Diffusion Semigroups and Diffusion Processes Corresponding to Degenerate Divergence Form Operators

J. Quastel^{1} and S. R. S. Varadhan^{2}

1. Introduction. In this article we study existence and uniqueness problems for semigroups and processes determined by time dependent second order linear partial differential operators of the form

L_{t}u = ∇ · a∇u + σc · ∇u  (1.1)

For simplicity we will take the space to be the d-dimensional torus T^{d}. The matrix a : [0, T] × T^{d} → R^{d} ⊗ R^{d} is assumed to be symmetric and non-negative definite. We define σ to be the symmetric non-negative definite square root of a, and c : [0, T] × T^{d} → R^{d} is a function about which assumptions will be made later. The focus of our study is on cases where the coefficients a and c have minimal regularity, and a is allowed to degenerate.

From the point of view of partial differential equations one considers parabolic equations of the form

\frac{∂u}{∂t} + L_{t}u = 0  (1.2)

on [0, T]×T^{d}, with a (final) boundary condition u(T, x) = f(x), and tries to prove existence and uniqueness in some suitable class. The class of potential solutions is to be chosen carefully. Formally one can try to obtain an apriori energy estimate by multiplying the solution by u and integrating. From the maximum principle one expects the solution to be in L^{∞}([0, T] × T^{d}) if we start with an f in L^{∞}(T^{d}). It is therefore not unnatural to look for solutions in the class H_{1,a} consisting of u such that

∫_{0}^{T} dt ∫_{T^{d}} dx ∇u · a∇u < ∞  (1.3)

provided we impose some control on c. But a serious question that arises is the meaning of a condition like (1.3) especially when a is degenerate. If a = σ^{2} with a symmetric σ, what we have is

∫_{0}^{T} dt ∫_{T^{d}} dx |σ∇u|^{2} < ∞

and we can try to define σ∇u as a distribution. It is therefore natural to impose on σ the condition

∫_{0}^{T} dt ∫ dx ∑_{i,j=1}^{d} |∇σ_{ij}|^{2} < ∞.

1 Department of Mathematics, University of California, Davis, CA 95616. Partially supported by U. S. National Science Foundation grant DMS-9504791.

2 Courant Institute, 251 Mercer St., New York, NY 10012. Partially supported by U. S. National Science Foundation grant DMS-9503419 and Army Research Office grant ARO-DAAH04 -95-1-0666.

Another issue that has to be faced is the control of large values of a or equivalently σ. In any case we have assumed no regularity of a in t and some control on a is needed. Our motivation comes from an effort to understand the motion of a tagged particle in an interacting particle system away from equilibrium. Such systems in their hydrodynamic scaling limits can be described approximately by their particle density ρ(t, x) which is a function of time and space. A tagged particle in equilibrium at a constant density ρ will diffuse like a Brownian Motion with a diffusion matrix a(ρ). One expects that, to within first order corrections that are due to density gradients, the leading term in the generator for the tagged particle in nonequilibrium will given by

\frac{1}{2}∇a(ρ(t, x))∇u.

In other words one expects the generator to be

L_{t}u = \frac{1}{2}∇a(ρ(t, x))∇u + σc∇u

for some decent choice of c. Self consistency dictates that ρ itself should be a solution of

\frac{∂ρ}{∂t} = L_{t}^{∗}ρ

We cannot expect a(t, x) = a(ρ(t, x)) to have any more regularity than ρ even when a(ρ) is a fairly regular function of ρ. While a(ρ) is nondegenerate or elliptic, the ellipticity could degenerate at the fringes, especially at high density. For what are known as simple exclusion models, there is a natural upper bound of 1 for the density and a(ρ) → 0 as ρ → 1. If the regularity of ρ is not strong enough to force continuity of ρ we do not even have local ellipticity. In the context of large deviation theory the only regularity one has on ρ is the finiteness of some energy-like rate function and so, for wide applicability, it is important to deal with a satisfying minimal regularity. It will be useful to investigate the case with minimal control on the size of a, namely a ∈ L^{1}([0, T] × T^{d}). We will therefore assume that

∫_{0}^{T} dt ∫ dx { ∑_{i,j=1}^{d} |∇σ_{ij}|^{2} + ∑_{i,j=1}^{d} |σ_{ij}|^{2}} < ∞.  (A1)

If one can establish that a(ρ) is Lipshitz in ρ then any minimal regularity that one has on ρ(t, x) can be transferred to a(t, x).

From the probabilistic point of view one is intersted in proving a uniqueness theorem for nonnegative weak solutions of the adjoint equation

\frac{∂p}{∂t} = L_{t}^{∗}p  (1.4)

with the initial condition p(0, x) = p_{0}(x). Section 2 contains some basic approximation tools that are needed and in section 3 we establish the existence and uniqueness theorems for parabolic equations under suitable conditons on a and c. The applications that we have in mind have a peculiar twist to the usual martingale problem corresponding to a given generator. We are given L_{t} as well as a nonnegative weak solution p of the equation (1.4). We want to show the existence and uniqueness of a measure P on the space of continuous functions that has p for its marginals and is a solution to the martingale problem for L_{t}. The vector field c can have singularities and in fact need not even be defined where p = 0, which can be a set of positive Lebesgue measure. In our application c will involve some sort of logarithmic gradient of p. In a much simpler context this is like taking an arbitrary nonnegative function p(x) and writing it as a solution of

0 = \frac{∂p}{∂t} = \frac{1}{2}Δp − ∇.(\frac{∇p}{2p})p

and asking if there is a unique process with generator

L_{t} = L = \frac{1}{2}Δ + \frac{∇p}{2p}.∇

that has p for an invariant measure. The first order term \frac{∇p}{2p} can be quite bad. This is the motivation for sections 4 and 5. For some of the results in these sections we need to strengthen the hypothesis (A1) by replacing it with (A2).

∫_{0}^{T} dt ∫ dx T race (ȧa^{−1}ȧ) < ∞.  (A2)

Here ȧ denotes \frac{∂a}{∂x_{i}} for any i = 1, …, d and

T race (ȧa^{−1}ȧ) = ∑_{i,j,k,ℓ=1}^{d} \frac{∂a_{jk}}{∂x_{i}} (a^{−1})_{kℓ} \frac{∂a_{ℓj}}{∂x_{i}} .

The condition (A2) is technical; we need it in order to prove the uniqueness of very weak solutions, which is the type of uniqueness that comes up when one studies the martingale problem for L_{t}. It implies both (A1) and the classical finite energy condition ∫_{0}^{T} dt ∫ dx(∇· a)a^{−1}(∇ · a) < ∞. In one dimension it is equivalent to either, while in higher dimensions it is considerably stronger in general. We do not claim that it is optimal. However when a(t, x) = a(ρ(t, x)) for a nice function a(ρ) the two conditions (A1) and (A2) are not too far apart and both demand the finiteness of some weighted quadratic norms on ∇ρ.

2. Some approximation results. We start with the bilinear form

⟨ϕ, ψ⟩_{1,a} = ∫_{0}^{T} dt ∫ dx (∇ϕ · a∇ψ + ϕψ)  (2.1)

that defines an inner product on C^{∞} ([0, T] × T^{d}). The Hilbert space H_{1,a} is defined as the completion of C^{∞} ([0, T] × T^{d}) under the norm

∥ϕ∥_{H_{1,a}} = ⟨ϕ, ϕ⟩_{1,a}^{1/2} .

