Stochastic partial differential equations: a rough paths view on weak solutions via Feynman–Kac

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

JOSCHADIEHL, PETERK. FRIZ ANDWILHELMSTANNAT

Stochastic partial differential equations: a rough paths view on weak solutions via Feynman–Kac

Tome XXVI, n^o4 (2017), p. 911-947.

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Stochastic partial differential equations: a rough paths view on weak solutions via Feynman–Kac

Joscha Diehl⁽¹⁾, Peter K. Friz⁽²⁾ and Wilhelm Stannat⁽³⁾

ABSTRACT. — We discuss regular and weak solutions to rough partial differential equations (RPDEs), thereby providing a (rough path-)wise view on important classes of SPDEs. In contrast to many previous works on RPDEs, our definition gives honest meaning to RPDEs as integral equations, based on which we are able to obtain existence, uniqueness and stability results. The case of weak “rough” forward equations, may be seen as robustification of the (measure-valued) Zakai equation in the rough path sense. Feynman–Kac representation for RPDEs, in formal analogy to similar classical results in SPDE theory, play an important role.

RÉSUMÉ. — Nous discutons des solutions régulières et faibles d’équa- tions aux dérivées partielles rugueuses (EDPR), fournissant ainsi un point de vue « chemins rugueux » sur des classes importantes d’ EDPS. Contrai- rement à de nombreux travaux antérieurs sur le sujet, notre définition donne un sens honnête aux EDPR en tant qu’équations intégrales, sur la base duquel nous sommes en mesure d’obtenir l’existence, l’unicité et la stabilité des résultats. Le cas d’équations forward faibles « rugueuses » peut être vu comme une robustification de l’équation de Zakai à va- leurs mesure, au sens des chemins rugueux. Des représentations de type Feynman–Kac pour EDPR, par analogie formelle avec les résultats clas- siques similaires dans la théorie des EDPS, jouent un rôle important.

Keywords:stochastic partial differential equations, Zakai equation, Feynman–Kac for- mula, rough partial differential equations, rough paths.

2010Mathematics Subject Classification:60H15.

(1) University of California San Diego, University of California San Diego, Dept of Mathematics - MC 0112, 9500 Gilman Drive La Jolla, CA 92093, USA —

[email protected]

(2) TU Berlin, Institut für Mathematik, MA 7-2, Strasse des 17. Juni 136, 10623 Berlin, Germany — [email protected] — Weierstrass-Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117 Berlin, Germany —

[email protected]

(3) TU Berlin, Institut für Mathematik, MA 7-2, Strasse des 17. Juni 136, 10623 Berlin, Germany — [email protected]

1. Introduction

Consider a diffusion processX on R^d with generator given by a second order differential operatorL. In its simplest form, the Feynman–Kac formula asserts that, for suitable datag,

u(t, x) =E^t,x[g(XT)], t6T, x∈R^d, (1.1) solves a parabolic partial differential equation, namely the terminal value problem

(−∂_tu_t=Lu_t u(T,·) =g .

(Below we will consider slightly more general operators including zero order terms, causing additional exponential factors in the Feynman–Kac formula.) On the other hand, the law ofXtstarted atX0=x, solves the forward (or Fokker–Planck) equation

( ∂_tρ_t=L^∗ρ_t ρ(0,·) =δx.

Formally at least, an infinitesimal version of (1.1) is given by

∂_thu_t, ρ_ti=h−Lu_t, ρ_ti+hu_t, L^∗ρ_ti= 0,

and indeed the resulting dualityhuT, ρ_Ti=hu0,ρ₀iis nothing than restate- ment of (1.1), att= 0.

In both cases, forward and backward, there may not exist a classicalC^1,2 solution. Indeed, it suffices to consider the case of degenerateX so that ρt

remains a measure; in the backward case considerg /∈C². In both cases one then needs a concept of weak solutions. A natural way to do this, consists in testing the equation in space; that is, to consider the evolution forhut, φi andhρt, fiwhereφandf are suitable test functions defined onR^d.

Applications from filtering theory lead to (backward) SPDEs of the form ( −dut=L[ut]dt+ Γ[ut]◦dWt

u(T,·) =g ,

where W = (W¹, . . . , W^e) and Γ = (Γ₁, . . . ,Γe) are first order differential operators,⁽¹⁾ in duality with the forward (or Zakai) equation

( dρ_t=L^∗[ρ_t]dt+ Γ^∗[ρ_t]◦dW_t ρ(0,·) =δx.

(1)Write Γ[u]◦dW=Pe

k=1Γk[u]◦dW^k.

