Sticky-reflected stochastic heat equation driven by colored noise

UDC 519.21
We prove the existence of a sticky-reflected solution to the heat equation on the spatial interval driven by colored noise. The process can be interpreted as an infinite-dimensional analog of the sticky-reflected Brownian motion on the real line, but now the solution obeys the usual stochastic heat equation except for points where it reaches zero. The solution has no noise at zero and a drift pushes it to stay positive. The proof is based on a new approach that can also be applied to other types of SPDEs with discontinuous coefficients.


Introduction
In this paper, we study the existence of a continuous function X : [0, 1] × [0, ∞) → [0, ∞) that is a weak solution to the following SPDE or Dirichlet X t (0) = X t (1) = 0, t ≥ 0, (1.3) boundary conditions and the initial condition whereẆ is a space-time white noise, the functions g ∈ C[0, 1] and λ ∈ L 2 := L 2 [0, 1] are non-negative, f is a continuous function from [0, ∞) to [0, ∞) which has a linear growth and f (0) = 0, and Q is a non-negative definite self-adjoint Hilbert-Schmidt operator on L 2 . We will also assume that in the case of the Dirichlet boundary conditions g(0) = g(1) = 0. The equation appears as a sticky-reflected counterpart of the reflected SPDE introduced in [HP89,NP92]. We assume that a solution obeys the stochastic heat equation being strictly positive, but reaching zero, its diffusion vanishes and an additional drift at zero pushes the process to be positive. The form of equation (1.1) is similar to the form of the SDE for a sticky-reflected Brownian motion on the real line (1.5) and we expect that the local behaviour of X at zero is very similar to the behaviour of the sticky-reflected Brownian motion x. Remark that SDE (1.5) admits only a weak unique solution because of its discontinuous coefficients, see e.g. [Chi97,KSS11,EP14]. The approaches which are applicable to sticky processes in finite-dimensional spaces can not be used for solving of SPDE (1.1). For instance, Engelbert and Peskir in [EP14] showed that equation (1.5) admits a weak (unique) solution, where their approach was based on the time change for a reflected Brownian motion. This method is very restrictive and can be applied only for the sticky dynamics in a one-dimensional state space. An equation for sticky-reflected dynamics for higher (finite) dimensions was considered by Grothaus and co-authors in [GV17,FGVh16,GV18], where they used the Dirichlet form approach [FOT11,MR92]. This approach was based on a priori knowledge of the invariant measure. Since the space is infinite-dimensional in our case, finding of the invariant measure seems a very complicated problem (see e.g. [FO01,Zam01] for the form of invariant measure for the reflected stochastic heat equation driven by the white noise).
In this paper, we propose a new method for the proof of existence of weak solutions to equations describing sticky-reflected behaviour. This approach is is a modification of the method proposed by the author in [Kon17], and is based on a property of quadratic variation of semimartingales.
The paper leaves a couple of important open problems. The first problem is the uniqueness of a solution to SPDE (1.1)-(1.4). Similarly to the one-dimensional SDE for sticky-reflected Brownian motion (1.5), where the strong uniqueness is failed [Chi97,EP14], we do not expect the strong uniqueness for the SPDE considered here. However, we believe that the weak uniqueness holds.
Another interesting question is the existence of solutions to a similar stickyreflected heat equation driven by the white-noise. It seems that the method proposed here can be adapted to the case of such an SPDE. For this we need a similar statement to Theorem 3.5, that remains an open problem.

Definition of solution and main result
For convenience of notation we introduce a parameter α 0 which equals to 1 in the case of the Neumann boundary conditions (1.2) and 0 in the case of Dirichlet boundary conditions (1.3). Let us also introduce for k ≥ 1 the space C k [0, 1] of k-times continuously differentiable functions on (0, 1) which together with their derivatives up to the order k can be extended to continuous functions on [0, 1]. We will write ϕ ∈ C k α 0 [0, 1] if additionally ϕ (α 0 ) (0) = ϕ (α 0 ) (1) = 0, where ϕ (0) = ϕ and ϕ (1) = ϕ ′ .
Denote the inner product in the space L 2 by ·, · and the corresponding norm by · . Let us give a definition of a weak solution to SPDE (1.1).
Definition 1.1. We say that a continuous function X : [0, 1] × [0, ∞) → [0, ∞) is a (weak) solution to SPDE (1.1)-(1.4), if for every ϕ ∈ C 2 α 0 [0, 1] the process is an (F X t )martingale with quadratic variation Hereinafter {e k , k ≥ 1} will denote the basis in L 2 consisting of eigenvectors of the non-negative definite self-adjoint operator Q. Let {µ k , k ≥ 1} be the corresponding family of eigenvalues of Q. We note that ∞ k=1 µ 2 k < ∞, since Q is a Hilbert-Schmidt operator. Introduce the function where the series trivially converges in L 1 [0, 1] and a.e. The main result of this paper reads as follows. Remark 1.3. Condition (1.7) means that the drift λ has to be equal to zero for those u for which the noise vanishes.
Remark 1.4. The equation can admit a solution even if condition (1.7) does not hold. The reason is that the existence can be failed if X t (u) = 0 for u ∈ [0, 1] such that λ(u) > 0 and χ(u) = 0 because of the term λI {Xt=0} and the absence of the noise for such u. However, if the initial condition is strictly positive for such u, then the solution could stay always strictly positive for such u, by the comparison principle for the classical heat equation. Therefore, the solution will exist. Take for example Q = 0 and f = 0. Then a weak solution to the heat equation considered with corresponding boundary and initial conditions is a solution to SPDE (1.1)-(1.4) if X t (u) > 0, t > 0, u ∈ (0, 1). But the strong positivity of X is valid e.g. under the assumption the strong positivity of the initial condition. Hence, SPDE (1.1)-(1.4) has a weak solution even if λ > 0 for e.g. Q = 0, f = 0 and g > 0.
We will construct a solution to equation (1.1) as a limit of polygonal approximation similarly to the approach done in [Fun83]. The main difficulty here is that coefficients are discontinuous. So, we cannot pass to the limit directly. In the next section, we will explain the key idea which allows to overcome this difficulty.

