s. the map u → A P t (u) is continuous for every t ∈ R + .Further, thanks to (3.1), the dominated convergence theorem implies that u → t 0 A u (g(s, X))ds is continuous for every t ∈ R + .Hence, the equality in (3.2) even holds for all u ∈ R d and we may use the uniqueness lemma of Gnedenko and Kolmogorov ([28, Lemma II.2.44]) to conclude that P -a.s.(B P , C P , ν P ) ≪ λ \ with (λ \ ⊗ P )-a.e.(b P , c P , F P ) = g( • , X).
This completes the proof of the first direction.
Conversely, if P is the law of a semimartingale (for its canonical filtration), starting at x 0 ∈ R d , with absolutely continuous characteristics whose densities (b P , c P , F P ) are such that (λ \ ⊗ P )-a.e.(b P , c P , F P ) ∈ Θ, then [28, Theorem II.2.42] yields that P ∈ R(x 0 ).

Nonlinear Markov Chains.
A (continuous-time) Markov chain is a strong Markov process on some countable discrete state space F .Typically, Markov chains are modeled via Q-matrices, which encode their infinitesimal dynamics.In particular, provided the Markov chain is a Feller-Dynkin process, the Q-matrix corresponds to the generator of the chain (see [50]).In the following we adapt the underlying martingale problem (see, e.g., [51,Theorem IV.20.6] or [54,Lemma 2.4]) to a nonlinear setting.For simplicity, we will restrict our attention to nonlinear Markov chains with a finite state space F .
Let U F {1, . . ., N } with N ∈ N, let Ω be the space of all càdlàg functions R + → F , and set Y u = 1 {X=u} for u ∈ U .Denote by M q the set of all Q-matrices on F , i.e., the set of all matrices Q = (q(k, n)) N k,n=1 ⊂ R N ×N such that q(k, n) ≥ 0 for all k = n and Q1 = 0. Let G be a set, let Q = (q(k, n)) N k,n=1 : G → M q be a map and define, for (t, ω) In case G is the singleton {g 0 }, then R(x 0 ) is also a singleton whose only element is the law of the continuous-time Markov chain with starting value x 0 and Q-matrix Q(g 0 ), see, e.g., [ where W is a cylindrical Brownian motion over H 1 and µ and σ are suitable measurable coefficients.To define our setting, we rely on the cylindrical martingale problem associated to semilinear SPDEs, see, e.g., [6,11] for more details.
We denote the set of linear bounded operators from let Ω be the space of all continuous functions R + → F , take a set G and let µ : . By virtue of [7,Lemma 7.3], there exists a countable set D ⊂ D(A * ) ⊂ H 2 such that, for every y ∈ D(A * ), there exists a sequence (y n ) ∞ n=1 ⊂ D such that y n → y and A * y n → A * y.In other words, D is a countable subset of D(A * ) which is dense in D(A * ) for the graph norm.We set In Lemma 11.2 below, we show that, under suitable assumptions on the coefficients and the parameter space G, any P ∈ R(x 0 ) is the law of a mild solution processes (see [13,Section 6.1]) to an SPDE of the form This observation confirms our interpretation as an SPDE with uncertain coefficients.

Nonlinear Markov Processes
In this section we investigate a class of nonlinear Markov processes.That is, we study the family {E x : x ∈ F } under the following standing assumption.
The program of this section is the following.First, we explain the Markov property of {E x : x ∈ F }, which further leads to a sublinear Markovian semigroup (T t ) t≥0 on the cone of bounded upper semianalytic functions.Afterwards, we establish conditions for (T t ) t≥0 to be a sublinear Markovian semigroup on the cone of bounded upper semicontinuous functions and we derive a strong Markov selection principle.
To the best of our knowledge, in the context of nonlinear stochastic processes, the nonlinear Markov property was first observed in [24,Lemma 4.32] for nonlinear Markovian semimartingales, see also [9,Proposition 2.8] for a setting with continuous paths.
Similar to the fact that linear Markov processes have a canonical relation to linear semigroups, nonlinear Markov processes can be related to nonlinear semigroups.Definition 4.3.Let H be a convex cone of functions f : F → R containing all constant functions.A family of sublinear operators T t : H → H, t ∈ R + , is called a sublinear Markovian semigroup on H if it satisfies the following properties: (i) (T t ) t≥0 has the semigroup property, i.e., T s T t = T s+t for all s, t ∈ R + and For a bounded upper semianalytic function ψ : The following proposition should be compared to [24,Remark 4.33] and [9, Proposition 2.9], where it has been established for nonlinear Markovian semimartingale frameworks.Its proof can be found in Section 9.
Proposition 4.4.The family (T t ) t≥0 defines a sublinear Markovian semigroup on the set of bounded upper semianalytic functions from F into R.
Linear semigroups with suitable regularity properties are well-known to be uniquely characterized by their (infinitesimal) generators.For nonlinear Lévy processes, a unique characterization of their nonlinear semigroups via (pointwise) generators has been established in [33,Proposition 6.5].It is very interesting to prove such a result also for other classes of nonlinear processes such as NSPDEs.We leave this question for future investigations.
In the following section we establish a regularity preservation property of (T t ) t≥0 in a continuous path setting.More precisely, we extend Proposition 4.4 via conditions for (T t ) t≥0 to be a sublinear Markovian semigroup on the space of bounded upper semicontinuous functions from F into R.