Such SPDEs were studied extensively in classical works [22, 24, 25]. It is a natural question, studied for instance in a series of papers by Gyöngy [17, 18], to what extent such SPDEs are approximated by (random) PDEs, upon replacing the (Stratonovich) differential dW = dW(ω) by ˙W^ε(ω) dt, given a suitable family of smooth approximation (W^ε) to Brownian motion. In recent works [10, 14], also [12, Ch. 12], it was shown that the backward solutions u^ε, interpreted as viscosity solution (assuming g ∈ Cb) actually converge locally uniformly, with limituonly depending on the rough path limit of (W^ε). WritingW= (W,W) for such a (deterministic!) rough path (see the Appendix and [12] for notation) say,α-Hölder for 1/3 < α <1/2, the question arises if one can give an honest meaning to the equations

−dut=L[ut]dt+ Γ[ut]dWt,

dρ_t=L^∗[ρ_t]dt+ Γ^∗[ρ_t]dW_t. (1.2) In the aforementioned works, these “rough partial differential equations”

(RPDEs) had only formal meaning. The actual definition was then given ei- ther in terms of a (flow)transformed equation in the spirit of Kunita (e.g. [14], also [12, p. 177]) or in terms of a unique continuous extension of the PDE solution as function of driving noise, [5, 14].

There are two difficulties with such rough partial differential equations.

The first one is the temporal roughness ofW, a problem that has been well- understood from the rough path analysis of SDEs. Indeed, following Davie’s approach to RDEs [8], the (rough) pathwise meaning of

dX =β(X) dW

is, by definition, and writingX_s,t=X_t−X_sfor path increments, X_s,t=β(X_s)W_s,t+β⁰β(X_s)Ws,t+o(|t−s|).

Under suitable assumptions onβ, uniqueness and local/global existence re- sults are well-known. This quantifies the statement thatX is controlled by W, with “Gubinelli derivative”β(X), and in turn implies the integral rep- resentation in terms of a bona-fide rough integral (cf. [12, Ch. 4])

Xt−Xs= Z t

s

β(X) dW= lim X

[u,v]∈P

β(Xu)Wu,v+β⁰β(Xu)Wu,v. This suggests that the meaning of the backward equation (1.2) is

u(s, x)−u(t, x) = Z t

s

L[ur] dr+ Z t

s

Γ[ur] dWr,

provideduis sufficiently regular (in space) such as to makeL[u],Γ[u] mean- ingful,andprovided the last term makes sense as rough integral. The other difficulty is exactly that u may not be regular in space so that L[u],Γ[u]

require a weak meaning. More precisely, we propose the following spatially weak⁽²⁾ formulation, of the form

hus, φi − hut, φi= Z t

s

hur, L^∗φidr+ Z t

s

hur,Γ^∗φidWr,

where, again, we can hope to understand the last term as rough integral.

(Everything said for backward equations translates, mutatis mutandis, to the forward setting.)

The main result of this paper is that — in all cases — one has existence and uniqueness results. Loosely speaking (and subject to suitable regularity assumptions on the coefficients of L,Γ; but no ellipticity assumptions) we have

Theorem 1.1. — For nice terminal datag there exists a unique (spa- tially) regular solution to the backward RPDE. Similarly, for nice initial data ρ0(with nice densityp0, say) the forward RPDE has a unique (spatially) reg- ular solution.

If the terminal data g is only bounded and continuous, we have exis- tence and uniqueness of a weak solution to the backward RPDE. Similarly, if the initial dataρ0 of the forward RPDE is only a finite measure, we have existence and uniqueness of a weak (here: measure-valued) solution to the forward RPDE.

In all cases, the (unique) solution depends continuously on the driving rough path and we have Feynman–Kac type representation formulae.

Let us briefly discuss the strategy of proof. In all cases (regular/weak, for- ward/backward) existence of a solution is verified via an explicit Feynman–

Kac type formula, based on a notion of “hybrid” Itô/rough differential equa- tion, which already appeared in previous works [7, 10], see also [12]. We then use regular forward existence to show weak backward uniqueness (Theo- rem 2.8), which actually requires us to work with exponentially decaying test functions. Next, regular backward existence leads to weak (actually, measure- valued) forward uniqueness (Theorem 3.5), here we just need boundedness and some control in the sense of Gubinelli. Then weak (measure-valued) forward existence gives regular backward uniqueness. At last, we note that, subject to suitable smoothness assumptions on the coefficients, regular for- ward equations can be reformulated as regular backward equations, from which we deduce regular forward uniqueness.

(2)There is no probability here, forWis adeterministicrough path. Nevertheless, with a view to later applications to SPDEs and to avoid misunderstandings, let us emphasize that in this paper “weak” is always understood as “analytically weak”.

It is a natural question what the above RPDE solutions have to do with classical SPDE solutions. To this end, recall [12, Ch. 9] consistency of RDEs with SDEs in the following sense: RDE solutions driven byW=W^Strato(ω), the usual (random) geometric rough path associated to Brownian motion W via iterated Stratonovich integration, are solutions to the corresponding (Stratonovich) SDEs. Consider now - for the sake of argument - a regular backward RPDE solution; that is, the unique solutionu=u(t, x;W) to

−dut=L[ut]dt+ Γ[ut]dWt

(with fixedC_b² time-T terminal data). We expect that

˜

u(t, x) = ˜u(t, x;ω) =u t, x;W^Strato(ω)

(1.3) is also a (and hopefully: the unique) solution to the (backward) SPDE, again with fixed terminal data,

−d˜u_t=L[˜u_t]dt+ Γ[˜u_t]◦dW_t.