Key idea of passing to the limit
We demonstrate our idea of passing to the limits in the case of discontinuous coefficients using the equation for a sticky-reflected Brownian motion in R where w is a standard Brownian motion in R and λ, σ, x 0 are positive constants. It is well known that this equation has only a unique weak solution (see e.g. [EP14]). Let us show that a solution to SDE (1.8) can be constructed as a weak limit of solutions to equations with "good" coefficients. The first three steps proposed here are rather standard and the last step shows how one can overcome the problem of the discontinuity of the coefficients.
Step III. Properties of the limit process. One can see that for every T > 0 there exist a random elementρ in L 2 [0, T ] and a subsequence N such that and η is a continuous square-integrable martingale with quadratic variation ρ. We may assume that N = N.
Step IV. Identification of quadratic variation and drift. Because of the discontinuity of the coefficients of equation (1.8), we cannot conclude directly that ρ(t) = t 0 σ 2 I {x(s)>0} ds and a(t) = λ t 0 I {x(s)=0} ds, t ∈ [0, T ], which would imply that x is a weak solution to SDE (1.8). We propose to overcome this problem as follows. Let us use the following facts: a) if x(t), t ≥ 0, is a continuous non-negative semimartingale with quadratic variation then κ 2 m (s m )I (0,+∞) (s) → I (0,+∞) (s) in R as m → ∞. So, using (1.10), a), b) and the dominated convergence theorem, we get almost surely Hence, (1.11) implies Consequently, where η is a continuous square-integrable martingale with quadratic variation that means that x is a weak solution to (1.8).
Content of the paper. To show the existence of a weak solution to SPDE (1.1)-(1.4), we will follow the argument above.
Step I will be done in Section 2.1. More precisely, we will construct a family of processes which will approximate a solution to SPDE (1.1)-(1.4). The approximating sequence is similar to one considered in [Fun83]. Section 2.2 is devoted to the tightness that is Step II of our argument.
Step III is made in Section 3.1, where we show that the limit process satisfies equalities similar to (1.11) (see Proposition 3.1 there). An analog of property a) above is stated for some infinite-dimensional semimartingales in Theorem 3.5 in Section 3.2. The proof of the existence theorem is given in Section 3.3, where we use the approach described in Step IV. Auxiliary statements are proved in the appendix.

Preliminaries
We will denote the inner product and the corresponding norm in a Hilbert space H by ·, · H and · H , respectively.
Denote the space of Hilbert-Schmidt operators on L 2 by L 2 . Remark that L 2 furnished with the inner product Ae k , Be k , A, B ∈ L 2 , is a Hilbert space, where the norm does not depend on the choice of basis in L 2 . The family of operators {e k ⊙ e l , k, l ≥ 1} form a basis in L 2 . Here, for every ϕ, ψ ∈ L 2 ϕ ⊙ ψ denotes the operator on L 2 defined as (ϕ ⊙ ψ)g = g, ψ ϕ, g ∈ L 2 .
We will consider the set R n×n of all n × n-matrices with real enters as a Hilbert space with the Hilbert-Schmidt inner product A, B R n×n = n k,l=1 A k,l B k,l . The indicator function will be defined as usually If φ : E 1 → E 2 is a function and S is a subset of E 2 , then I {φ∈S} will denote the function x → I S (φ(x)) from E 1 to E 2 .
One can show that the space H T equipped with the inner product is a Hilbert space.
Considering a sequence Z n in H T , we will say that Z n → Z a.e. as n → ∞, if Let L ∈ L T 2 , Z ∈ L T 2 , and S be a Borel measurable subset of R. It is easily seen that L t I {Zt∈S} · , t ∈ [0, T ], where L t I {Zt∈S} · is the composition of two operators, is well-defined and belongs to L ∈ L T 2 . We will denote such a function shortly by L · I {Z·∈S} · .