If ϕ_{n} is a Cauchy sequence of smooth functions in the space H_{1,a}, that converges to ϕ ∈ L_{2}([0, T]×T^{d}) the corresponding sequence f_{n} = σ∇ϕ_{n} has a limit f in L_{2}([0, T]×T^{d}) and it is natural to call f = σ∇ϕ. On the other hand for ϕ ∈ L_{2}([0, T], T^{d}) we can try to define the norm by duality as given by

N^{2}(ϕ) = sup {2 ∫_{0}^{T} dt ∫ dxϕ (∇ · σg + f) − ∫_{0}^{T} dt ∫ dx(|g|^{2} + |f|^{2})}  (2.2)

with the supremum being taken over f, g_{1}, … g_{d} ∈ C^{∞}([0, T]×T^{d}). As long as (A1) holds, the above definition makes sense and for ϕ ∈ H_{1,a}

N(ϕ) = ∥ϕ∥_{H_{1,a}}

A key point is that if (A1) holds, the converse is also true. If N(ϕ) < ∞ then ϕ ∈ H_{1,a} and

∥ϕ∥_{H_{1,a}} = N(ϕ).

This requires a certain amount of work, because the the way in which the usual mollification process of convolution by a smooth function works on H_{1,a} is far from clear. It will be proved below.

We first prove the easy fact that condition (A2) implies condition (A1).

Lemma 2.1. Let a : [0, T] × T^{d} → R^{d} ⊗ R^{d} and let σ denote the non-negative definite square root of a. Then

∑_{i,j} \left|{ \atop σ̇_{ij}}\right|^{2} ≤ T race (ȧa^{−1}ȧ) .

Proof. Suppose that a is diagonal at some point (t, x). Then

σ̇_{ij} = \frac{ȧ_{ij}}{\sqrt{a_{ii}} + \sqrt{a_{jj}}} .

Therefore at (t, x),

∑_{i,j} \left|{ \atop σ̇_{ij}}\right|^{2} = ∑_{i,j} (\frac{ȧ_{ij}}{\sqrt{a_{ii}} + \sqrt{a_{jj}}} )^{2} ≤ ∑_{i,j} \frac{ȧ_{ij}ȧ_{ji}}{a_{jj}} = T race (ȧa^{−1}ȧ) .

But T race(MM^{∗}) = ∑_{ij} |M_{ij}|^{2} and T race(M) are invariant under orthogonal change of basis, and a is diagonalizable at every point. The lemma follows.

Let G_{ϵ} be the fundamental solution of the heat equation at time ϵ^{2}, and let ϕ_{ϵ} := ϕ∗G_{ϵ} be the spatial convolution. The following lemma provides a crucial estimate that will be used repeatedly.

Lemma 2.2. Let f ∈ W^{1,2}. Then the family of functions

g_{ϵ}(x) = ∫ dy |f(x + y) − f(x)|^{2} \frac{|∇G_{ϵ}|^{2}}{G_{ϵ}} (y).

for ϵ > 0 is uniformly integrable.

Proof. Recall that a family of functions g_{γ}, γ ∈ Γ is uniformly integrable if and only if there exists a function Ψ : R_{+} → R_{+} such that

(a) lim_{x→∞} \frac{Ψ(x)}{x} = ∞

(b) sup_{γ∈Γ} ∫ dx Ψ (|g_{γ}(x)|) < ∞.

Furthermore if g is any function in L^{1} then there exists an increasing, convex function Ψ satisfying (a) such that Ψ(|g|) ∈ L^{1}. Choose such a Ψ for |∇f|^{2}. For points x ∈ T^{d} and y ∈ R^{d} the point x + y is of course well defined on T^{d} and the convolution can be performed by integrating with respect to y on R^{d}. For x ∈ T^{d} and y ∈ R^{d}, let us define

h_{y}(x) = \frac{|f(x + y) − f(x)|^{2}}{∥y∥^{2}}

By Jensen’s inequality,

∫ dx Ψ(h_{y}(x)) = ∫ dx Ψ (\frac{|f(x + y) − f(x)|^{2}}{∥y∥^{2}} )

= ∫ dx Ψ (| ∫_{0}^{1} dt \frac{⟨y · ∇f(x + ty)⟩}{∥y∥}|^{2})

≤ ∫ dx ∫_{0}^{1} dt Ψ (|∇f(x + ty)|^{2})

= ∫ dx Ψ (|∇f|^{2})

The last term is finite, and therefore the family h_{y}, y ∈ R^{d} is uniformly integrable. Let

c_{ϵ} = ∫_{R^{d}} dy∥y∥^{2} \frac{|∇G_{ϵ}|^{2}}{G_{ϵ}} (y)

and note that c_{ϵ} = c < ∞, is independent of ϵ. Therefore

∫ dx Ψ (c^{−1}g_{ϵ}(x)) = ∫ dx Ψ (∫ dy |f(x + y) − f(x)|^{2} c^{−1} \frac{|∇G_{ϵ}|^{2}}{G_{ϵ}} (y))

≤ ∫ dx ∫ dy Ψ (h_{y}(x)) c^{−1}∥y∥^{2} \frac{|∇G_{ϵ}|^{2}}{G_{ϵ}} (y)

≤ sup_{y∈R^{d}} ∫ dx Ψ (h_{y}(x)) .

We have just shown that the last term is bounded. Therefore the family {c^{−1}g_{ϵ}, ϵ > 0} is uniformly integrable and so is the family {g_{ϵ}, ϵ > 0}.

The basic fact regarding convolutions in H_{1,a} is contained in the next lemma. Although the lemma is an easy consequence of the earlier lemma, its conclusion is far from obvious. When σ is not invertible, control over |σ∇u| can hide a lot of cancellations. Convolving u while leaving σ alone could royally mess up these cancellations. The assumption (A1) prevents this. Lemma 2.3. Suppose that (A1) holds.

(i) If ϕ ∈ L^{2} ([0, T] × T^{d}) then ∥σ∇ϕ_{ϵ} − (σ∇ϕ)_{ϵ} ∥_{L^{1}([0,T]×T^{d})} → 0
(ii) If ϕ ∈ L^{∞} ([0, T] × T^{d}) then ∥σ∇ϕ_{ϵ} − (σ∇ϕ)_{ϵ} ∥_{L^{2}([0,T]×T^{d})} → 0

In particular for ϕ ∈ H_{1,a} with σ∇ϕ = f, ∥σ∇ϕ_{ϵ} − f∥_{L^{1}([0,T]×T^{d})} → 0; if we assume in addition that ϕ ∈ L^{∞} ([0, T] × T^{d}), then ∥σ∇ϕ_{ϵ} − f∥_{L^{2}([0,T]×T^{d})} → 0

Proof. Let us prove (i) first. After an integration by parts we can write

\begin{aligned}|σ∇ϕ_{ϵ} & − (σ∇ϕ)_{ϵ}|(x) \\ & = |∫ dy (σ(x + y) − σ(x)) ∇ϕ(x + y)G_{ϵ}(y)| \\ & = |∫ dy [∇ · σ(x + y) + (σ(x + y) − σ(x)) \frac{∇G_{ϵ}}{G_{ϵ}} (y)] ϕ(x + y)G_{ϵ}(y)| \\ & ≤ \frac{1}{2} ∫ dy |∇ · σ(x + y) + (σ(x + y) − σ(x)) \frac{∇G_{ϵ}}{G_{ϵ}} (y)|^{2} G_{ϵ}(y) \\ & + \frac{1}{2} ∫ dy ϕ^{2}(x + y)G_{ϵ}(y) \end{aligned}  (2.1)