(Similar for weak backward and weak/regular forward equations.) Unfortu- nately, we cannot hope for a general RPDE/SPDE consistency statement for the simple reason that the choice of spaces in which SPDE existence and uniqueness statements are proven are model-dependent and therefore vary from paper to paper. In other words, checking that ˜u(t, x;ω) is a — and then the (unique) — SPDE solution within a given SPDE setting will necessarily require to check details specific to this setting. Luckily, there are arguments which do not force us into such a particular setting.

• Consider a notion of (Stratonovich) SPDE solution for which there are existence, uniqueness results and Wong–Zakai stability, by which we mean that the (unique bounded, or finite-measure valued) solu- tions to the random PDEs obtained by replacing dW(ω) by the mollified W˙ ^ε(ω)dt converge to the unique SPDE solution. (Such Wong–Zakai results are found e.g. in the works of Gyöngy.) Assume also that our regularity assumptions fall within the scope of these existence and uniqueness results. Then, for fixed terminal (resp.

initial) data, our unique RPDE solution, with driving rough path W=W^Strato(ω), coincides with (and in fact, may be a very pleas- ant version of) the unique SPDE solution. (This follows immediately from continuous dependence of our RPDE solutions on the driving rough paths, together with well-known rough path convergence of mollifier approximations [16].) In a context of viscosity solutions, this argument was spelled out in [14].

• Consider a notion of (Stratonovich) SPDE solution for which there are existence, uniqueness results and a Feynman–Kac representation formula. (This is the case in essentially every classical work on linear

SPDEs, especially in the filtering context.) Recall that such SPDE Feynman–Kac formulas are conditional expectations, given W(ω) (the observation, in the filtering context). In contrast, the Feynman–

Kac formula alluded to in Theorem 1.1, is of unconditional form E^t,x(. . .), the expectation taken over some hybrid Itô-rough process (with rough driver dW). By a stochastic Fubini argument (similar to the one in [10]) one can show that the Feynman–Kac formula, evaluated atW=W^Strato(ω), indeed yields the SPDE Feynman–

Kac formula. In particular, our unique RPDE solution, with driving rough pathW=W^Strato(ω), then coincides with the unique SPDE solution.

• At last, we consider an immediate consequence of our (rough path-) wise definition in case ofW=W^Strato(ω). For the sake of argument, let us now focus on the weak backward equation,

hus, φi − hut, φi= Z t

s

hur, L^∗φidr+ Z t

s

hur,Γ^∗φidWr.

With ˜u(t, x;ω) =u(t, x;W^Strato(ω)), as before it follows from con- sistency of rough with classical (backward) Stratonovich integra- tion [12, Ch. 5] that

h˜u_s, φi − h˜u_t, φi= Z t

s

h˜u_r, L^∗φidr+ Z t

s

h˜u_r,Γ^∗φi ◦dW,

for the same class of spatial test functions. Such notion of weak (or distributional) SPDE solutions appear for instance in the works of Krylov, e.g. [21, Def. 4.6]. Hence, whenever such a notion of SPDE solution comes with uniqueness results, it is straight-forward to see that ˜u, i.e. our solution constructed via rough paths, must coincide with the unique SPDE solution.

1.1. Notation

The second resp. first oder operators we shall consider are of the following form,

Lu:=1

2Tr σ(x)σ^T(x)D²u

+hb(x), Dui+c(x)u Γku:=hβk(x), Dui+γk(x)u;

withσ= (σ₁, . . . , σ_dB), β= (β₁, . . . , β_e) andbvector fields onR^dand scalar functionsc, γ₁, . . . , γ_e. We note that the formal adjoints are given as,

L^∗ϕ= 1

2Tr[˜a(x)D²ϕ] +h˜b(x), Dϕi+ ˜c(x)ϕ, Γ^∗_kϕ=hβ˜k(x), Dϕi+ ˜γk(x)ϕ

where

˜

a(x) :=a(x) :=σσ^T(x)

˜b_i(x) :=∂_ja_ji(x)−b_i(x)

˜

c(x) := 1

2∂ijaij(x)−div(b)(x) +c(x) β˜k(x) :=−βk(x)

˜

γk(x) :=−div(βk)(x) +γk(x).

(1.4)

Precise assumptions on the coefficients will appear in the theorems below.

Let us remark, however, that we did not push for optimal assumptions.

As is typical in rough path theory,C_bⁿ-regularity (bounded, with bounded derivatives up to ordern) can often be improved to Lip^γ-regularity (in the sense of Stein) withγ∈(n−1, n), depending on the Hölder exponent of the driving rough path.

2. The backward equation

Replacing the rough path by a smooth path, sayW ∈C¹([0, T],R^e) we certainly want to recover a solution to the PDE

(−∂tut=Lut+Pe

k=1ΓkutW˙_t^k (≡Lut+ ΓutW˙t)

u(T,·) =g . (2.1)

For the precise statement of the following lemma, let us now introduce a suitable class of test functions with exponential decay, that will become important in the concept of weak solutions.