4.2.
The USC b -Feller Property.It is natural to ask whether the sublinear Markovian semigroup (T t ) t≥0 preserves some regularity.In case F is endowed with the discrete topology, as it is the case for the class of nonlinear Markov chains from Section 3.2, it is trivial that T t is a selfmap on the space of bounded continuous functions for every t ∈ R + .In general, however, such a preservation property is non-trivial to establish.In the following we provide general conditions for a type of regularity preservation property of nonlinear stochastic processes with continuous paths, namely the so-called USC b -Feller property.Before we formulate our conditions, let us introduce a last bit of notation.We fix an R + -valued continuous adapted process L = (L t ) t≥0 on (Ω, F , F) and, for M > 0, we set When Ω is the Wiener space of continuous paths, which will be assumed below, a typical choice for L could be d F (X, y 0 ), where d F is a metric on F and y 0 ∈ F is an arbitrary reference point.(i) The underlying path space Ω is the space of all continuous functions R + → F , for every u ∈ U , the process Y u has continuous paths and are continuous (where the image and the inverse image spaces are endowed with the local uniform topology).Furthermore, for every u ∈ U, M > 0 and any compact set K ⊂ F , the process ) is upper hemicontinuous and compact-valued, and, for every t ∈ R + and m ∈ N, the correspondence is upper hemicontinuous and compact-valued.Here, co denotes the closure of the convex hull.(v) For every compact set K ⊂ F , the set x∈K R(x) ⊂ P(Ω) is relatively compact.
For nonlinear one-dimensional diffusions, a version of the following theorem was established in [9].Moreover, for nonlinear semimartingales with jumps some conditions were proved in [24,Theorem 4.41,Lemma 4.42]. 3In the context of controlled diffusions, upper semicontinuity of the value function has been established in [16].We provide a result which reaches beyond semimartingale settings.Its proof can be found in Section 10.
Theorem 4.6.Suppose that Condition 4.5 holds.Then, (T t ) t≥0 is a sublinear Markovian semigroup on the space USC b (F ; R) of bounded upper semicontinuous functions F → R.
Remark 4.7.It is interesting to note that Condition 4.5 is not sufficient for the C b -Feller property of (T t ) t≥0 , i.e., it does not imply [53,Exercise 12.4.2].In fact, the counterexample reveals an interesting effect when passing from linear to nonlinear settings.To explain this, we recall the example.Let b : R → R be a bounded continuous function such that b(x) = sgn(x) |x| for |x| ≤ 1 and which is continuously differentiable off (−1, 1).For x ∈ R, define In other words, the set R(x) contains all laws of solutions to the (deterministic) ordinary differential equation Notice that this is a special case of the NSPDE framework discussed in Section 3.3.We say that a family (P . By the linearity of the expectation map P → E P [ • ], it is easy to see that a family (P x ) x∈R of probability measures is a C b -Feller family if and only if it is a USC b -Feller family in the sense that x → E Px [f (X t )] is upper semicontinuous for every f ∈ USC b (R; R) and t > 0. Thus, [53, Exercise 12.4.2]also tells us that it is not possible to select a USC b -Feller family (P x ) x∈R such that P x ∈ R(x).However, Theorem 4.6 (or Corollary 5.4 below) shows that the nonlinear semigroup has the USC b -Feller property (in the sense that (T t ) t≥0 is a nonlinear semigroup on the cone USC b (R; R)).Broadly speaking, passing from the linear to the nonlinear setting provides a smoothing effect in the sense that the nonlinear semigroup has the USC b -Feller property although it is not possible to select a linear semigroup with this property from the uncertainty set.
In Section 5 below, we return to the class of NSPDEs from Section 3.3 and we provide some general parametric conditions which imply Condition 4.5.From a practical point of view, (i) -(iv) from Condition 4.5 are mainly structural assumptions, while checking part (v) requires an argument.4.3.The strong Markov selection principle.For a probability measure P on (Ω, F ), a kernel Ω ∋ ω → Q ω ∈ P(Ω), and a finite stopping time τ , we define the pasting measure ) is Borel and the strong Markov property holds, i.e., for every (s, x) ∈ R + × F and every finite stopping time τ ≥ s, We introduce a correspondence K : R + × F ։ P(Ω) by where (t, x) ∈ R + × F .The following theorem is proved in Section 7.
For every ψ ∈ USC b (R; F ) and every t > 0, there exists a strong Markov family {P (s,x) : (s, x) ∈ R + × F } such that, for all (s, x) ∈ R + × F , P (s,x) ∈ K(s, x) and In particular, for every x ∈ F , In a relaxed framework for finite-dimensional controlled diffusions, a strong Markov selection principle has been established in [16].A strong Markov selection principle for nonlinear one-dimensional diffusions was proved in [9].Theorem 4.9 covers the result from [9] as a special case (see Section 5 below).
In the next section we tailor the Theorems 4.6 and 4.9 to the NSPDE setting from Section 3.3.Furthermore, we are going to discuss the strong USC b -Feller property, which provides a smoothing effect.

NSPDEs: USC b -Feller Properties and Strong Markov Selections
We return to the class of nonlinear SPDEs from Section 3.3 and provide explicit conditions (in terms of the drift and diffusion coefficients) for the USC b -Feller property and the strong Markov selection principle.