Integrating both side with respect to x,

sup_{ϵ>0} ∥σ∇ϕ_{ϵ} − (σ∇ϕ)_{ϵ}∥_{L^{1}([0,T]×T^{d})} ≤ \frac{1}{2}[C^{2} + ∥ϕ∥_{L^{2}([0,T]×T^{d})}^{2}]

which in turn implies

sup_{ϵ>0} ∥σ∇ϕ_{ϵ} − (σ∇ϕ)_{ϵ}∥_{L^{1}([0,T]×T^{d})} ≤ C∥ϕ∥_{L^{2}([0,T]×T^{d})}  (2.2)

We note that for smooth ϕ the claim is obvious and given the estimate (2.2), the result follows for ϕ in L^{2}([0, T] × T^{d}).

We now turn to (ii). Since we already have convergence in L^{1}([0, T] × T^{d}), we only need to establish the uniform integrability of the functions |σ∇ϕ_{ϵ} − (σ∇ϕ)_{ϵ}|^{2}. They are dominated by

∥ϕ∥_{L^{∞}([0,T]×T^{d})}^{2} ∫ dy |∇ · σ(x + y) + (σ(x + y) − σ(x)) \frac{∇G_{ϵ}}{G_{ϵ}} (y)|^{2} G_{ϵ}(y).  (2.3)

Since we have assumed in (A1) that |∇ · σ|^{2} ∈ L^{1}([0, T] × T^{d}) the family {(|∇ · σ|^{2})_{ϵ} : ϵ > 0} is uniformly integrable. The remaining piece in (2.3) is uniformly integrable as a consequence of Lemma 2.2.

The following is a useful variant of Lemmas 2.2 and 2.3.

Lemma 2.4. Let H_{ϵ} be any sequence of functions such that \frac{∥y∥^{2}}{ϵ^{2}} H_{ϵ}(y) is (a bounded multiple of) an approximate identity on R^{d}.

(i) For any f ∈ W^{1,2}, the family of functions

g̃_{ϵ}(x) = ϵ^{−2} ∫ dy |f(x + y) − f(x)|^{2} H_{ϵ}(y)

for ϵ > 0 is uniformly integrable. (ii) Suppose that (A1) holds and ϕ ∈ L^{∞} ([0, T] × T^{d}). Then

∫ dx ∫ dz |σ(x + z)∇ϕ_{ϵ}(x) − (σ∇ϕ)_{ϵ} (x)|^{2}H_{ϵ}(z) → 0.

Proof. To prove (i), recalling the notation of the proof of Lemma 2.2, if

g̃_{ϵ}(x) = ∫ dy h_{y}(x)∥y/ϵ∥^{2}H_{ϵ}(y).

then

sup_{ϵ>0} ∫ dxΨ (g̃_{ϵ}(x)) ≤ sup_{ϵ>0} ∫ dx ∫ dy Ψ (h_{y}(x)) ∥y/ϵ∥^{2}H_{ϵ}(y)

≤ sup_{y∈R^{d}} ∫ dx Ψ(h_{y}(x)).

Since {h_{y}(x), y ∈ R^{d}} is uniformly integrable, so is {g̃_{ϵ}(x), ϵ > 0}.

To prove (ii) from (i) we use exactly the same argument as in the proof of Lemma 2.3 to reduce it to the uniform integrability of

∫ dz ∫ dy |f(x + y) − f(x + z)|^{2} \frac{|∇G_{ϵ}|^{2}}{G_{ϵ}} (y)H_{ϵ}(z)

for any f ∈ W^{1,2}. Now expand |f(x + y) − f(x + z)| ≤ |f(x + y) − f(x)| + |f(x + z) − f(x)|. For the first term appeal to the previous lemma. For the second term use (i) and the fact that

∫ dy \frac{|∇G_{ϵ}|^{2}}{G_{ϵ}} (y) ≤ Cϵ^{−2}.

Lemma 2.5. Suppose that (A1) holds and N(ϕ) < ∞ for some ϕ ∈ L_{2}([0, T]×T^{d}). Then there exists f = (f_{1}, ⋯, f_{d}) ∈ L^{2}([0, T] × T^{d}) such that for all ψ = (ψ_{1}, ⋯, ψ_{d}) ∈ C^{∞},

∫_{0}^{T} dt ∫ dx ϕ∇ · (σψ) = − ∫_{0}^{T} dt ∫ dx fψ,

and N^{2}(ϕ) = ∥f∥_{L^{2}([0,T]×T^{d})}^{2} + ∥ϕ∥_{L^{2}([0,T]×T^{d})}^{2}. Furthermore we have the following chain rule: let F : R → R satisfy |F (x)| ≤ C|x| and |F′(x)| ≤ C for all x ∈ R and some C < ∞. Then for all ψ ∈ C^{∞},

∫_{0}^{T} dt ∫ dx F (ϕ)∇ · (σψ) = − ∫_{0}^{T} dt ∫ dx F′(ϕ)fψ.  (2.4)

Proof. The first statement is obvious. To prove (2.4), note first that it holds for ϕ ∈ C^{∞}. In particular

∫_{0}^{T} dt ∫ dx F (ϕ_{ϵ})∇ · (σψ) = − ∫_{0}^{T} dt ∫ dx F′(ϕ_{ϵ})σ∇ϕ_{ϵ}ψ.

Since ϕ ∈ L^{2}, ϕ_{ϵ} → ϕ strongly in L^{2}. Since |F (x)| ≤ C|x| and ∇ · (σψ) ∈ L^{2} we conclude

∫_{0}^{T} dt ∫ dx F (ϕ_{ϵ})∇ · (σψ) → ∫_{0}^{T} dt ∫ dx F (ϕ)∇ · (σψ).

By lemma 2.3, we have σ∇ϕ_{ϵ} → f strongly in L^{1}. Since F′ is bounded we conclude that

∫_{0}^{T} dt ∫ dx F′(ϕ_{ϵ})σ∇ϕ_{ϵ}ψ → ∫_{0}^{T} dt ∫ dx F′(ϕ)fψ.

This completes the proof.

Theorem 2.6. Suppose that (A1) holds, and ϕ ∈ L_{2}([0, T] × T^{d}) with N(ϕ) < ∞. Then ϕ ∈ H_{1,a}.

Proof. First let us suppose that ϕ is in L^{∞}([0, T] × T^{d}). Let, as before, ϕ_{ϵ} denote the convolution of ϕ with the heat kernel G_{ϵ}. For ϵ_{1}, ϵ_{2} > 0,

\begin{aligned}∥ϕ_{ϵ_{1}} − ϕ_{ϵ_{2}}∥_{H_{1,a}} & ≤ ∥σ∇ϕ_{ϵ_{1}} − (σ∇ϕ)_{ϵ_{1}}∥_{L^{2}([0,T]×T^{d})} + ∥σ∇ϕ_{ϵ_{2}} − (σ∇ϕ)_{ϵ_{2}}∥_{L^{2}([0,T]×T^{d})} \\ & + ∥f_{ϵ_{1}} − f_{ϵ_{2}}∥_{L^{2}([0,T]×T^{d})} + ∥ϕ_{ϵ_{1}} − ϕ_{ϵ_{2}}∥_{L^{2}([0,T]×T^{d})} . \end{aligned}

All terms on the right tend to 0 as ϵ_{1}, ϵ_{2} → 0. For the first two terms we use Lemma 2.3 and the others are obvious.