Definition 2.1. — For n>0 denote with C_expⁿ (R^d) the class of func- tionsφ∈Cⁿ(R^d)such that there exists c >0such that

|D^kφ(x)|6ce^−1c|x|, k= 0,1, . . . , n . Define the quasinorm⁽³⁾ k · k_Cn

exp(R^d) as the infimum over the values of c satisfying the bound. Define moreover the spaceC_exp^m,n([0, T]×R^d) to be the

(3)which we shall need in order to speak of bounded sets inC_expⁿ (R^d).

class of functions φ ∈ C^m,n([0, T]×R^d) such that there exists c >0 such that

|D^j,kφ(t, x)|6ce^−1c|x|, j= 0, . . . , m, k= 0,1, . . . , n.

We then recall the following Feynman–Kac representation for solutions to the classical equation (2.1).

Lemma 2.2. — Assumec, b, σi, γj, βk∈C_b²,i= 1, . . . , dB,j, k= 1, . . . , e.

Letube given as u(t, x) =E^t,x

"

g(XT) exp Z T

t

c(Xr) dr+ Z T

t

γ(Xr) ˙Wrdr

!#

(2.2) with

dX_t=σ(X_t) dB(ω) +b(X_t) dt+β(X_t) ˙W_tdt ,

whereB is ad_B-dimensional Brownian motion and W ∈C¹([0, T],R^e).

(1) Ifg∈C_b²(R^d)thenuis the uniqueC_b^1,2([0, T]×R^d)solution to(2.1).

If moreoverg∈C_exp² (R^d)thenu∈C_exp^1,2([0, T]×R^d).

(2) Ifg∈Cb(R^d)thenu∈Cb([0, T]×R^d)and it is the unique bounded analytically weak solution to (2.1), that is, forϕ∈ D(R^d)

hut, ϕi=hg, ϕi+ Z T

t

hur, L^∗ϕidr+ Z T

t

hur,Γ^∗ϕidWr. (2.3) Proof. — Let us first note that the expectation actually exists, since g, c,γ and|W˙ |are bounded.

(1): The proof amounts to taking derivatives under the expectation, see for example Theorem V.7.4 in [19], which shows that u is a C_b^1,2 solu- tion.⁽⁴⁾ Uniqueness follows from the maximum principle, see for example Theorem 8.1.4 in [20].

Ifg ∈C_exp² (R^d) then one can show that actuallyu∈C_exp^1,2([0, T]×R^d).

This is similar to the rough case in Theorem 2.8, so we omit the proof here.

(2): Take somegⁿ∈C_b²(R^d) converging toglocally uniformly, uniformly bounded by 2kgk_∞. Let uⁿ be the corresponding classical solution from part (1). Thenuⁿsatisfies (2.3) withgreplaced bygⁿ. Now by the Feynman–

Kac representation, we get for everyN >0,

|uⁿ(t, x)−u(t, x)|.E[|gⁿ(X_T^t,x)−g(X_T^t,x)|²]^1/2 6 sup

|y|6N

|gⁿ(y)−g(y)|+ 2kgk_∞E[1_[−N,N]C(|X_T^t,x)].

(4)In [19] it is assumed that the term in the exponential is non-positive, but a term bounded from below poses no additional difficulty: just replaceu(t, x) byu(t, x)e^−c(T−t) forcsufficiently large.

Hence for everyR >0 sup

|x|6R

|uⁿ(t, x)−u(t, x)|. sup

|y|6N

|gⁿ(y)−g(y)|+ 1 N sup

|x|6R

E[|X_T^t,x|], from which the locally uniform convergence of uⁿ_t to ut follows, uniformly in t 6 T. Taking the limit in the integral equation, we then see that u satisfies (2.3).

To show uniqueness, letu∈C_b([0, T]×R^d) be any solution to (2.3). It is immediate that the equation then also holds for test functionsϕ∈C_c²(R^d).

It is straightforward to show that forϕ∈C_c^1,2([0, T]×R^d) we have hut, ϕti=hg, ϕTi+

Z T t

hur,−∂tϕr+L^∗ϕridr+ Z T

t

hur,Γ^∗ϕridWr. (2.4) Finally, via dominated convergence, (2.4) also holds forϕ∈C_exp^1,2([0, T]×R^d).

Now, an application of Lemma 3.1 (2) yields, for every t ∈ [0, T) and everyφ∈C_exp⁴ (R^d), aϕ∈C_exp^1,2([t, T]×R^d) that satisfies

∂_sϕ_s=L^∗ϕ_s+ Γ^∗ϕ_sW˙_s, ϕt=φ .

Then, by (2.4),

hut, φi=hut, ϕti=hg, ϕTi.

So, tested against φ ∈ C_c⁴(R^d), all solutions coincide at every t ∈ [0, T], which gives uniqueness inCb([0, T]×R^d).

In (2.1), replacing W by a rough pathW, we are interested in the fol- lowing formal equation

−du=Ludt+ ΓudW

u(T,·) =g . (2.5)

We will next introduce two solution concepts, weak and regular in nature (see Definitions 2.3 and 2.6 below). Both rely on the (standard) notion of a controlled rough path spaceDW^2α- see Appendix, Definition 4.1, for a recall.