We recall the precise setting.Let (H 1 , • H1 , •, • H1 ) and (H 2 , • H2 , •, • H2 ) be two separable Hilbert spaces over the real numbers, and let (A, D(A)) be the generator of a C 0 -semigroup (S t ) t≥0 on the Hilbert space H 2 .Moreover, let Ω be the space of all continuous functions R + → F , let G be a topological space and let µ : be Borel functions, where, for the latter, we mean that σh 1 is a Borel function for every h 1 ∈ H 1 .Further, let D ⊂ D(A * ) be a countable set such that for every y ∈ D(A * ) there exists a sequence (y n ) ∞ n=1 ⊂ D such that y n → y and A * y n → A * y.Recall that such a set exists by [7,Lemma 7.3].We set f i (x) We denote the operator norm on L(H 1 , H 2 ) by • L(H1,H2) and the Hilbert-Schmidt norm by • L2(H1,H2) .
(i) G is a compact metrizable space.
(iii) For every y ∈ D, the functions y, µ H2 and σ * y H1 are continuous.
(iv) There exists a constant C > 0 such that µ(g, x) H2 + σ(g, x) L(H1,H2) ≤ C(1 The semigroup S is compact, i.e., S t is compact for every t > 0, and there exists an α ∈ (0, 1/2) and a Borel function and, for all t > 0, x ∈ H 2 and g ∈ G, In particular, Finally, we establish a smoothing effect for stochastic nonlinear Cauchy problems with drift uncertainty, which are infinite-dimensional processes of the form with drift uncertainty.Definition 5.5.We say that (T t ) t≥0 has the strong USC b -Feller property if To the best of our knowledge, the strong USC b -Feller property of sublinear Markovian semigroups was discovered in [9] for nonlinear one-dimensional elliptic diffusions.In [10] it was established for nonlinear multidimensional diffusions with fixed (strongly) elliptic diffusion coefficients.Adapting some ideas from [9,10,53], we establish the seemingly first result for infinite-dimensional nonlinear stochastic processes.Condition 5.6.
(i) G is a compact metrizable space and H 1 ≡ H 2 ≡ H.
(iii) For every y ∈ D, y, µ H is continuous and σ equals constantly the identity operator on H, i.e., σ(g, x) ≡ id for all (g, x) ∈ G × H. (iv) There exists a constant C > 0 such that Remark 5.7 ( [6]).Part (v) of Condition 5.6 implies that S is a compact semigroup.Moreover, the condition holds for all T > 0 once it holds for some T > 0.
The proof for the next theorem is given in Section 12.
Theorem 5.8.If Condition 5.6 holds, then (T t ) t≥0 has the strong USC b -Feller property.
Under suitable additional conditions (see [13,Hypothesis 9.1]), Theorem 5.8 can be extended to NSPDEs with certain diffusion coefficient, i.e., to the case σ(g, x) ≡ σ(x).Recall that the Standing Assumptions 2.1 and 2.2 are in force.The proof of Theorem 2.5 is based on an application of the general [17, Theorem 2.1], which provides three abstract conditions on the set C implying the DPP.In the following we verify these conditions.For reader's convenience, let us restate them. ( Here, the pasting measure P ⊗ τ Q was defined in (4.3).In the following three sections we check these properties.In the fourth (and last) section, we finalize the proof of Theorem 2.5.Hereby, although technically different, we follow a strategy from [39] and we also use some technical ideas from [8] about the measurability of the semimartingale property in time.The following lemma is a version of [8, Lemma 3.3] for F -valued càdlàg, instead of real-valued continuous, processes.The proof is verbatim the same and omitted here.Lemma 6.3.Let Y = (Y t ) t≥0 be an F -valued càdlàg process and set The following lemma is a restatement of [27, Lemma 2.9 a)].Lemma 6.4.Take two filtered probability spaces B * = (Ω * , F * , F * = (F * t ) t≥0 , P * ) and B ′ = (Ω ′ , F ′ , F ′ = (F ′ t ) t≥0 , P ′ ) with right-continuous filtrations and the property that there is a map φ : Proof.For ω, ω ′ ∈ Ω and t ∈ R + , we define the concatenation and we notice that (ω ⊗ t X) a stopping time τ with t * ≤ τ < ∞, and a probability measure P on (Ω, F ) such that P (X = ω * on [0, t * ]) = 1.We denote by P ( • |F τ ) a version of the regular conditional P -probability given F τ .We recall some simple observations.Lemma 6.7.
(i) There exists a P -null set N ∈ F τ such that In particular, there exists a P -null set N ∈ F τ such that In the following we use the standard notation M s = M •∧s for a process M and a time s.The next lemma is implied by [53, Theorem 1.2.10].Lemma 6.8.Let G = (G t ) t≥0 be the filtration on (Ω, F ) that is generated by a càdlàg process with values in a Polish space.Furthermore, let ρ be a G-stopping time such that ρ ≥ t * .If M − M t * is a P -G-martingale, then there exists a P -null set N such that M − M ρ(ω) is a P ( • |G ρ )(ω)-G-martingale for all ω ∈ N .Lemma 6.9.Suppose that Y ∈ S sp (t * , P ).Then, there exists a P -null set N such that, for every ω ∈ N , Y ∈ S sp (τ (ω), P ( • |F τ )(ω)) and P ( • |F τ )(ω)-a.s.