If N(ϕ) < ∞ but ϕ is only in L^{2}([0, T] × T^{d}), we truncate it at level C and define

ϕ_{C}(x) = \{ \begin{matrix}ϕ if |ϕ| ≤ C, \\ C if |ϕ| ≥ C. \end{matrix}

By the previous lemma, for all ψ ∈ C^{∞},

∫_{0}^{T} dt ∫ dx ϕ_{C} ∇ · (σψ) = − ∫ ∫_{|ϕ|≤C} dxdt fψ.

Then for 0 < C_{1} < C_{2} < ∞,

∥ϕ_{C_{1}} − ϕ_{C_{2}}∥_{1,a}^{2} = ∫ ∫_{C_{1}≤|ϕ|≤C_{2}} dxdt |f|^{2} + ∫_{0}^{T} ∫ dtdx |ϕ_{C_{1}} − ϕ_{C_{2}}|^{2}

In particular, ϕ_{C} has a limit in H_{1,a} as C → ∞ which is of course ϕ.

3. Existence and uniqueness for a class of degenerate parabolic equations. If we are given the basic divergence form operator

L_{t}^{0} = ∇ · ∇

there are several natural problems involving the operator that one can consider. For some suitable c one can study

\{ \begin{matrix}\frac{∂u}{∂t} + ∇ · a∇u + σc · ∇u = g for 0 ≤ t ≤ T, \\ u(T, x) = f(x) \end{matrix}  (3.1)

Because of the maximim principle if f, g ∈ L^{∞} we can look for a solution u ∈ L^{∞}.

We could also exploit the energy relation and consider, for suitable c_{1}, c_{2}, g and h, solutions in L^{2} of

\{ \begin{matrix}\frac{∂u}{∂t} + ∇ · a∇u + σc_{1} · ∇u + ∇ · σc_{2}u = g + ∇ · σh for 0 ≤ t ≤ T, \\ u(T, x) = f(x) \end{matrix}  (3.2)

The adjoint problem

\{ \begin{matrix}\frac{∂p}{∂t} = ∇ · a∇p − ∇ · σcp for 0 ≤ t ≤ T, \\ p(0, x) = p_{0}(x) \end{matrix}  (3.3)

is important in probability theory as a Fokker-Planck equation and p is a nonnegative probability density, usually with a total integral equal to 1.

We will have something to say about each one of these problems. The minimal regularity that our solutions satisfy will be that they are in H_{1,a} and

sup_{0≤t≤T} ∫_{T^{d}} |u(t, x)|^{2} dx < ∞.

All of our equations have the form

\frac{∂u}{∂t} = ∇ · σF + G  (3.4)

and this is to be interpreted in a suitable sense. We say that (3.4) is satisfied in the sense of distributions if, for any C^{∞} test function,

\frac{d}{dt} ∫_{T^{d}} ϕ(t, x)u(t, x)dx = ∫_{T^{d}}[−F · σ∇ϕ + ϕ G + u \frac{∂ϕ}{∂t} ]dx  (3.5)

It does not make any difference if we limit ourselves to using C^{∞} functions ϕ(x) that do not depend on t. We begin with a lemma that will be useful in dealing with such solutions.

Lemma 3.1. Suppose u_{1}, u_{2}, ⋯, u_{k} are functions such that for each i = 1, …, k,

\{ \begin{matrix}u_{i} ∈ L^{∞}([0, T] × T^{d}) ∩ H_{1,a} \\ \frac{∂u_{i}}{∂t} = ∇ · σf_{i} + g_{i} \end{matrix}

in the sense of (3.5), with f_{i} ∈ L^{2}([0, T] × T^{d}) and g_{i} ∈ L^{1}([0, T] × T^{d}). Then for any smooth function F : R^{k} → R, if v = F (u) = F (u_{1}, ⋯, u_{k}),

\frac{∂v}{∂t} = ∑_{i=1}^{k} ∇ · \frac{∂F}{∂x_{i}} (u) σf_{i} − ∑_{i,j=1}^{k} \frac{∂^{2}F}{∂x_{i}∂x_{j}} (u)f_{i} · σ∇u_{j} + ∑_{i=1}^{k} \frac{∂F}{∂x_{i}} (u)g_{i}

again in the sense of (3.5). We can assume without loss of generality that for every i, f_{i} is in the range of σ for almost all (t, x).

Proof. Let us take the case k = 1. We convolute u by the heat kernel G_{ϵ} at time ϵ^{2} to get u_{ϵ} and let v_{ϵ} = F (u_{ϵ}). It is not hard to see that

\begin{aligned}\frac{∂v_{ϵ}}{∂t} & = F′(u_{ϵ})(\frac{∂u}{∂t} ∗ G_{ϵ}) = F′(u_{ϵ})[∇(σf)_{ϵ} + g_{ϵ}] \\ & = ∇F′(u_{ϵ})(σf)_{ϵ} + F′(u_{ϵ})g_{ϵ} − F′′(u_{ϵ})∇u_{ϵ} · (σf)_{ϵ}. \end{aligned}

We use a test function ϕ that depends only on x, and write

\frac{d}{dt} ∫ dx v_{ϵ}ϕ = − ∫ dx F′(u_{ϵ})∇ϕ · (σf)_{ϵ} + ∫ dx F′(u_{ϵ})ϕg_{ϵ} − ∫ dx ϕF′′(u_{ϵ})∇u_{ϵ} · (σf)_{ϵ}

We want to let ϵ → 0 and conclude that

\frac{d}{dt} ∫ dx vϕ = − ∫ dx F′(u)∇ϕ · (σf) + ∫ dx F′(u)ϕg − ∫ dx ϕF′′(u)∇u · (σf)

which establishes the lemma. The only tricky part is the convergence of ∇u_{ϵ} · (σf)_{ϵ} to ∇u · σf in L^{1}([0, T] × T^{d}). To this end we write

∇u_{ϵ} · (σf)_{ϵ} = σ_{ϵ}∇u_{ϵ} · (σ_{ϵ})^{−1}(σf)_{ϵ}

and Lemma 2.4 assures us that σ_{ϵ}∇u_{ϵ} → σ∇u in L^{2}([0, T] × T^{d}). It suffices to check that (σ_{ϵ})^{−1}(σf)_{ϵ} → f in L^{2}([0, T] × T^{d}). By the convexity of the map ξ, A → ⟨ξ, A^{−1}ξ⟩ it follows that

∥(σ_{ϵ})^{−1}(σf)_{ϵ}∥_{L^{2}} ≤ ∥f∥_{L^{2}}

and with a little computation one can show the convergence of (σ_{ϵ})^{−1}(σf)_{ϵ} to f in L^{2}([0, T] × T^{d}). Remark. In principle, if we assume that g_{i} are in L^{2}, there is no apriori reason why we need u_{i} to be in L^{∞} for the lemma. The final formula makes sense if u_{i} are just in H_{1,a}. But we find that we have to strengthen the definition of (3.4) with a larger class of test functions in order to drop the L^{∞} assumption. For bounded functions in H_{1,a} the validity for the larger class follows already from the validity for smooth test functions, whereas for a general function in H_{1,a} one has to assume it. The culprit is Lemma 2.3 where ϕ has to be L^{∞} in order to imply convergence in L^{2}. In fact if u_{i} ∉ L^{∞} the conclusion of the lemma is false as shown by the following counterexample: In dimension d = 1 take

σ(t, x) = t^{−α}|x|

and

u(t, x) = t^{1/4}|x|^{−1/2+\sqrt{t}}.