Definition 2.3(Analytically weak backward RPDE solution). — Given anα-Hölder rough pathW= (W,W),α∈(1/3,1/2], we say that a bounded, measurable function u = u(t, x;W) = ut(x;W) is an analytically weak solution to (2.5), if for all functions ϕ∈C_exp³ (R^d), we have (Y^ϕ,(Y^ϕ)⁰)∈ DW^2αwith

(Yt)^ϕ_i :=hut,Γ^∗_iϕi, (Y_t^ϕ)⁰_ij:=−

ut,Γ^∗_jΓ^∗_iϕ ,

that is

kY^ϕ,(Y^ϕ)⁰k_W,2α<∞, (2.6) and the following equation is satisfied:

hut, ϕi=hg, ϕi+ Z T

t

hur, L^∗ϕidr+ Z T

t

hur,Γ^∗ϕidWr, 06t6T. (2.7) Here,R

YdWis the rough integral against (Y, Y⁰).

Remark 2.4. — Different from the smooth case, Lemma 2.2, we work with test functions in the larger classC_exp³ here. This is necessary, since with presence of the rough integral we were not able to automatically enlarge the space of functions for which the integral equation holds, as was done in the proof of Lemma 2.2.

Remark 2.5. — Heuristically, the origin of the compensator term Y_t⁰= hu_t,Γ^∗Γ^∗ϕican be seen as follows. One certainly expects that

Z t s

hu_r,Γ^∗ϕidW_r≈ hu_s,Γ^∗ϕiW_s,t wherea≈bmeansa−b=O

|t−s|^2α

. Hence, in view of (2.7), hut, ϕi − hus, ϕi ≈ −

Z t s

hu,Γ^∗ϕidW≈ − hus,Γ^∗ϕiWs,t

Replacingϕby Γ^∗ϕ(note that the latter is not inC_exp³ though) gives hut,Γ^∗ϕi − hus,Γ^∗ϕi=− hus,Γ^∗Γ^∗ϕiWs,t+O

|v−u|^2α

so that t 7→ hu_t,Γ^∗ϕi is controlled by W, with Gubinelli derivative

− hu_t,Γ^∗Γ^∗ϕi.

Definition 2.6 (Regular backward RPDE solution). — Given an α- Hölder rough path W = (W,W), α ∈ (1/3,1/2], we say that a function u=u(t, x;W)∈C^0,2 (with respect to t, x) is a solution to (2.5)if, for all x∈R^d,(Γu(·, x),ΓΓu(·, x))is controlled byW (Definition 4.1) and

u(t, x) =g(x) + Z T

t

Lu(r, x) dr+ Z T

t

Γu(r, x) dWr. (2.8) Remark 2.7. — If a regular solution in the sense of Definition 2.6 pos- sesses a uniform bound on the control (see for example (2.10) below) then it is also a weak solution in the sense of Definition 2.3.

Recall thatgeometric rough paths are limits of smooth paths under the appropriate rough path metric. While rough path integration does not rely on this assumption (indeed, (2.7), resp. (2.8) were formulated for a general

α-Hölder rough path), it is a very natural assumption when it comes to stability results.

Theorem 2.8. — Throughout, W is a geometricα-Hölder rough path, α ∈ (1/3,1/2]. Assume σ_i, β_j ∈ C_b³(R^d), b ∈ C_b¹(R^d), c ∈ C_b¹(R^d), γ_k ∈ C_b²(R^d). Considerg∈C_b⁰(R^d).

(1) Stability. Let u = u^W be the solution to (2.1) as given by the Feynman–Kac representation(2.2), wheneverW∈C¹. PickW∈C¹ convergent in rough path sense to W. Then there exists a bounded, continuous function u^W, independent of the choice of the approxi- mating sequence, so that u^W →u^W uniformly. The resulting map W 7→ u^W is continuous. Moreover, the following Feynman–Kac representation holds:

u^W(t, x) =E^t,x

"

g(XT) exp Z T

t

c(Xr) dr+ Z T

t

γ(Xr) dWr

!#

, whereX solves the rough SDE (see Appendix, Lemma 4.19)

dX_t=σ(X_t) dB(ω) +b(X_t) dt+β(X_t) dWt, (2.9) whereB is adB-dimensional Brownian motion.

(2) Analytically weak backward RPDE solution. Let u = u^W be the function constructed in (1). Then u= u^W ∈ Cb([0, T]×R^d) is a bounded solution to (2.5)in the sense of Definition 2.3. Moreover, (2.6)is bounded, uniformly over bounded sets ofϕinC_exp³ (R^d), and u^W is the only solution in the class ofC_b functionsusatisfying this uniform bound on (2.6).