).Moreover, when ν P denotes the third P -characteristic of Y •+t * , then, for every ω ∈ N , the third Proof.Let G t * = (G t * t ) t≥0 be the filtration generated by X •+t * and let (τ n ) ∞ n=1 be a localizing sequence for the P -G t * -local martingale By Lemma 6.8, there exists a P -null set N such that By Lemma 6.7, possibly making N again larger, we have Hence, for all ω ∈ N , using that G τ (ω) ⊂ G t * •+τ (ω)−t * and Lemma 6.7 (iii), the tower rule shows that the process -stopping time by Galmarino's test ([14, Theorem IV.100]).Finally, since we can enlarge N a last time such that also P (τ n → ∞|F τ )(ω) = 1 for all ω ∈ N .Of course, τ n → ∞ implies that ρ n → ∞.In summary, by virtue of Lemma 6.7 (iii), for all ω ∈ N , the process

This finishes the proof of the first claim.
To prove the second claim, let g : R → R be a bounded Borel function which vanishes around the origin.Then, arguing as above, possibly making N a bit larger, for every ω ∈ N , the process The third characteristic can be characterized by these local martingale properties for countably many test functions g, see [28, Theorem II.2.21].Thus, the formula for the third P ( • |F τ )(ω)-characteristic follows.The proof is complete.
for P -a.a.ω ∈ Ω.To simplicity our notation, we set P P ⊗ τ Q.The following corollary is an immediate consequence of part (ii) of Lemma 6.7.
Lemma 6.13.Suppose that Y ∈ S sp ac (t * , P ).If Y ∈ S sp ac (τ (ω), Q ω ) for P -a.a.ω ∈ Ω, then Y ∈ S sp ac (t * , P ).Proof.Step 1: Semimartingale Property.Let T > t * , and take a sequence (H n ) ∞ n=0 of simple predictable processes on [[t * , T ]] such that H n → H 0 uniformly in time and ω.Then, by Lemma 6.12, we get The first term converges to zero as Y ∈ S sp (t * , P ) and thanks to the Bichteler-Dellacherie (BD) Theorem ([49, Theorem III.43]).The second term converges to zero by dominated convergence, the assumption that P -a.s.Y ∈ S sp (τ, Q) and, by virtue of part (iii) of Lemma 6.7, again the BD Theorem.Consequently, invoking the BD Theorem a third time, but this time the converse direction, yields that Y ∈ S(t * , P ).
7. Proof of the Strong Markov Selection Principle: Theorem 4.9 The proof of the strong Markov selection principle is based on some fundamental ideas of Krylov [31] on Markovian selections as worked out in the monograph [53] by Stroock and Varadhan.The main technical steps in the argument are to establish stability under conditioning and pasting for a suitable sequence of correspondences.Hereby, we adapt the proof of [9,Theorem 2.19] to our more general framework.
This section is split into two parts.In the first, we study some properties of the correspondence K and thereafter, we finalize the proof.7.1.Preparations.Let us stress that in this section we do not assume Condition 4.5.In each of the following results, we indicate precisely which prerequisites are used (except for our Standing Assumptions 2.1, 2.2 and 4.1, which are always in force).
Proof.In the first part of this proof, we adapt an argument from [53,Lemma 11.1.2].The map ω → ρ M (ω) is lower semicontinuous.To see this, recall that and notice that the set on the right hand side is closed, as Since, for every t > 0, where the set on the right hand side is open by the continuity of ω → L(ω), we conclude that ρ + M is upper semicontinuous.Notice that ρ + M = ρ M+ lim KցM ρ K .Define Ψ : Ω × C(R + ; R) → Ω by Ψ(ω, α) = ω, and φ : R + → [0, 1] by φ(M ) E P [e −ρM •Ψ ], and set The dominated convergence theorem yields that The set R + \D is countable, as monotone functions, such as φ, have at most countably many discontinuities.Since ρ M ≤ ρ + M , for every M ∈ D we have P -a.s.
To see this, notice that The proof is complete.
Proposition 7.6.Assume that (i) -(iv) from Condition 4.5 hold.Then, for every closed set K ⊂ F , the set R(K) x∈K R(x) is closed (in P(Ω) endowed with the weak topology).Proof.We adapt the proof strategy from [9,Proposition 3.8].Let (P n ) ∞ n=1 ⊂ R(K) be such that P n → P weakly.By definition of R(K), for every n ∈ N, there exists a point x n ∈ K such that P n ∈ R(x n ).Since P n → P and {δ x : x ∈ K} is closed ([1, Theorem 15.8]), there exists a point x 0 ∈ K such that P • X −1 0 = δ x 0 .In particular, x n → x 0 .We now prove that P ∈ R(x 0 ).Before we start our program, let us fix some auxiliary notation.We set Ω * Ω×C(R + ; R) U and endow this space with the product local uniform topology.In this case, the Borel σ-field B(Ω * ) F * coincides with σ(Z t , t ≥ 0), where Z = (Z (1) , (Z (2,u) ) u∈U ) denotes the coordinate process on Ω * .Furthermore, we define F * (F * t ) t≥0 to be the right-continuous filtration generated by the coordinate process Z.