Finite sticky reflected particle system
In this section, we construct a sequence of random processes which will be used for the approximation of a solution to SPDE (1.1)-(1.4).
Let n ≥ 1 be fixed. We set π n k = I [ k−1 n , k n ) , k ∈ [n] := {1, . . . , n}. Let W t , t ≥ 0, be a cylindrical Wiener process in L 2 . Define the following Wiener processes on R as follows and note that their joint quadratic variation [w n k , w n l ] t = n Qπ n k , Qπ n l t =: q n k,l t, t ≥ 0.

Consider the following SDE
(2.1) satisfying the initial condition where ∆ n x n k = (∆ n x n ) k = n 2 x n k+1 + x n k−1 − 2x n k and We will construct a solution to SPDE (1.1)-(1.4) as a weak limit in C ([0, ∞), . Remark that equation (2.1) has discontinuous coefficients. So the classical theory of SDE cannot be applied in our case. The existence of the solution will follow from Theorem 2.1 which we state below.

SDE for sticky-reflected particle system
The aim of this section is to prove the existence of solutions to (2.1), (2.2). We will formulate the problem in slightly general form. So, let n ∈ N and g k , λ k , k ∈ [n], be non-negative numbers. We also consider a family of Brownian motions w k (t), t ≥ 0, k ∈ [n], (with respect to the same filtration) with joint quadratic variation [w k , w l ] t = q k,l t, t ≥ 0.
(2.5) with initial condition y k (0) = g k , k ∈ [n], (2.6) and the boundary conditions y 0 (t) = α 0 y 1 (t), y n+1 (t) = α 0 y n (t), t ≥ 0. (2.7) Theorem 2.1. Let q k,k = 0 imply λ k = 0 for every k ∈ [n]. Then there exists a family of non-negative (real-valued) continuous processes y k (t), t ≥ 0, k ∈ [n], in R which is a weak (martingale) solution to (2.1), (2.2), that is, y k (0) = g k , for each is an (F y t )-martingale, and the joint quadratic variation of N k and N l , k, l ∈ [n], equals We are going to construct a solution to the SDE approximating the coefficients by Lipschitz continuous functions and using the method described in Section 1.2.
Let us take a non-decreasing function (2.8) Since equation (2.8) has locally Lipschitz continuous coefficients with linear growth, it has a unique strong solution.
Our goal is to to show that the sequence {y ε = (y ε k ) n k=1 } ε>0 has a subsequence which converges in distribution to a week solution to (2.1). We denote for every where σ ε k,l (s) = κ ε (y ε k (s))κ ε (y ε l (s))q k,l . Remark 2.2. According to the choice of the approximating sequence for a, the equality holds for every k ∈ [n] satisfying q k,k > 0.
Consider the following metric space Proof. In order to prove the statement, it is sufficient to show that each family of coordinate processes of (y εm , a εm , η εm , [η εm ]), m ≥ 1, is tight in the corresponding space. We will only show the tightness for {y εm , m ≥ 1}. The tightness for other families can be obtained similarly.
According to the Aldous tightness criterion [Ald78, Theorem 1], it is enough to show that for every T > 0, any family of stopping times τ m , m ≥ 1, bounded by T and any sequence δ m decreasing to zero The conditions above trivially follow from the convergence for all t ∈ [0, T ], the Itô formula and Gronwall's lemma, one can check that for every p ≥ 1 there exists a constant C p,T,n , depending on p, T and n, such that (2.9) Next, by the Itô formula and the optional sampling Theorem 7.12 [Kal02], we have for all m ≥ 1. Using Hölder's inequality, and estimate (2.9) one can conclude that This completes the proof of the lemma.
By Lemma 2.3 and Prokhorov's theorem, there exists a sequence {ε m } m≥1 converging to zero such that the sequence y εm := (y εm , a εm , η εm , [η εm ]) converges to a random element y := (y, a, η, ρ) in W R n in distriburion. By the Skorokhod representation Theorem 3.1.8 [EK86], one can choose a probability space (Ω,F ,P) and determine there a family of random elementsỹ,ỹ εm , m ≥ 1 taking values in W R n such that Lawỹ = Law y, Lawỹ εm = Law y εm , m ≥ 1, andỹ εm → y in W R n a.s. So, without loss of generality, we will assume that y εm → y in W R n a.s. as m → ∞.
Since the sequence {ε m } m≥1 will be fixed to the end of this section, we will write m instead of ε m in order to simplify the notation. Let , of y are nonnegative and (iii) For every k ∈ [n] and T > 0 there exists a random elementȧ k in L 2 ([0, T ], R) such that almost surely (iv) For every k, l ∈ [n] and T > 0 there exists a random elementρ k,l in L 2 ([0, T ], R) such that almost surely Proof. We remark that for every k ∈ [n] and for every m ≥ 1 and k ∈ [n] almost surely Passing to the limit and using the dominated convergence theorem, we obtain the equality (i).
The equality in (ii) follows from Remark 2.2 and the convergence in distribution of (a m k , [η m k ]) to (a k , ρ k,k ) in (C ([0, +∞), R)) 2 . We next prove (iii). Let T > 0 be fixed. Denote the ball in L 2 ([0, T ], R) with center 0 and radius r > 0 by B T r and furnish it with the weak topology of the and Theorem V.5.1 ibid, B T r is a compact metric space.
We fix k ∈ [n] and take r : Thenȧ m k is a random element in B T r for every m ≥ 1. By the compactness of B T r , the in R in distribution along N . Since the family of functions Let us show that there exists a random elementȧ k in L 2 ([0, T ], R) such that a k = · 0ȧ k (s)ds a.s. We define the map Φ : By the Kuratowski Theorem A.10.5 [EK86], the set Im Φ is Borel measurable in C[0, T ] and the map Φ −1 is Borel measurable. By (2.11), a k ∈ Im Φ a.s. Thus, we can defineȧ k = Φ −1 (a k ). This completes the proof of (iii).
Similarly, one can prove (iv). Statement (v) follows from the fact that the limit of local martingales is a local martingale and the uniform boundedness of E (η m k (t)) 2 in m. Indeed, for every This finishes the proof of the lemma.
Proposition 2.5. Let y(t) = (y(t), a(t), η(t), ρ(t)), t ≥ 0, be as in Lemma 2.4. Let additionally λ k = 0 if q k,k = 0, k ∈ [n]. Then 1) for every k, l ∈ [n] almost surely Proof. We take the sequence {y m n } n≥1 as in the proof of Lemma 2.4. Again without loss of generality we may assume that it converges to y a.s. We first show that almost surely Recall that almost surely where σ m k,l (s) = q k,l κ m (y m k (s))κ m (y m l (s)), and for each T > 0, k, l ∈ [n] there exist random elementsρ k,l in L 2 ([0, T ], R) such that almost surely s., the uniform boundedness of σ m k,l , and the density of span By Lemma 2.4, y k and y l are non-negative continuous semimartingales with The latter equality and (2.12) yield that almost surely for every t ∈ [0, T ] where we have used the convergence κ m (x m )I (0,+∞) (x) → I (0,+∞) (x) as x m → x in R and the dominated convergence theorem. Hence, a.s.
and, consequently, according to Lemma 2.4 (ii), almost surely for all k ∈ [n] such that q k,k = 0. If q k,k = 0, then λ k = 0, by the assumption of Proposition 2.5. Therefore, a m k = 0 implies that a k = 0. This finishes the proof of the propostion.
Proof of Theorem 2.1. The statement of the theorem directly follows from Lemma 2.4 and Proposition 2.5.