(3) Regular backward RPDE solution. Assume σ_i, β_j ∈ C_b⁶(R^d), b ∈ C_b⁴(R^d), c ∈ C_b⁴(R^d), γk ∈ C_b⁶(R^d) and g ∈ C_b⁴(R^d). Then u = u^W ∈C_b^0,4([0, T]×R^d) is a bounded solution to (2.5)in the sense of Definition 2.6. It is the only solution in the class of functions in C_b^0,4([0, T]×R^d)that satisfies

sup

x

kΓu(·, x),ΓΓu(·, x)k_W,2α<∞. (2.10) If moreoverg∈C_exp⁴ (R^d), thenu∈C_exp^0,4([0, T]×R^d).

Remark 2.9. — We consider solutions in C_b^0,4, instead of the obvious choice C_b^0,2, because of two reasons. First, in order to show that u is controlled by W we need g ∈ C_b⁴(R^d) which automatically gives us u ∈ C_b^0,4([0, T]×R^d). Second, this additional regularity is needed for the unique- ness proof via duality.

Remark 2.10. — Results of the type in Theorem 2.8 (1), even in nonlin- ear situations, were obtained in [4, 5, 9, 7, 14]. However, in all these refer- ences, the only intrinsic meaning of these equations was given in terms of a transformed equation, somewhat in the spirit of the Lions–Souganidis [23]

theory of stochastic viscosity solutions. On the contrary, parts (2) and (3) of the above theorem present a direct intrinsic characterization. See also [3, Ch. 3].

Proof. — (1): This follows from stability of “rough SDEs”, see Lem- ma 4.19.

(2):Existence. — For simplicity only, we takec=γ=b= 0 so that u(s, x) =E[g(X_T^s,x)],

dXt=σ(Xt) dBt(ω) +β(Xt) dWt.

(ByX^s,xwe mean the unique solution started atX_s=x.) In the following we consider the above SDE as an RDE w.r.t. the joint lift Z = (Z,Z) of Wand the Brownian motionB (see Lemma 4.18 and Lemma 4.19 below).

Denote with Φ its associated flow.

Recall Yt = hut,ϕi,¯ Y_t⁰ =−hut,Γ^∗ϕi, where ¯¯ ϕ:= Γ^∗ϕ. Since ϕ ∈C_exp³ andβj ∈C_b²,j= 1, . . . , e, we have that ¯ϕ∈C_exp² . Then

Yt−Ys−Y_t⁰Ws,t

=E Z

R^d

{g(Φt,T(x))−g(Φs,T(x))}ϕ(x) +¯ g(Φt,T(x))Γ^∗ϕ(x)W¯ s,tdx

=E Z

R^d

g(y)n

¯

ϕ(Φ⁻¹_t,T(y)) det(DΦ⁻¹_t,T(y))−ϕ(Φ¯ ⁻¹_s,T(y)) det(DΦ⁻¹_s,T(y)) + Γ^∗ϕ(Φ¯ ⁻¹_t,T(y)) det(DΦ⁻¹_t,T(y))W_s,to

dy

=E Z

R^d

g(y)n

¯

ϕ(Φ⁻¹_t,T(y)) det(DΦ⁻¹_t,T(y))−ϕ(Φ¯ ⁻¹_s,T(y)) det(DΦ⁻¹_s,T(y)) + Γ^∗ϕ(Φ¯ ⁻¹_t,T(y)) det(DΦ⁻¹_t,T(y))W_s,t + Γ^∗ϕ(Φ¯ ⁻¹_t,T(y)) det(DΦ⁻¹_t,T(y))B_s,to

dy

. Note that Γ^∗ϕ=−div (bϕ). Hence the term in curly brackets is bounded in absolute value, using Lemma 4.9, by a constant times

kϕk¯ _C³

b(M(y))exp(CN_1;[0,T](Z)) (kZkα+ 1)^17+3d|t−s|^2α.

Hence

|Yt−Ys−Y_t⁰Ws,t| .

Z

R^d

E hkϕk_C³

b(M(y))exp(CN_1;[0,T](Z)) (kZkα+ 1)^17+3di

dy|t−s|^2α. Next observe that

E hkϕk_C³

b(M(y))exp(CN_1;[0,T](Z)) (kZkα+ 1)^17+3di 6E

hkϕk²_C3 b(M(y))

i^1/2 E

hexp(2CN_1;[0,T](Z)) (kZkα+ 1)^34+6di^1/2 , and Lemma 4.18 now implies that the last term is bounded and Lemma 4.21 implies that the first term decays exponentially iny. Therefore

|Y_t−Y_s−Y_t⁰W_s,t|.|t−s|^2α, as desired.

The estimate|Y_t⁰−Y_s⁰|6C|t−s|^αis shown analogously, and then

|Y_t−Y_s−Y_s⁰W_s,t|6|Y_t−Y_s−Y_t⁰W_s,t|+|Y_s,t⁰ W_s,t|=O(|t−s|^2α), as desired.

It remains to show that the integral equation (2.7) is satisfied. For this let Wⁿ be a sequence of smooth paths converging to W in α-rough path metric. Letuⁿ be the solution to (2.3) as given by Lemma 2.2 (2).