If α < 1/2 then σ satisfies (A1),

u ∈ H_{1,a}

and

\frac{∂u}{∂t} = ∇ · σf, f ∈ L^{2}([0, T] × T^{d}).

However for this example

\frac{∂}{∂t}u^{2} ≠ 2u\frac{∂u}{∂t}

in the sense of (3.5).

The simplest situation is equation (3.1). Here F = σ∇u ∈ L^{2}, and if we assume that c ∈ L^{∞}, then G = −σc · ∇u + g ∈ L^{2} . In this case it will suffice to interpret (3.4) in the sense of (3.5) with C^{∞} test functions.

Lemma 3.2. If c ∈ L^{∞}, equation (3.1) has at most one bounded solution.

Proof. Since we can take the difference of two solutions to get a new solution, it amounts to proving that any bounded solution of

\{ \begin{matrix}\frac{∂u}{∂t} + ∇ · a∇u + σc · ∇u = 0 for 0 ≤ t ≤ T, \\ u(T, x) = 0 \end{matrix}

is identically zero. From lemma 3.1

\frac{∂ϕ(u)}{∂t} = −∇ · ϕ′(u)a∇u − ϕ′(u)σc · ∇u = ϕ′′(u)∇u · a∇u

It is not difficult to see that ϕ(u) can be replaced by a more general function ϕ(t, u) and this produces an extra term which is \frac{∂ϕ}{∂t}(t, u). Since u is bounded the choice ϕ(t, u) = e^{kt}u^{2} is allowed

\frac{∂e^{kt}u^{2}}{∂t} = −2e^{kt}[∇ua∇u + 2uσc∇u − ∇u · a∇u] + ke^{kt}u^{2}

integrating over space we get

\begin{aligned}\frac{d}{dt}e^{kt} ∫ u^{2}(t, x) dx & = e^{kt} ∫ [−2uσc∇u + 2 < ∇u , a∇u > +ku^{2}] dx \\ & ≥ 0 \end{aligned}

provided k > \frac{1}{2}C^{2} where C is an upper bound for ∥c∥. In particular ∫ u^{2}(t, x) dx ≤ ∫ u^{2}(T, x) dx = 0 and we are done.

For proving existence we have to approximate the equation by more regular ones.

Lemma 3.3. Suppose that (A1) holds. Then there exists a family of positive definite matrices {a_{ϵ}} defined for ϵ > 0 with positive definite square roots {σ_{ϵ}} such that for some c_{ϵ} > 0

\begin{matrix}(i) & c_{ϵ} |ξ|^{2} ≤ ξ · a_{ϵ}(t, x)ξ ≤ c_{ϵ}^{−1}|ξ|^{2}, (t, x) ∈ [0, T] × T^{d}, ξ ∈ R^{d}, \\ (ii) & ∥(σ_{ϵ})_{ij} − σ_{ij}∥_{L^{2}([0,T]×T^{d})} → 0, i, j = 1, …, d, \\ (iii) & ∥ ˙ (σ_{ϵ})_{ij} − σ̇_{ij}∥_{L^{2}([0,T]×T^{d})} → 0 i, j = 1, …, d. \end{matrix}

The dot denotes the partial derivative in any spatial direction. We can assume without loss of generality that σ_{ϵ} is smooth for each ϵ > 0.

Proof. For instance

σ_{ϵ} = (I + ϵσ)^{−1}(σ + ϵI)

clearly works. We can always mollify it somewhat to get smoothness. Another choice is to convolute a with G_{ϵ} to smoothen it and take the positive definite square root. While it is not obvious, it is nevertheless true that ∥∇σ_{ϵ} − ∇σ∥_{L^{2}}^{2} → 0. If we need it we can always add a small δ(ϵ)I to a or a_{ϵ} to make it elliptic.

We now prove existence.

Lemma 3.4. Suppose (A1) holds and c, f and g ∈ L^{∞}. Then there is a u ∈ H_{1,a} which is a solution to the equation (3.1), in the sense of (3.5) .

Proof. For each ϵ > 0, it is known that a regular solution u_{ϵ} of

\{ \begin{matrix}\frac{∂u_{ϵ}}{∂t} + ∇ · a_{ϵ}∇u_{ϵ} + σ_{ϵ}c · ∇u_{ϵ} = g for 0 ≤ t ≤ T, \\ u_{ϵ}(T, x) = f(x) \end{matrix}  (3.6)

exists satisfying

\{ \begin{matrix}sup_{ϵ} ∥u_{ϵ}∥_{L^{∞}} ≤ C \\ sup_{ϵ} ∫_{0}^{T} dt ∫ dx |σ_{ϵ}∇u_{ϵ}|^{2} ≤ C \end{matrix}  (3.7)

It is clear from the above estimates that we can choose a subsequence, along which u_{ϵ} ⇀ u and σ_{ϵ}∇u_{ϵ} ⇀ v weakly in L^{2}([0, T] × T^{d}). We will show that u is a solution of (3.1). Clearly u ∈ L^{∞}. From

\frac{∂u_{ϵ}}{∂t} = ∇ · σ_{ϵ}F_{ϵ} + G_{ϵ}  (3.8)

with F_{ϵ} = −σ_{ϵ}∇u_{ϵ} and G_{ϵ} = −c · σ_{ϵ}∇u_{ϵ} + g, by taking limits we get

\frac{∂u}{∂t} = ∇ · σF + G  (3.9)

with F = −v and G_{ϵ} = −c · v + g. It remains only to show that u ∈ H_{1,a} and σ∇u = v. But we have the relation

∫_{0}^{T} ∫_{T^{d}} u_{ϵ}(t, x)∇ · σ_{ϵ}ψ dxdt = − ∫_{0}^{T} ∫_{T^{d}} σ_{ϵ}∇u_{ϵ} · ψdxdt  (3.10)

The strong convergence of σ_{ϵ} and its gradient along with the weak convergence in L^{2} of u_{ϵ} allows us to pass to the limit to conclude that

∫_{0}^{T} ∫_{T^{d}} u(t, x)∇ · σψ dxdt = − ∫_{0}^{T} ∫_{T^{d}} v · ψ dxdt  (3.11)

holds for all smooth test functions. This proves that u ∈ H_{1,a} and σ∇u = v. The boundary value is preserved, at least in the sense of a weak limit as t → T and we are done.

The next step is to establish the energy identity and some other estimates on the solution. The same argument that we used to prove uniqueness in lemma 3.2 yields the following identities for each ϵ > 0 as well as for the limit.