Tightness
Let a family of non-negative continuous processes {x n k (t), t ≥ 0, k ∈ [n]} be a weak solution to SDE (2.1)-(2.3), which exists according to Theorem 2.1. Let the continuous processX n t , t ≥ 0, taking values in C[0, 1] be defined by (2.4). We note thatX n t (u) ≥ 0 for all u ∈ [0, 1], t ≥ 0 and n ≥ 1. The aim of this section is to prove the tightness of the family X n , n ≥ 1 in C ([0, ∞), C[0, 1]). The similar problem was considered in [Fun83, Section 2], where the author study the existence of solutions to an SPDE with Lipschitz continuous coefficients. The tightness argument there is based on properties of fundamental solution to the discrete analog of the heat equation and the fact that coefficients of the equation has at most linear growth. The Lipschitz continuity was not needed for the proof of the tightness. Since the proof in our case repeats the proof from [Fun83], we will point out only its main steps. The main statement of this sections reads as follows.
For the proof of the proposition it is enough to show that the family X n , n ≥ 1 is tight in C ([0, T ], C[0, 1]) = C ([0, T ] × [0, 1], R) for every T > 0. So, we fix T > 0, and use Corollary 16.9 [Kal02] which yields the tightness if X n , n ≥ 0 satisfies the following conditions:
The family X n , n ≥ 1 trivially satisfies the first condition becauseX n 0 (0) = g n 1 is uniformly bounded in n ≥ 1. In order to check the second condition, we first write equation (2.1) in the integral form. Let {p n k,l (t), t ≥ 0, k, l ∈ [n]} be the fundamental solution of the system of ordinary differential equations 5 with the initial condition p n k,l (0) = nI {k=l} , k, l ∈ [n], and the boundary conditions where the operator ∆ n (k) = ∆ n is applied to the vector (p k,l (t)) n k=1 for every l ∈ [n]. Noting that { W t , √ nπ n k , t ≥ 0, k ∈ [n]} is a family of standard Brownian motions, it is easily seen thatX n has the same distribution as the solution to the integral equatioñ and ⌈v⌉ = ⌈v⌉ n := l n for v ∈ π n l , l ∈ [n]. We will denote byX n,i t (u) the i-th term of the right hand side of equation (2.13).
To prove lemmas 2.7 and 2.8, one needs to repeat the proofs of lemmas 2.1 and 2.2 from [Fun83] which are based on properties of the fundamental solution p n (t, u, v), t ∈ [0, T ], u, v ∈ [0, T ], and the fact that the coefficients of the equation has at most the linear growth. We omit the proof of those lemmas here.
Remark 2.9. LetX t , t ≥ 0, be a limit point of the sequence X n , n ≥ 1 in i.e X is a limit in distribution of a subsequence of X n , n ≥ 1 .