Part (1) of the theorem now implies thatuⁿ converges locally uniformly tou, hence the convergence of all the terms in (2.7) except the rough integral is immediate. For the rough integral, in view of Theorem 4.16 in [12], it is enough to show that

sup

n

kY⁰ⁿk_α<∞ sup

n

kY⁰ⁿ−Y⁰k_∞→0 sup

n

kRⁿk2α<∞ sup

n

kRⁿ−Rk∞→0, withY_tⁿ:=huⁿ_t,Γ^∗ϕi, Y⁰ⁿ:=huⁿ_t,Γ^∗Γ^∗ϕiand

Rⁿ_s,t=huⁿ_t −uⁿ_s,Γ^∗ϕi − huⁿ_s,Γ^∗Γ^∗ϕiW_s,tⁿ .

The first two statements follow from the fact that the preceding considera- tions were uniform forWbounded in rough path norm. Finally, convergence in supremum norm ofY⁰ⁿ_t −Y_t⁰ =huⁿ_t −ut,Γ^∗Γ^∗ϕi andRⁿ_s,t−Rs,t follows from local uniform convergence ofuⁿ.

Uniqueness. — Letφ∈C_exp^0,3([t, T],R^d) be such that⁽⁵⁾ φ(t, x) =ϕ(x) +

Z T t

α(r, x) dr+ Z T

t

ηi(r, x) dWⁱ_r,

withα∈C_exp^0,3([0, T]×R^d), and (η_i=1,...,e, ηi,j=1,...,e⁰ ) controlled byW. Assume moreover for someδ >0

kη(x), η⁰(x)kW,2α.e^−δ|x|, kDη(x), Dη⁰(x)kW,2α.e^−δ|x|. Then by Lemma 4.17

huT, φTi=hut, φti − Z T

t

hur, L^∗φridr− Z T

t

hur,Γ^∗_kφridW^k_r +

Z T t

hu_r, α(r)idr+ Z T

t

hu_r, η_k(r)idW^k_r.

So it remains to find, for givenϕ, such aφwithα(r) =L^∗φ(r), ηi(r) = Γ^∗_iφ(r) and η_i,j⁰ (r) = Γ^∗_jΓ^∗_iφ(r). But this is exactly what Theorem 3.5 (3) gives us forϕ∈C_exp⁴ (R^d). Then

hg, φTi=huT, φTi=hut, φti=hut, ϕi,

which gives uniqueness ofu_t. This holds for allt∈[0, T], which gives unique- ness ofu.

(3): Again, for simplicity only, we takec=γ=b= 0 so that u(t, x) =E

g X_T^t,x . Then

Du(t, x) =E[Dg(X_T^t,x)DX_T^t,x].

Indeed, using the integrability ofDX given by Lemma 4.19 and the fact that Dgis bounded, the statement follows from interchanging differentiation and integration, see for example Theorem 8.1.2 in [11].

(5)Notation dWⁱ,dW^k, . . . is somewhat abusive, see Remark 4.4.

Then by Lemma 4.14 Du(t, x)−Du(s, x)

=E[Dg(X_T^t,x)DX_T^t,x−Dg(X_T^s,x)DX_T^s,x]

=E Z t

s

Dg(X_T^r,x)dDX_T^r,x+ Z t

s

D²g(X_T^r,x)hdX_T^r,x, DX_T^r,x· i

=E hZ t

s

Dg(X_T^r,x)D²X_T^r,xhV(x)dZ_r,· i+ Z t

s

Dg(X_T^r,x)DV(x)DX_T^r,xdZ_r +

Z t s

D²g(X_T^r,x)hDX_T^r,xV(x)dZr, DX_T^r,x· ii

+O(|t−s|)

=E[Dg(X_T^s,x)D²X_T^s,xhβ(x)Ws,t,· i] +E[Dg(X_T^s,x)DX_T^s,xDβ(x)]Ws,t

+E[D²g(X_T^s,x)hDX_T^s,xV(x)Ws,t, DX_T^s,x· i] +O(|t−s|^2α)

=∂x[β(x)Du(s, x)]Ws,t+O(|t−s|^2α).

So Γuis controlled as claimed and (2.10) is satisfied. Showing thatu∈C_b^0,4 also follows from differentiation under the expectation and the proof that the integral equation is satisfied now follows by using smooth approximations to W, as in part (2).

Uniqueness follows from existence of the measure-valued forward equa- tion. The argument is dual to the one that will be used in the proof of Theorem 3.5 (2), so we omit the proof here.

Finally, the exponential decay ofu, ifg∈C_exp⁴ , follows from Lemma 4.20.

3. The forward equation

We now consider the forward equation (∂tρt=L^∗ρt+Pe

k=1Γ^∗_kρtW˙_t^k (≡L^∗ρt+ Γ^∗ρtW˙t)

ρ₀=p₀. (3.1)

on the spaceM(R^d) of finite measures onR^d.

Equation (3.1) is dual to the backward equation — considered in the previous section — in a sense that will be made precise in the following (see in particular Corollary 3.7 below).