∫_{T^{d}} e^{kT} u_{ϵ}^{2}(T, x)dx − ∫_{T^{d}} e^{kt}u_{ϵ}^{2}(t, x)dx

= ∫_{t}^{T} ∫_{T^{d}} e^{kt} [2∇u_{ϵ} · a_{ϵ}∇u_{ϵ} − 2u_{ϵ}c.σ_{ϵ}∇u_{ϵ} + 2u_{ϵ}g + ku_{ϵ}^{2}] dxds

= 2 ∫_{t}^{T} ∫_{T^{d}} e^{kt}∥σ_{ϵ}∇u_{ϵ} − \frac{1}{2}cu_{ϵ}∥^{2} dxds + ∫_{t}^{T} ∫_{T^{d}} e^{kt}[k − \frac{1}{2}∥c∥^{2}]u_{ϵ}^{2} dxds

+ 2 ∫_{t}^{T} ∫_{T^{d}} u_{ϵ}g dxds

We can rewrite this as

∫_{T^{d}} e^{kt}u_{ϵ}^{2}(t, x)dx + 2 ∫_{t}^{T} ∫_{T^{d}} e^{ks}∥σ_{ϵ}∇u_{ϵ} − \frac{1}{2}cu_{ϵ}∥^{2} dxds + ∫_{t}^{T} ∫_{T^{d}} e^{ks}[k − \frac{1}{2}∥c∥^{2}]u_{ϵ}^{2} dxds

= ∫_{T^{d}} e^{kT} f^{2}(x)dx + 2 ∫_{t}^{T} ∫_{T^{d}} e^{ks}u_{ϵ}g dxds  (3.12)

and a similar equation with ϵ = 0 in the limit. Since the left hand side is lower semicontinuous under weak convergence and the right hand side is continuous and equlaity holds in the limit, we conclude that

lim_{ϵ→0} ∥u_{ϵ} − u∥_{L^{2}([0,T]×T^{d})} = 0

and

lim_{ϵ→0} ∥σ_{ϵ}∇u_{ϵ} − σ∇u∥_{L^{2}([0,T]×T^{d})} = 0

The mappings t → u_{ϵ}(t, ·) are uniformly weakly continuous as a maps into L^{2}(T^{d}) and the above arguments actually tell us that for each t ∈ [0, T],

lim_{ϵ→0} ∥u_{ϵ}(t, ·) − u(t, ·)∥_{L^{2}(T^{d})} = 0

Moreover, we can read off from equation (3.12), the equicontinuity of ||u_{ϵ}(t, ·)∥^{2} as a family of function of t. We conclude that

lim_{ϵ→0} sup_{0≤t≤T} ∥u_{ϵ}(t, ·) − u(t, ·)∥_{L^{2}(T^{d})} = 0

We combine these observations in the form of Theorem 3.5. Under the assumptions (A1), if c, f, g ∈ L^{∞}, there equation (3.1) exists a unique u ∈ H_{1,a} ∩ L^{∞} that solves (3.1) in the sense of (3.5). Moreover the solution admits the following energy identity and estimate.

∫_{T^{d}} f^{2}(x)dx − ∫_{T^{d}} u^{2}(t, x)dx = ∫_{t}^{T} ∫_{T^{d}} [2∇u · a∇u − 2uc · σ∇u + 2ug] dxds (3.13)

sup_{0≤t≤T} ∥u(t, ·)∥_{L^{2}(T^{d})}^{2} + ∫_{0}^{T} ∫_{T^{d}} |σ∇u|^{2} dxdt ≤ C_{T} [∥f∥_{L^{2}(T^{d})}^{2} + ∥g∥_{L^{2}([0,T]×T^{d})}^{2}] (3.14)

where the constant C_{T} depends on T and the L^{∞} norm of c.

Proof. Except for the estimate (3.14) everything else has already been established. From the identity (3.13), using Schwarz’s inequality, we get

sup_{0≤t≤T}∥u(t, ·)∥_{L^{2}(T^{d})}^{2} + ∫_{0}^{T} ∫_{T^{d}} |σ∇u|^{2} dxdt

≤ ∥f∥_{L^{2}(T^{d})}^{2} + C_{T} [∥f∥_{L^{2}(T^{d})}^{2} + ∥g∥_{L^{2}([0,T]×T^{d})}^{2}] + \frac{1}{2} sup_{0≤t≤T} ∥u(t, ·)∥_{L^{2}(T^{d})}^{2}

and we are done.

We now turn to the study of equation (3.2). If we assume that everything in sight is smooth and the problem is elliptic then (3.2) clearly has a smooth solution. If we multiply by u and integrate we get the energy identity

\begin{aligned}_{∫T^{d}} & u^{2}(t, x)dx + ∫_{t}^{T} ∫_{T^{d}} |σ∇u|^{2} dxds \\ & = ∫_{T^{d}} f^{2}dx + ∫_{t}^{T} ∫_{T^{d}}(uc_{1} − uc_{2} + h)σ∇u dxds − ∫_{t}^{T} ∫_{T^{d}} g u dxds \end{aligned}  (3.15)

It is but one step from this to get an apriori estimate

sup_{0≤t≤T} ∥u(t, ·)∥_{L^{2}(T^{d})}^{2} + ∫_{0}^{T} ∫_{T^{d}} |σ∇u|^{2} dxdt
≤ C_{T} [∥f∥_{L^{2}(T^{d})}^{2} + ∥g∥_{L^{2}([0,T]×T^{d})}^{2} + ∥h∥_{L^{2}([0,T]×T^{d})}^{2}]

(3.16)

where the constant depends on T and L^{∞} bounds for c_{1}, c_{2}

In an exactly similar fashion as before we can pass to the limit via solutions of a mollified equation involving smooth a_{ϵ}, σ_{ϵ}, c_{1,ϵ}, c_{2,ϵ}, g_{ϵ}, h_{ϵ} and f_{ϵ}. When the dust settles what we have is

Theorem 3.6. For any c_{1}, c_{2} in L^{∞} and f, g, h in L^{2} there exists a function u ∈ H_{1,a} satisfying the bounds (3.16), that is a solution of (3.2) in the sense of (3.5).

The difficult question is that of uniqueness. The sticky point is lemma 3.1 which requires that the functions u_{i} be in L^{∞} in order to prove the energy identity and we know by counterexamples that this is necessary. We can save the situation by demanding a little bit more from our potential solutions. If we have a function u ∈ H_{1,a} with \frac{du}{dt} ∈ H_{−1,a}, what we have is a function u ∈ H_{1,a} that satisfies (3.4) in the sense of (3.5). We could try smooth ϕ or test against ϕ ∈ H_{1,a} that satisfy

\frac{∂ϕ}{∂t} = ∇ · σξ + η

verifying

\frac{d}{dt} ∫_{T^{d}} u(t, x)ϕ(t, x)dx = ∫_{T^{d}} [−F · σ∇ϕ + Gϕ − ξ · σ∇u + uη] dx

While it is sufficient to use smooth test functions for u ∈ L^{∞}, for u ∈ L^{2} we need a larger class. We can get away with using test functions ϕ ∈ L^{∞}. So let us adopt the convention that we will say that (3.4) holds in the distributional sense if we test with smooth functions and in the weak sense if we enlarge the class of test functions to include L^{∞} functions. The distinction is real as can be seen from our counterexample: Dimension d = 1,

σ(t, x) = t^{−α}|x|

with 0 < α < 1/2 and

u(t, x) = t^{1/4}|x|^{−1/2+\sqrt{t}}.