Step 1.In this first step, we show that the family {P n • (X, (A P n (Y u )) u∈U ) −1 : n ∈ N} is tight when seen as a family of probability measures on (Ω * , F * ).As Ω * is endowed with a product topology and since P n → P , it suffices to prove that, for every fixed u ∈ U , the family {P n • A P n (Y u ) −1 : n ∈ N} is tight when seen as a family of Borel probability measures on C(R + ; R), endowed with the local uniform topology.First, for every M > 0, we deduce tightness of {P n • A P n •∧ρM (Y u ) −1 : n ∈ N} from Kolmogorov's tightness criterion ( [29,Theorem 23.7]).By the first part from Condition 4.5 (iii), there exists a constant C > 0 such that, for all s < t, We conclude the tightness of Using the Arzelà-Ascoli tightness criterion given by [29,Theorem 23.4], we transfer this observation to the global family {P n • A P n (Y u ) −1 : n ∈ N}.For a moment, fix ε, N > 0. By virtue of Condition 4.5 (iii), more precisely (4.2), there exists an M o > N such that

Thanks to the tightness of the family {P
Consequently, we obtain that This observation shows that the family {P n • A P n N (Y u ) −1 : n ∈ N} is tight.For r > 0, ω ∈ C(R + ; R) and h > 0, we define [29,Theorem 23.4] yields that sup n∈N as h → 0. Using [29,Theorem 23.4] once again, but this time the converse direction, we conclude tightness of : n ∈ N} (on the respective probability space).Up to passing to a subsequence, we can assume that (P n • (X, (A P n (Y u )) u∈U ) −1 ) ∞ n=1 converges weakly (on the space (Ω * , F * )) to a probability measure Q.
Step 2. Recall that Z = (Z (1) , (Z (2,u) ) u∈U ) denotes the coordinate process on Ω * and fix some u ∈ U .Next, we show that Z (2,u) is Q-a.s.locally Lipschitz continuous.Thanks to Lemma 7.5, there exists a dense set D ⊂ R + such that, for every M ∈ D, the map ζ M is Q • (Z (1) , Z (2,u) ) −1 -a.s.lower semicontinuous.By part (iii) of Condition 4.5, for every M > 0, there exists a constant all n ∈ N. Hence, for every M ∈ D, using the Q • (Z (1) , Z (2,u) ) −1 -a.s.lower semicontinuity of ζ M , we deduce from [47,Example 17,p. 73 Further, since D is dense in R + , we can conclude that Z (2,u) is Q-a.s.locally Lipschitz continuous and hence, in particular, locally absolutely continuous and of finite variation.
This proves that (λ \ ⊗ Q)-a.e. (dZ (2,u) Step 4. In the final step of the proof, we show that Y u ∈ S sp ac (P ) and we relate B 0,u to A P (Y u ).Notice that Q • Φ −1 = P .For a moment, we fix u ∈ U .As in the proof of Lemma 7.5, we obtain the existence of a dense set s. bounded by a constant independent of n, which, in particular, implies that it is a true is a Q-F * -martingale.Since Z (2,u) is Q-a.s.locally absolutely continuous by Step 2, this means that Y u • Φ is a Q-F * -semimartingale whose first characteristic is given by Z (2,u) .Next, we relate this observation to the probability measure P and the filtration F + .For a process A on (Ω * , F * ), we denote by A p,Φ −1 (F+) its dual predictable projection to the filtration Φ −1 (F + ).Recall from [26,Lemma 10.42] that, for every t ∈ R + , a random variable V on (Ω * , F * ) is Φ −1 (F t+ )-measurable if and only if it is F * t -measurable and V (ω (1) , ω (2) ) does not depend on ω (2) .Thanks to Stricker's theorem ([27, Lemma 2.7]), the process (2,u) /dλ \ ) u∈U ∈ Θ • Φ.By virtue of (iii) from Condition 4.5, for every M ∈ D, we have where Var( • ) denotes the total variation process.Hence, we get from [26, Proposition 9.24] that the first F+) .Lemma 6.4 yields that Y u is a P -F + -semimartingale whose first characteristic A P (Y u ) satisfies Consequently, we deduce from the Steps 2 and 3 that P -a.s.
where we use (ii) and (iii) from Condition 4.5 and [52, Theorems II.4.3 and II.6.2] for the final equality.This observation completes the proof.
Given the previous observations, the following result can be proved similar to [9, Proposition 5.7].For reader's convenience, we provide the details.Proposition 7.7.Suppose that Condition 4.5 holds.The correpondence (t, x) → K(t, x) is upper hemicontinuous with nonempty and compact values.
Proof.The correspondence x → K(0, x) has nonempty values by Standing Assumption 2.2.Further, it has compact values by Proposition 7.6 and part (v) of Condition 4.5.Thus, Lemmata 7.1 and 7.4 yield that the same is true for (t, x) → K(t, x).
It remains to show that K is upper hemicontinuous.Let G ⊂ P(Ω) be closed.We need to show that For each n ∈ N, there exists a probability measure By virtue of Lemma 7.4, we conclude that {P n : n ∈ N} ⊂ K • is relatively compact.Hence, passing to a subsequence if necessary, we can assume that P n → P weakly for some P ∈ K • ∩ G. Let d F be a metric on F which induces its topology.For every ε ∈ (0, t 0 ), the set {d F (X s , x 0 ) ≤ ε for all s ∈ [0, t 0 −ε]} ⊂ Ω is closed.Consequently, by the Portmanteau theorem, for every ε ∈ (0, t 0 ), we get deduce from Lemma 6.4 that Y u ∈ S sp ac (t, P) and that P-a.s.A P (Y u •+t ) = A Pt (Y u ) • θ t .Hence, using Standing Assumption 4.1, we get that In summary, P ∈ C(t, ω).The converse implication follows by symmetry.