The spaceM(R^d) is endowed with the weak topology; that isµn→µif µn(f)→ µ(f) for all f ∈Cb(R^d). It is metrizable with compatible metric

given by the Kantorovich–Rubinstein metricd, defined as d(µ, ν) := sup

kfk_C1 b(Rd)61

Z

R^d

f(x)ν(dx)− Z

R^d

f(x)µ(dx)

(see Chapter 8.3 in [2]). A compatible metric on the space of continuous finite-measure-valued paths is then given byd(µ_·, ρ_·) := sup_t6Td(µt, ρt).

Lemma 3.1. — Assumec, b, σ_i, γ_j, β_k∈C_b²,i= 1, . . . , d_B,j, k= 1, . . . , e.

Define forW ∈C¹the measure valued processρvia its action onf ∈Cb(R^d) as

ρt(f) :=E^0,ν

f(Xt) exp Z t

0

c(Xs) ds+ Z t

0

γ(Xs) ˙Wsds

, (3.2) whereν ∈ M(R^d)is the law of the initial condition of the diffusion X with dynamics

dXt=σ(Xt) dB(ω) +b(Xt) dt+β(Xt) ˙Wtdt, whereB is ad_B-dimensional Brownian motion.

(1) Thenρis the unique, continuousM(R^d)-valued path which satisfies, for allf ∈C_b²(R^d),

ρt(f) =ν(f) + Z t

0

ρs(Lf) ds+ Z t

0

ρs(Γkf) dW_s^k. (3.3) (2) Assume moreoverσ_i ∈C_b⁴(R^d),i= 1, . . . , d_B, b, β_k ∈C_b³(R^d),k=

1, . . . , e.

If ν has a density p0 ∈ C_exp² (R^d) then ρt has a density p ∈ C_exp^1,2([0, T] ×R^d) which is the unique bounded classical solution to(3.1).

Remark 3.2. — We choose p₀∈C_exp² (R^d) in part (2) since this is what we shall work with in the rough case. In the smooth case, the assumptions on the densityp₀can be weakened. Assume for example thatν has a density p0∈C_b²∩L¹. Then ρthas a C_b² densityptfor allt>0 andp∈C_b^1,2 is the unique bounded classical solution to (3.1). Moreover,

kptk_L1(R^d)=ρ_t(1) =E^0,ν

exp Z t

0

c(X_s) ds+ Z t

0

γ(X_s) ˙W_sds

. (3.4) Indeed, by the smoothness assumptions on the coefficients, (3.1) has a unique solution inC_b^1,2(this can again be seen by a Feynman–Kac argument, as in Lemma 2.2).

We have to show that the unique classical solution pt ∈ C_b² of (3.1) with non-negative initial condition p0 ∈ C_b²∩L¹ is integrable. First recall that from the maximum principle, pt > 0 for all t > 0 (see for example

Theorem 8.1.4 in [20]). Note that (3.1) implies that_dt^d R

ϕp_tdx=RL˜_tϕp_tdx, where ˜L_tφ:=Lφ+ Γ_kφW˙_t^k, hence

Z

ϕptdx= Z

ϕp0dx+ Z t

0

Z L˜sϕpsdxds (3.5) for any smooth and compactly supported function ϕ. Our aim now is to extend this equality to the constant functionϕ≡1. To this end consider for ε >0 the function

ϕε(x) :=ϕ εkxk² ,

whereϕ(r) = (1 +r)^−d+12 ,r>0. It is easy to check that bothϕεand Lϕε(x)

=−ε(d+ 1) 1 +εkxk²−^d+3₂ X

ij

(σσ^T)_ij(x) +X

i

(˜b_i)_t(x)x_i

!

+cϕ_ε(x)

+ε²(d+ 1)(d+ 3) 1 +εkxk²−^d+5₂ X

ij

(σσ^T)ij(x)xixj

!

are integrable. Since the coefficientsσσ^T and ˜b_t:=b+ Γ_kW˙_t^k have at most linear growth andcis bounded, there exists a finite constantM, independent ofε, such that

L˜_sϕ_ε6M ϕ_ε.

Next fix a smooth compactly supported test functionχ on Rsatisfying 1_[−1,1] 6χ 61_[−2,2] and let χ_N(x) := χ_kxk2

N²

. Then χ_Nϕ_ε is compactly supported and

L˜t(χNϕε)

=χ_NL˜_tϕ_ε−4εe+ 1 N² χ⁰

kxk² N²

1 +εkxk²−^d+3₂ X

ij

(σσ^T)(x)x_ix_j + ((L0)tχN)ϕε

where ( ˜L0)tu= ˜Ltu−cu. Again due to the assumptions on the coefficients of L (resp. L0) we obtain that L0χN is uniformly bounded in N, so that

|L˜_t(χ_Nϕ_ε)| is uniformly bounded in N in terms of ϕ_ε and |L˜_tϕ_ε|. Since L˜t(χNϕε) → L˜tϕε pointwise, Lebesgue’s dominated convergence now im- plies that (3.5) extends to the limitN → ∞, hence

Z

ϕεptdx= Z

ϕεp0dx+ Z t

0

Z

L˜sϕεpsdxds 6

Z

ϕ_εp₀dx+M Z t

0

Z

ϕ_εp_sdxds .

(3.6)