Then σ satisfies (A1) and u ∈ H_{1,a} is a distributional solution but not a weak solution of

\frac{∂u}{∂t} − ∇ · a∇u = ∇ · σh

with h ∈ L^{2} and initial condition

u(0, x) = 0.

We will now prove existence and uniqueness of weak solutions, so for this example there is nonuniqueness of distributional solutions.

Just for the record,

Lemma 3.7. If u ∈ L^{∞} and (3.4) holds in the distributional sense, then it holds in the weak sense.

Proof. This is basically the same as lemma 3.1. Start with u, v ∈ L^{∞} satisfying

\frac{∂u}{∂t} = ∇ · σf + g, \frac{∂v}{∂t} = ∇ · σξ + η

mollify both by G_{ϵ}, compute

\frac{d}{dt} ∫_{T^{d}} u_{ϵ}(t, x)v_{ϵ}(t, x) dx

integrate by parts the obvious terms and pass to the limit as ϵ → 0.

We now strengthen Theorem 3.6 to get the slightly stronger version of existence, and then prove uniqueness.

Theorem 3.8. Let c_{1}, c_{2} ∈ L^{∞} and f, g, h ∈ L^{2}. Then the equation (3.2) admits a solution in the weak sense. The weak solution is unique and satisfies the energy identity and estimate.

Proof. We do not have to work any harder to construct a solution in the weak sense. We use the special chioce

a_{ϵ} = a ∗ G_{ϵ}

to solve our approximate problems. If we mollify our bounded test function v by G_{ϵ} as well, by lemma 3.1 we get

\frac{∂v_{ϵ}}{∂t} = ∇σ_{ϵ} ξ_{ϵ} + η_{ϵ}

and we calculate

\frac{d}{dt} ∫_{T^{d}} u_{ϵ}(t, x)v_{ϵ}(t, x)dx = ∫_{T^{d}} [−F_{ϵ} · σ_{ϵ}∇v_{ϵ} + G_{ϵ}v_{ϵ} − ξ_{ϵ} · σ_{ϵ}∇u_{ϵ} + η_{ϵ}u_{ϵ}] dx  (3.18)

where

F_{ϵ} = −σ_{ϵ}∇u_{ϵ} − c_{2,ϵ}u_{ϵ} + h_{ϵ}, G_{ϵ} = −c_{1,ϵ} · σ_{ϵ}∇u_{ϵ} + g_{ϵ}
ξ_{ϵ} = ξ ∗ G_{ϵ}, and η_{ϵ} = η ∗ G_{ϵ}

As ϵ → 0, F_{ϵ}, G_{ϵ}, σ_{ϵ}∇u_{ϵ} and u_{ϵ} converge weakly to F = σ∇u − c_{2}u + h, G = c_{1}σ∇u + g, σ∇u, and u respectively. On the other hand σ_{ϵ}∇v_{ϵ}, v_{ϵ}, ξ_{ϵ} and η_{ϵ} converge strongly to σ∇v, v, ξ and η respectively. We can therefore take the limit as ϵ → 0 in (3.18) and we are done.

We now turn to the proof of uniqueness as well as the energy identity. Let us look at the special case of (3.2) with c_{1} = c_{2} = 0.

\frac{∂u}{∂t} + ∇ · a∇u = g + ∇σ̇h  u(T, x) = f(x)  (3.19)

with f, g, h in L^{2}(T^{d}). We can now use the solution of

\frac{∂v}{∂s} = ∇ · a∇v in [t, T] × T^{d}, v(t, x) = ϕ(x)

in L^{∞} ∩H_{1.a}, as a test functions to conclude that the difference w of two solutions of (3.19) in the weak sense, satisfies

∫_{T^{d}} ϕ(x)w(t, x) = ∫_{t}^{T} ∫_{T^{d}}[−w∇ · a∇v + v∇ · a∇w]dxds = 0

proving the uniqueness of such solutions in the special case. If we approximate f, g in L^{2} by smooth functions f_{n}, g_{n} respectively and take h = 0 we can solve (3.19) with f_{n}, g_{n}, 0 replacing f, g, h. From the estimate (3.16) it follows that the solutions u_{n} have a limit in a very strong sense and the limit will be a solution of (3.19) in the weak sense. Because of uniqueness, the limit has to equal u. In particular u as the limit of u_{n} will satisfy the energy identity

∫_{T^{d}} u^{2}(t_{2}, x)dx − ∫_{T^{d}} u^{2}(t_{1}, x)dx = 2 ∫_{t_{1}}^{t_{2}} ∫_{T^{d}}[|σ∇u|^{2} + ug]dxdt

We can replace g by g + ∇σh so long as h is smooth. Again we can approximate h ∈ L^{2} by smooth h, and we conclude that any solution in the weak sense of (3.19) will have to satisfy

∫_{T^{d}} u^{2}(t_{2}, x)dx − ∫_{T^{d}} u^{2}(t_{1}, x)dx = 2 ∫_{t_{1}}^{t_{2}} ∫_{T^{d}}[|σ∇u|^{2} + ug − h∇u]dxdt  (3.21)

Now if we take any solution of (3.2) in the weak sense, we can think of it as a solution in the weak sense of

\frac{∂u}{∂t} + ∇ · a∇u = g̃ + ∇ · σh̃  u(T, x) = f(x)  (3.20)

with

g̃ = g − c_{1} · σ∇u  and  h̃ = h − ∇ · σc_{2}u

and (3.21) will hold with g̃ and h̃ replacing g and h. This is nothing else but the energy identity for (3.2). Once we have the energy identity for all solutions of (3.2) in the weak sense, uniqueness is immediate.

Remark. One of the consequences of uniqueness is that if u ∈ H_{1,a}, and

\frac{∂u}{∂t} = ∇ · σF + G

in the weak sense, with F, G ∈ L^{2} then we can rewrite

\frac{∂u}{∂t} + ∇ · a∇u = ∇ · σ[F + σ∇u] + G

and as we saw in the earlier argument u is a limit of bounded solutions. In other words there are bounded functions u_{n} such that

\frac{∂u_{n}}{∂t} = ∇ · σF_{n} + G_{n}

with u_{n}, F_{n} and G_{n} converging in L^{2} respectively to u, F and G. In particular lemma 3.1 is now valid and we can use test functions ϕ that satisfy

\frac{∂ϕ}{∂t} = ∇ · σξ + η

in the weak sense.

Remark. At this point one may have the impression that to get the weak solution one has to be careful in the choice of a_{ϵ} used to approximate the problem. In fact it is not necessary. If we use arbitrary a_{ϵ} satisfying lemma 3.3, we can solve at the same time equations of the form

\frac{∂v_{ϵ}}{∂t} = ∇ · a_{ϵ}∇v_{ϵ} v_{ϵ}(0, x) = f(x)

and v_{ϵ} → v very nicely, as ϵ → 0 according to Theorem 3.3. where v solves

\frac{∂v}{∂t} = ∇ · a∇v v(0, x) = f(x)

and it is easy to check that such v’s can be used as test functions. But that is enough to establish uniqueness.