Passing to the limit
The goal of the present section is to show that there exists a solution to SPDE (1.1)-(1.4). The solution will be constructed as a limit point of the family of processes X n , n ≥ 1 from Proposition 2.6, which exists by Prokhorov's theorem. Since the coefficients of the equation are discontinuous, we cannot pass to the limit directly.
In the next section, we will show that there exists a subsequence of X n , n ≥ 1 whose weak limit in C ([0, ∞), C[0, 1]) is a heat semimartingale 6 . After that we will prove an analog of the Itô formula and state a property similar to one for usual Rvalued semimartingales, stated in Lemma A.1, for such heat semimartingales. Then, using the argument described in Section 1.2, we show thatX solves equation (1.1)-(1.4). In this section, T > 0 will be fixed.

Martingale problem for limit points of the discrete approximation
We start from the introduction of a new metric space where we will consider the convergence. Denote and consider the following balls in the Hilbert spaces L T 2 and L T 2 , respectively. We furniture this sets with the induced weak topologies. By Theorem V.5.1 [DS88], those spaces are metrizable. Moreover, by Alaoglu's Theorem V.4.2 [DS88], they are compact metric spaces.
We can trivially estimate λ n ≤ λ and The latter inequality follows from Lemma A.3. Hence λ n I {X n =0} and Γ n are random elements in L T 2 and L T 2 , respectively. Let us consider the random element X n := X n , X n , λ n I {X n =0} , I {X n >0} , Γ n , n ≥ 1, (3.1) in the complete separable metric space The following statement is the main result of this section.
Proposition 3.1. There exists a subsequence of {X n , n ≥ 1} which converges to X = X , X, a, σ, Γ in W L 2 in distribution. Moreover, the limit X satisfies the following properties:

3)
and Proof. We first remark that the families λ n I {X n =0} , n ≥ 1 , I {X n >0} , n ≥ 1 and {Γ n , n ≥ 1} are tight due to the compactness of the spaces where they are defined. Consequently, by Proposition 2.6 and Proposition 3.2.4 [EK86], the family X n , λ n I {X n =0} , I {X n >0} , Γ n , n ≥ 1 is also tight. By the Prokhorov theorem, there exists a subsequence N ⊂ N such that in distribution along N . Without loss of generality, let us assume that N = N. Since it is easily to see that max The existence of a convergent subsequence of {X n } n≥1 , and statement (i) are proved.
The statement (ii) directly follows from Lemma A.5. In order to prove statement (iii) of the proposition, we first define the following L 2 -valued martingale Set for ϕ ∈ L 2∆ n ϕ := n 3 n k=1 ∆ n ϕ n k π n k , (3.5) where ϕ n k = ϕ, π n k , ϕ n 0 = α 0 ϕ n 1 and ϕ n n+1 = α 0 ϕ n n . Since X n = n k=1 x n k π n k and the family {x n k , k ∈ [n]} solves SDE (2.1)-(2.3), we get that for every ϕ ∈ L 2 M n t , ϕ = M n t , pr n ϕ = X n t , pr n ϕ − g n , pr n ϕ − 1 2 t 0 ∆ n X n s , pr n ϕ ds (3.6) and the quadratic variation of the (F X n )-martingale M n t , ϕ equals T ] for every k ≥ 1, by the Aldous tightness criterion. According to the tightness of {X n , n ≥ 1}, we also have that { X n · ,ẽ k , n ≥ 1} is tight in C[0, T ] for each k ≥ 1. Using Proposition 2.4 [EK86] and Prokhorov's theorem, we can choose a subsequence N ⊂ N such that in (C[0, T ]) 3 N in distribution along N . In particular, we have that M n · ,ẽ k 2 − [ M n · ,ẽ k ], n ≥ 1, is a sequence of martingales which converges toM 2 k −V k in C[0, T ] in distribution along N for all k ≥ 1.
We fix m ≥ 1 and let (FX ,M,V ,m ) be the complete right continuous filtration generated by (X k ,M k ,V k ), k ∈ [m]. By Proposition IX.1.17 [JS03], we can conclude thatM k andM 2 we have that E M 2 k (T ) < +∞, by Lemma 4.11 [Kal02]. HenceM 2 k is a continuous square-integrable (FX ,M,V ,m )-martingale with quadratic variation M k =V , k ∈ , we may assume that this sequence converges almost surely. Therefore, for every t ∈ [0, T ] and k ≥ 1 a.s. as n → ∞. Using Taylor's formula and the fact thatẽ k ∈ C 3 α 0 [0, 1], k ≥ 1, it is easy to see that for every t ∈ [0, T ] and k ≥ 1 Consequently, for every t ∈ [0, T ] the sequence M n t ,ẽ k , n ≥ 1, converges to a.s. as n → ∞. Thus, for every m ∈ N and t i ∈ [0, T ], i ∈ [m], in R 3m N a.s. as n → ∞. This and convergence (3.7) imply that Consequently, for every k ≥ 1 the process M k is a continuous square-integrable (F X,M )-martingale with quadratic variation where (F X,M t ) t∈[0,T ] is the complete right continuous filtration generated by X k , M k , k ≥ 1.