Definition 7.10.A correspondence U : R + × F ։ P(Ω) is said to be (i) stable under conditioning if for any (t, x) ∈ R + × F , any stopping time τ with t ≤ τ < ∞, and any P ∈ U(t, x), there exists a P -null set )) for all ω ∈ N ; (ii) stable under pasting if for any (t, x) ∈ R + ×F , any stopping time τ with t ≤ τ < ∞, any P ∈ U(t, x) and any F τ -measurable map Ω ∋ ω → Q ω ∈ P(Ω) the following implication holds: Lemma 7.11.The correspondence K is stable under conditioning and pasting.
Proof.Stability under conditioning follows from Lemmata 6.10 and 7.9, and stability under pasting follows from Lemmata 6.14 and 7.9.
Recall from [1, Definition 18.1] that a correspondence U : R The proof of the following result is similar to those of [9, Lemma 5.12].We added it for reader's convenience and because it explains why both stability properties, i.e., stability under conditioning and pasting, are needed, see also Remark 7.13 below.
Lemma 7.12.Suppose that U : R + × F ։ P(Ω) is a measurable correspondence with nonempty and compact values such that, for all (t, x) ∈ R + × F and P ∈ U(t, x), P (X = x on [0, t]) = 1.Suppose that U is stable under conditioning and pasting.Then, for every φ ∈ USC b (F ; R), the correspondence As U is stable under conditioning, we have P (N ) = 0.By [53, Lemma 12.1.9],N, A ∈ F τ .
We now show that U ∞ is singleton-valued.Take P, Q ∈ U ∞ (t, x) for some (t, x) ∈ R + ×F .By definition of U ∞ , we have for all s ∈ R + .Next, we prove that We use induction over n.For n = 1 the claim is implied by the equality for all s ∈ R + .Suppose that the claim holds for n ∈ N and take test functions g 1 , . . ., g n+1 ∈ C b (F ; R) it suffices to show that P -a.s.
As U ∞ is stable under conditioning, there exists a null set N 1 ∈ F tn such that δ ω(tn) ⊗ tn P ( • |F tn )(ω) ∈ U ∞ (t n , ω(t n )) for all ω ∈ N 1 .Notice that, by the tower rule, there exists a P -null set N 2 ∈ G n such that, for all ω ∈ N 2 and all A ∈ F , Clearly, E P [P (N 1 |G n )] = P (N 1 ) = 0, which implies that P (N 3 ) = 0.For a moment, take ω ∈ N 2 ∪ N 3 .Using that U ∞ has convex and compact values and that δ ω ′ (tn) Consequently, by virtue of (7.4), we also have δ ω(tn) As P = Q on G n , we get that P (N ) = 0.For all ω ∈ N , the induction base implies that The induction step is complete and hence, P = Q.
We proved that U ∞ is singleton-valued and we write U ∞ (s, y) = {P (s,y) }.By the measurability of U ∞ , the map (s, y) → P (s,y) is measurable.It remains to show the strong Markov property of the family {P (s,y) : (s, y) ∈ R + × F }. Take (s, y) ∈ R + × F .As U ∞ is stable under conditioning, for every finite stopping time τ ≥ s, there exists a P (s,x) -null set N such that, for all ω ∈ N , This yields, for all ω ∈ N , that This is the strong Markov property and consequently, the proof is complete.Remark 7.13.Notice that the strong Markov property of the selection {P (s,y) : (s, y) ∈ R + × F } follows solely from the stability under conditioning property of U ∞ .We emphasis that the stability under pasting property of each U N , N ∈ Z + , is crucial for its proof.Indeed, in Lemma 7.12, the fact that U is stable under pasting has been used to establish that U * is stable under conditioning.Proof.The inclusion K(0, ω(t)) ⊃ {P t : P ∈ C(t, ω)} follows from Lemma 7.2.For the converse inclusion, take P ∈ K(0, ω(t)).Then, by Lemmata 7.3 and 7.9, we have Q δ ω ⊗ t P t ∈ C(t, ω).Since, for every A ∈ F , it holds that We can now follow the proof of [9, Proposition 2.8] to deduce Proposition 4.2 from the DPP (Theorem 2.5).

Proof of Proposition 4.2. For every upper semianalytic function
which means nothing else than Now, the DPP (Theorem 2.5) yields that The proof is complete.9. Proof of the Sublinear Semigroup Property: Proposition 4.4 First, let us show that T t , for every t ∈ R + , is a selfmap on the space of bounded upper semianalytic functions.Take a bounded upper semianalytic function ψ : F → R and t ∈ R + .Clearly, T t (ψ) is bounded.It remains to show that T t (ψ) is also upper semianalytic.Define a function φ Evidently, Ω ∋ ω → ω(t) and F × Ω ∋ (x, ω) → 1 {ω(0)=x} are Borel.Hence, by [3,Lemma 7.30], the function φ is upper semianalytic.We set K P ∈ P(Ω) : P ∈ C(0, ω) for some ω ∈ Ω .