Now we introduce the following process in
(3.8) Remark that M t , t ∈ [0, T ], is a well-defined continuous process in L 2 . Indeed, by the Burkholder-Davis-Gundy inequality, Lemma A.4, (3.2) and the dominated convergence theorem, for every n, m ≥ 1 E max L t pr n,n+mẽ k , L t pr n,n+mẽ as n, m → ∞, where pr n,n+m is the orthogonal projection in L 2 onto span{ẽ k , k = n + 1, . . . , n + m}. This implies the convergence of series (3.8) and the continuity This implies statement (iii). The proposition is proved.

A property of quadratic variation of heat semi martingales
In this section, we will assume that (Ω, F, (F t ) t≥0 , P) is a filtered complete probability space, where the filtration (F t ) t≥0 is complete and right continuous. Let T > 0 be fixed. Consider a continuous (F t )-adapted L 2 -valued process Z t , t ∈ [0, T ], such that there exist random elements a and L in L T 2 and L T 2 , respectively, such that for every is a local (F t )-martingale with quadratic variation Note that the assumptions on L implies that the continuous process t 0 L s 2 L 2 ds, t ∈ [0, T ], is well-defined and (F t )-adapted.
Let Z n t = pr n Z t , t ≥ 0, and a n t = pr n a t , t ∈ [0, T ]. We also introducė and note thatŻ n , n ≥ 1, is a sequence of random elements in L T 2 . Proof. Set z k (t) := Z t ,ẽ k , t ∈ [0, T ], k ≥ 1. Then, by the definition of Z, for every k ≥ 1 the process is a continuous local (F t )-martingale with quadratic variation where a k (s) := a s ,ẽ k . Denote σ 2 k,l (t) := L tẽk , L tẽl , t ∈ [0, T ], and note that z k (t)ẽ k and a n t = n k=1 a k (t)ẽ k , t ∈ [0, T ], n ≥ 1.
By the Itô formula and the polarisation equality, we get A simple computation gives that Trivially, Z n t 2 → Z t 2 a.s. as n → ∞ for all t ∈ [0, T ]. Using the dominated convergence theorem, we can conclude that t 0 a n s , Z n s ds → t 0 a s , Z s ds a.s. as n → ∞. Next, by Lemma A.4 and the dominated convergence theorem, t 0 L s pr n 2 L 2 ds → t 0 L s 2 L 2 ds a.s. as n → ∞. Next, we will show that M n (t) converges in probability. Since M n is a local martingale, we need to choose a localization sequence of (F t )-stopping times defined as follows Then the processes M n (t ∧ τ k ), t ∈ [0, T ], n ≥ 1, are square-integrable (F t )martingales for every k ≥ 1, and τ k ↑ T as k → ∞. By the Burkholder-Davis-Gundy inequality (see e.g. [IW89, Theorem III.3.1]), for every k, n, m ≥ 1, n < m, where pr n,m is the orthogonal projection in L 2 onto span {ẽ k , k = n + 1, . . . , m}.
We have obtained that every term, except This implies the convergence of Ż n s (ω) n≥1 in L 2 for almost all s and ω. Hencė Z n , n ≥ 1, converges toŻ a.e. a.s. as n → ∞. The equality in the second part of the lemma follows from the monotone convergence theorem and (3.11). In particular, Ż n L 2 ,T → Ż L 2 ,T . Thus,Ż n →Ż in L T 2 a.s., according to Proposition 2.12 [Kal02].
Proposition 3.4. Let F ∈ C 2 (R) has a bounded second derivative and h ∈ C 1 [0, T ]. Then Proof. As in the proof of Lemma 3.3, we can compute for every n ≥ 1 Consequently, By the boundedness of the second derivative of F we have that there exists a constant C > 0 such that |F ′ (x)| ≤ C(1 + |x|) and |F (x)| ≤ C(1 + |x| 2 ). Therefore, e. a.s. as n → ∞. But this easily follows from the integration by parts formula.
Theorem 3.5. Let the process Z t , t ∈ [0, T ], and the random element L ∈ L T 2 be as above. Then the following equality holds.
Proof. In order to prove the theorem, we will use Proposition 3.4. We fix a function Then Let a non-negative function h ∈ C 1 [0, 1] be fixed. By Proposition 3.4, and the quadratic variation of the local (F t )-martingale equals Making ε → 0+, we can immediately conclude that for every t ∈ [0, T ] Similarly to the proof of Lemma 3.3, using the localization sequence, one can show that M Fε,h (t) → 0 in probability. By the dominated convergence theorem and Lemma A.4, Again, by the dominated convergence theorem and Lemma A.6, we have We have obtained that for every t ∈ [0, T ] t 0 L s I {Zs=0} h· , L s L 2 ds = 0 a.s.
Then taking h = 1 and applying Lemma A.3, it is easy to see that The proof of the theorem is completed.

Proof of the existence theorem
In this section, we will consider the random element X n defined in Section 3.1. According to Proposition 3.1, there exists a subsequence N ⊂ N such that X n = X n , X n , λ n I {X n =0} , I {X n >0} , Γ n → X , X, a, σ, Γ in W L 2 in distribution along N . As before, without loss of generality, we may assume that N = N. By the Skorokhod representation theorem, we can assume that this sequence converges almost surely. SinceX n →X in C ([0, T ], C[0, 1]) a.s., and a.s. for all t ∈ [0, T ] the qualityX t = X t in L 2 holds, the inequality e.] = 1. (3.13) I. We will first show that Γ = I {X·>0} · Q 2 I {X·>0} · a.s. Using Proposition 3.1 (ii), there exists a random element L in L T 2 such that Γ = L 2 a.s. Next, by Proposition 3.1 and Theorem 3.5, L · I {X·>0} = L a.s. Therefore, using the convergence of Γ n = pr n I {X n · >0} Q 2 I {X n · >0} pr n to Γ in B (L 2 ) a.s., we obtain that for evert t ∈ [0, T ] ∩ Q and k, l ≥ 1 almost surely In the last equality, we have used the fact that I (0,+∞) (x n )I (0,+∞) (x) → I (0,+∞) (x) as x n → x in R, convergence (3.13) and the dominated convergence theorem. Since the family I [0,t] e k ⊙ e l , t ∈ [0, T ] ∩ Q, k, l ≥ 1 is countable and its linear span is dense in L T 2 , we trivially get that (3.14) II. Let χ 2 be defined by (1.6). We next want to show that But this will directly follow from the following lemma.
Lemma 3.6. Let Z n t , t ∈ [0, T ], n ≥ 1, be a sequence of L 2 -valued measurable functions such that Z n t ≥ 0 for all t ∈ [0, T ] and n ≥ 1, and in B (L 2 ), and I {Z n >0} → σ in B (L 2 ) as n → ∞, then (3.17) We postpone the proof of the lemma to the end of this section.
Proof of Lemma 3.6. It is easily seen that convergence (3.16) is equivalent to the convergence as n → ∞. So, for every ϕ ∈ L T 2 , we have as n → ∞, where ϕ · ⊙ ϕ · is defined as ϕ t ⊙ ϕ t , t ∈ [0, T ]. Replacing ϕ by e k I {Z=0} for every k ≥ 1, we obtain that Taking into account the equalityĨ n t = Ĩ n t 2 , t ∈ [0, T ], we can conclude that for every k ≥ 1 such that µ k > 0. We claim that χĨ n , n ≥ 1, converges to 0 in L T 2 as n → ∞. Indeed, by convergence (3.21) and the dominated convergence theorem, Next, since I {Z n >0} → σ in the weak topology of L T 2 as n → ∞, and I {Z>0} , I {Z=0} are uniformly bounded, we trivially obtain that in the weak topology of L T 2 as n → ∞. Using the fact that I (0,+∞) (x n )I (0,+∞) (x) → I (0,+∞) (x) as x n → x in R, and the uniqueness of a weak limit, we get (3.24) Since χ ∈ L 2 , convergence (3.23) yields On the other hand side, χĨ n → 0 in L T 2 , by (3.22). Hence χσI {Z=0} = 0.
The latter equality and (3.24) yield that is equivalent to equality (3.17). This completes the proof of the lemma.
In this proof, functions from L 2 will be considered as random elements on the probability space We remark thatλ n is the conditional expectation E [λ|S n ] determined on that probability space, where S n = σ {π n k , k ∈ [n]}. By Proposition 1 [ABP98],λ n → λ in L 2 as n → ∞. In particular,λ n converges to λ in probability as n → ∞. Let n 2 e l , π n k 2 π n k = ∞ l=1 µ 2 l (pr n e l ) 2 , n ≥ 1.