By Lemma 6.5, the correspondence C has a Borel measurable graph.Thus, the set is Borel measurable, and consequently, the set K is analytic as it is the image of K under the projection to the second coordinate ([3, Proposition 7.39]).Notice that We conclude from [3, Propositions 7.47 and 7.48] that x → T t (ψ)(x) is upper semianalytic.Fix ψ ∈ USC b (F ; R) and t ∈ R + , and notice that ω → ψ(ω(t)) is upper semicontinuous and bounded.Thus, by [4,Theorem 8.10.61], the map P(Ω) ∋ P → E P [ψ(X t )] is upper semicontinuous, too.Thanks to Proposition 7.7, the correspondence x → R(x) is upper hemicontinuous and compact-valued.Thus, upper semicontinuity of x → T t (ψ)(x) follows from [1,Lemma 17.30].The proof is complete.

Proof of Proposition 5.3
The following section is devoted to the proof of Proposition 5.3, which is split into several parts.In this section, we presume that Condition 5.1 holds.11.1.Proof of parts (i) and (ii) from Condition 4.5.Take u = y, • i H2 ∈ U .By virtue of [19,Problem 13,p.151], the maps ω → Y u (ω) = y, ω i H2 and ω → L(ω) = ω H2 are continuous.Furthermore, for every R, M > 0, we have Before we prove this lemma, we record another useful observation which is used in the proof.
Lemma 11.2.For every x ∈ H 2 and P ∈ R(x), there exists a predictable map g : [[0, ∞]] → G such that, possibly on a standard extension of the filtered probability space (Ω, F , F, P ), the coordinate process X admits the dynamics where W is a cylindrical standard Brownian motion.
Proof.Take x ∈ H 2 and P ∈ R(x).Thanks to Proposition 2.4, there exists a predictable function L u,g(s,X) (X s )ds, u = y, • i H2 , y ∈ D(A * ), i = 1, 2, (11.1) are P -local martingales.Thanks to [43,Theorem 13], in our setting the concepts of analytically weak and mild solutions are equivalent for the SPDE where W is a cylindrical standard Brownian motion.Using the P -local martingale properties of the processes from (11.1), the claim of the lemma now follows as in the proof of [11,Lemma 3.6].We omit the details.
Proof of Lemma 11.1.Step 1: Recap on the Factorization Method.In this step, let P be a probability measure on some filtered probability space which supports a cylindrical standard P -Brownian motion W over the Hilbert space H 2 .Take 1/p < λ ≤ 1 and set, for Notice that R λ is well-defined, as, by Hölder's inequality,  We now recall the factorization formula from [12].Take α ∈ (0, 1/2) and let p > 2 be large enough such that 1/p < α.Moreover, let f : (0, T ] → [0, ∞] be a Borel function such that let φ be a predictable L 1 (H 1 , H 2 )-valued process and let ψ be a real-valued predictable process such that S t φ s L2(H1,H2) ≤ f(t)|ψ s |, for all t, s ∈ (0, T ], and The factorization formula given by [13,Theorem 5.10] shows that see also [7, Step 0 in Section 4] for more details.By Eq. (4.4) in [7], it also holds that (11.5)where the constant c p only depends on p.In particular, (11.3) and (11.5) yield, for all r ∈ [0, T ], that where C > 0 is a constant which only depends on T, S, f, α and p.
Step 2: A Gronwall Argument.We are in the position to prove Lemma 11.1.Take x ∈ K and P ∈ R(x) and let g be as in Lemma 11.2.Then, using the inequalities (11.3) and (11.6), for every M > 0 and all r ∈ [0, T ], we obtain For any P ∈ R(x 0 ), by Lemma 11.2 and the factorization formula (11.4), possibly on an enlargement of the filtered probability space (Ω, F , F, P ), we have a.s.
where The following proof adapts the idea from [10] that the strong USC b -Feller property can be deduced from the the strong Markov selection principle and a change of measure.Throughout this proof, we fix a function ψ ∈ USC b (F ; R) and a time T > 0. W.l.o.g., assume that |ψ| ≤ 1.We start with some auxiliary observations.Notice that Condition 5.6 implies Condition 5.1.Hence, by Corollary 5.4, there exists a strong Markov family {P (s,x) : (s, x) ∈ R + × H} such that P (s,x) ∈ K(s, x) and T T (ψ)(x) = E P (0,x) [ψ(X T )].To simplify our notation, we set P x P (0,x) .By Lemma 11.2, with some abuse of notation, for every x ∈ H, there exists a predictable map g x : [[0, ∞[[ → G and a cylindrical standard P x -Brownian motion W such that P x -a.s.X t = S t x + t 0 S t−s µ(g x (s, X), X s )ds + t 0 S t−s dW s , t ∈ R + .
We conclude that the map x → E Px [ψ(X T )] is continuous.This finishes the proof of Theorem 5.8.
Lemma A.1.The space L is a Polish space.
Proof.The proof is split into two steps.
Step 1.Let M f (R d ) be the set of all finite measures on (R d , B(R d )) endowed with the weak topology.This space is Polish by [48,Theorem 1.11].We now prove that is a G δ subset of M f (R d ) and consequently, a Polish subspace.Let (g n ) ∞ n=1 be a sequence of continuous functions from R d into [0, 1] such that g n ց 1 {0} .It is well-known that such a sequence exists.Now, set g m dK < 1/n .