Remark that pr n e l → e l in probability as n → ∞ for all l ≥ 1. We fix a subsequence N ⊂ N. Then, by Lemma 4.2 [Kal02], there exists a subsequence N ′ ⊂ N such thatλ n → λ a.s. along N ′ . Using Lemma 4.2 [Kal02] again and the diagonalisation argument, we can find a subsequence N ′′ ⊂ N ′ such that pr n e l → e l a.s. along N ′′ for all l ≥ 1. By Fatou's lemma, This implies the convergence λ n → λ a.s. along N ′′ , and hence, λ n → λ in probability as n → ∞, by Lemma 4.2 [Kal02]. We also remark that λ n ≤λ n , n ≥ 1, and λ n → λ in L 2 . Hence, dominated convergence Theorem 1.21 [Kal02] implies that λ n → λ . By Proposition 4.12 [Kal02], λ n → λ in L 2 as n → ∞.
Proof. Set A n := n l=1 ν l ε l ⊙ ε l , n ≥ 1. Then it is easily seen that the sequence {A n } n≥1 converges to √ A * A = ∞ l=1 ν l ε l ⊙ ε l in L 2 . Hence Lemma A.4. Let A ∈ L 2 , and a sequence of bounded operators B n , n ≥ 1, in L 2 converge pointwise to an operator B, that is, for every ϕ ∈ L 2 B n ϕ → Bϕ in L 2 as n → ∞. Then B is bounded and AB * n → AB * in L 2 as n → ∞.
Proof. We first note that norms B n , n ≥ 1, are uniformly bounded, by the Banach-Stheinhaus theorem. Consequently, B is a bounded operator on L 2 . Next, we will show that {AB * n } n≥1 converges to AB * in the weak topology of L 2 . Let {ε n , n ≥ 1} and {ν 2 n , n ≥ 1} are eigenvectors and eigenvalues of A * A. Then for every k, l ≥ 1 AB * n , ε k ⊙ ε l L 2 = AB * n ε l , ε k = ε l , B n A * ε k → ε l , BA * ε k = AB * , ε k ⊙ ε l L 2 as n → ∞.
Since span{ε k ⊙ ε l , k, l ≥ 1} is dense in L 2 and AB * n L 2 ≤ A L 2 B * n , n ≥ 1, is uniformly bounded, the sequence {AB * n } n≥1 converges to AB * in the weak topology of L 2 . By the dominated convergence theorem, the uniform boundedness of the norms of B n , n ≥ 1, and Lemma A.3, we obtain Remark that a non-negative definite self-adjoint operator A on L 2 has the square root, i.e. there exists a unique non-negative definite self-adjoint operator √ A on L 2 such that √ A 2 = A. This trivially follows from the spectral theorem. (iii) For every r > 0 the map Φ r : S r → B (L p,sa 2 ) defined as is Borel measurable.
Proof. Let L n , n ≥ 1, be a sequence from B (L p,sa 2 ) which converges to L in B (L 2 ). We take arbitrary t ∈ [0, T ] and ϕ, ψ ∈ L 2 and consider Due to the density of the set span I [0,t] ϕ ⊙ ψ, t ∈ [0, T ], ϕ, ψ ∈ L 2 in L T 2 , we obtain that L is self-adjoint a.e. Similarly, one can show that L is non-negative definite. Hence, B (L p,sa 2 ) is closed. Next we prove (ii). Take a sequence L n , n ≥ 1, from S r which converges to L in B (L 2 ). We remark that L ∈ B (L p,sa 2 ) due to (i). Then by Fatou's lemma and the fact that T 0 L n t e k , e k → T 0 L t e k , e k , n → ∞, for all k ≥ 1. Thus, S r is closed.
In order to check (iii), we first remark that it is enough to show that for every t ∈ [0, T ] and ϕ, ψ ∈ L 2 , the map S r ∋ L → is Borel measurable. By Theorem 1.2 [VTC87], the Borel σ-algebra on B (L 2 ) coincides with σ-algebra of all Borel measurable sets of L T 2 contained in the ball B (L 2 ). Consequently, it is enough to show that map (A.2) is Borel measurable as a map from S r to R, where S r is embedded with the strong topology of L T 2 . But then map (A.2) S r ∋ L → t 0 Φ r s (L)ϕ, ψ = t 0 L s ϕ, L s ψ ds is continuous, and, thus, Borel measurable. This finishes the proof of the lemma.