The map K → g m dK is continuous by the definition of the weak topology.Thus, {K ∈ M f (R d ) : g n dK < 1/m} is open and consequently, H is a G δ set.We now explain that M f 0 (R d ) = H.If K ∈ M f 0 (R d ), then lim m→∞ g m dK = K({0}) = 0, by the dominated convergence theorem.Hence, K ∈ H. Conversely, take K ∈ H.For every n ∈ N, there exists an m ∈ N such that g m dK < 1/n.As k → g k decreases, we get g k dK < 1/n for all k ≥ m.This proves that lim m→∞ g m dK = 0 and it follows from the dominated convergence theorem that K({0}) = 0.In summary, K ∈ M f 0 (R d ), which completes the proof of M f 0 (R d ) = H.Step 2. Let d be a metric that induces the weak topology on M f 0 (R d ).Then, (K 1 , K 2 ) → d L (K 1 , K 2 ) d(gdK 1 , gdK 2 ), g(x) 1 ∧ x 2 , induces the topology of L. In other words, the function K → gdK is an isometry between L and M f 0 (R d ).Further, it is a bijection whose inverse is given by K → 1 R d \{0} dK/g.Hence, L and M f 0 (R d ) are isometric, which implies that L is a Polish space.We also identify a class of continuous functions on L.

4. 1 .Proposition 4 . 2 .
The nonlinear Markov property.The following proposition provides the Markov property of {E x : x ∈ F }.The proof is given in Section 8.For every upper semianalytic function ψ

Example 5 . 9 ( 6 .
[21]).Let O ⊂ R d be a bounded domain with smooth boundary.Part (v) of Condition 5.6 holds when H = L 2 (O) and A is a strongly elliptic operator of order 2m > d, see [21, Remark 3, Example 3] for some details.In particular, this is the case when d = 1 and A is the Laplacian, i.e., Condition 5.6 holds for one-dimensional stochastic heat equations with drift uncertainty.Proof of the Dynamic Programming Principle: Theorem 2.5 28, Proposition IX.1.4]is stated for F = R d but the argument needs no change to work for more general state spaces.

Finally, we discuss 10 .
the properties (i) -(iii) from Definition 4.3.The properties (ii) and (iii) are trivially satisfied and the first property (i) is implied by Proposition 4.2.The proof is complete.Proof of the USC b -Feller Property: Theorem 4.6

E 2 − 1 ≤
Px Z Px t 1 {ρM >t} = lim M→∞ Q M x (ρ M > t) = 1,where we use the monotone convergence theorem for the first equality.This shows that Z Px is a true P x -martingale.Consequently, by a standard extension theorem ([29,Lemma 19.19]), there exists a probability measureQ x on (Ω, F ) such that Q x (G) = E Px [Z Px T 1 G ]for all G ∈ F T and T > 0. Using again Girsanov's theorem (cf.[35, Proposition I.0.6]),we obtain that the processB W + • 0 µ(g x (s, X), X s )ds is a cylindrical standard Q x -Brownian motion and, under Q x , X t = S t x + t 0 S t−s dB s , t ∈ R + .Now, we are in the position to prove Theorem 5.8.Let (x n ) ∞ n=0 ⊂ H be a sequence such that x n → x 0 and fix ε > 0. By virtue of (12.1), there exists an M ′ > 0 such thatsup n∈Z+ Q x n (ρ M ′ ≤ T ) ≤ ε. (12.2)Furthermore, by Lemma 11.1, there exists an M • > 0 such thatsup n∈Z+ P x n (ρ M • ≤ T ) ≤ ε. (12.3)We set M M ′ ∨ M • .By the linear growth part of Condition 5.6, there exists a constant C = C M > 0 such that µ(g, x) H ≤ C for all g ∈ G and x ∈ H : x H ≤ M .Take β ∈ (0, T ) and notice that, for all n ∈ Z + ,E P x n |1 − Z P x n β∧ρM | 2 ≤ E P x n |1 − Z P x n β∧ρM | 2 = E P x n Z P x n β∧ρM e βC 2 − 1.Now, let β ∈ (0, T ) small enough such that, for all n ∈ Z + , E P x n |1 − Z P x n β∧ρM | ≤ ε.(12.4) [23,G is a metrizable Souslin space, the measurable implicit function theorem[23, Theorem 7.2]yields the existence of a P t -measurable function g ).Thanks to Proposition 5.3, the Theorems 4.6 and 4.9 yield the following result.Corollary 5.4.Suppose that Condition 5.1 holds.Then, (T t ) t≥0 is a sublinear Markovian semigroup on USC b (H 2 ; R) and, for every ψ ∈ USC b (H 2 ; R) and every t > 0, there exists a strong Markov family {P (s,x) : (s, x) ∈ R + × H 2 } such that, for every (s, x) ∈ R + × H 2 , Proposition 5.3.Condition 5.1 implies Condition 4.5 with L ≡ X H2 .
[5,ce, as ucp (uniformly on compacts in probability) limits of continuous local martingales are again continuous local martingales (see[5, Lemma B.11]), we conclude that all processes of the form 2, are P -local martingales.Recall that D ⊂ D(A * ) has the property that for every y ∈ D(A * ) there exists a sequence (y n ) ∞ n=1 ⊂ D such that y n → y and A * y n → A * y.