If we want to go beyond bounded c in equation (3.1), it is not very natural to replace the condition c ∈ L^{∞} by c ∈ L^{2}. If we take the simplest case of L_{t} = L = Δ + b(x).∇, and impose the condition that b(x) ∈ L^{p}, there is an easy counterexample of a smooth solution to u_{t} − Lu ≡ 0 provided the dimension d > p. As for relaxing the condition ∇σ ∈ L^{2}, even in d = 1 the operator L = D_{x}|x|^{α}D_{x} has problems at 0 as soon as α < 1. It is easy to see that if we approximate by D_{x}a(x)D_{x} with smooth a(·) from below the Greens functions G(x, y) = 0 for x < 0, y > 0 or x > 0, y < 0 for all the approximations. On the other hand if we approximate by D_{x}(ϵ + |x|^{α})D_{x}, G(x, y) > 0 for all x, y even in the limit.

We now turn to the study of equation (3.3) which as we said earlier is important in probability theory. In the context that motivated this study, where the operator L_{t} comes from a specific solution of a non-linear problem we are given a particular solution of equation (3.3). In dealing with simple exclusion models the densities are all bounded by 1 and after renormalization they remain bounded. We are concerned with related existence and uniqueness results that are of interest in probability theory and are implied by the existence of a particular solution. We will deal exclusively with bounded densities and therefore lemma 3.1 applies. We will use relative entropy as our main tool for proving uniqueness. Let us recall the definition. If u_{1} and u_{2} are nonnegative integrable functions with ∫ u_{1}dx = ∫ u_{2}dx = 1,

h(u_{1}/u_{2}) = ∫ dx u_{1} log(u_{1}/u_{2}).

Lemma 3.9. Let (A1) hold. Suppose that u_{i}, i = 1, 2 are bounded, non-negative, solutions of the forward equations

\frac{∂u_{i}}{∂t} = L_{t}^{∗}u_{i} = ∇ · a∇u_{i} − ∇ · σc_{i}u_{i}, u_{i}(0) = u_{i}^{0},

satisfying

\{ \begin{matrix}∫_{0}^{T} dt ∫ dx\frac{∇u_{i}·a∇u_{i}}{u_{i}} < ∞, i = 1, 2, \\ ∫_{0}^{T} dt ∫ dx |c_{i}|^{2}u_{i} < ∞, i = 1, 2, \\ ∫_{0}^{T} dt ∫ dx |c_{2} − c_{1}|^{2}u_{1} < ∞, \\ h(u_{1}^{0}/u_{2}^{0}) < ∞. \end{matrix}

Then the relative entropy at time T > 0 is finite and h(u_{1}(T)/u_{2}(T)) − h(u_{1}^{0}/u_{2}^{0})

≤ −\frac{1}{2} ∫_{0}^{T} dt ∫ dx (\frac{∇u_{1}}{u_{1}} − \frac{∇u_{2}}{u_{2}} ) · a (\frac{∇u_{1}}{u_{1}} − \frac{∇u_{2}}{u_{2}} ) u_{1}

+ \frac{1}{2} ∫_{0}^{T} dt ∫ dx (c_{2} − c_{1})^{2}u_{1}.

Proof. We note first that our equations can be cast in the form

\frac{∂u_{i}}{∂t} = ∇ · σF_{i}.

with F_{i} in L^{2}([0, T] × T^{d}) We wish to apply Lemma 3.1 for the function H(u_{1}, u_{2}) = u_{1} log \frac{u_{1}}{u_{2}} . Due the singularity near 0 we have to do it with H_{δ}(u_{1}, u_{2}) = u_{1} log \frac{u_{1}+δ}{u_{2}+δ} and then let δ → 0. This can be done and after an integration by parts one obtains

\begin{aligned}h(u_{1}(T)/u_{2}(T)) − h(u_{1}^{0}/u_{2}^{0}) = & − ∫_{0}^{T} dt ∫ dx (\frac{∇u_{1}}{u_{1}} − \frac{∇u_{2}}{u_{2}} ) · a (\frac{∇u_{1}}{u_{1}} − \frac{∇u_{2}}{u_{2}} ) u_{1} \\ & − ∫_{0}^{T} dt ∫ dx (\frac{∇u_{1}}{u_{1}} − \frac{∇u_{2}}{u_{2}} ) · σ(c_{2} − c_{1})u_{1}. \end{aligned}

(3.21)

The Lemma follows by Schwarz’s inequality.

A consequence of the lemma is Corollary 3.10. If condition (A1) holds, a nonnegative bounded solution of

\frac{∂u}{∂t} = ∇ · a∇u − ∇ · σcu

in the class satisfying

\{ \begin{matrix}∫_{0}^{T} ∫_{T^{d}} \frac{|σ∇u|^{2}}{u} dxdt < ∞ \\ ∫_{0}^{T} ∫_{T^{d}} |c|^{2}udxdt < ∞ \end{matrix}

is unique.

Before we prove the next theorem on the existence and uniqueness of solutions for (3.3) in certain situations we prove the following lemma that we need. Lemma 3.11. Suppose that (A1) holds and c ∈ L^{2} ([0, T] × T^{d}). Then for each bounded non-negative u^{0} there exists a weak solution of

\{ \begin{matrix}\frac{∂u}{∂t} = ∇ · a∇u + c · σ∇u, in [0, T] × T^{d} \\ u(0) = u^{0}, \end{matrix}

satisfying 0 ≤ u ≤ ∥u^{0}∥_{∞} and

∫_{0}^{T} dt ∫ dx \frac{∇u · a∇u}{u} ≤ C  (3.22)

with C < ∞ depending only on ∥u^{0}∥_{∞} and ∥c∥_{L^{2}([0,T]×T^{d})}.

Proof. Let c^{n} be a truncation of c at level n. For each integer n by reversing time in theorem 3.8 we obtain a unique weak solution u_{n} ≥ 0, of

\frac{∂u_{n}}{∂t} = ∇ · a∇u_{n} + σc^{n} · ∇u_{n},  u_{n}(0) = u^{0}  (3.23)

satisfying 0 ≤ u_{n} ≤ ∥u^{0}∥_{∞}. If we take ϕ(u) = u log u in lemma (3.1) and integrate by parts ( we can replace u by u + δ and let δ → 0 at the end ), we get

∫_{0}^{T} dt ∫ dx \frac{|σ∇u_{n}|^{2}}{u_{n}} ≤ C,

and ∥u_{n}∥_{H_{1,a}} ≤ C where C < ∞ depends only on ∥u^{0}∥_{∞} and ∥c∥_{L^{2}([0,T]×T^{d})}. From the definition, for any ϕ ∈ C_{0}^{∞} ([0, T) × T^{d}),

\begin{aligned}0 = & − ∫_{0}^{T} dt ∫ dx u_{n} \frac{∂ϕ}{∂t} − ∫ dx fϕ(0, x) \\ & + ∫_{0}^{T} dt ∫ dx ∇u_{n} · a∇ϕ − ∫_{0}^{T} dt ∫ dx c^{n} · σ∇u_{n} ϕ. \end{aligned}  (3.24)

We can extract subsequnces u_{n} ⇀ u and σ∇u_{n} ⇀ v weakly in L^{2}([0, T] × T^{d}). We can identify v = σ∇u as in the proof of Lemma 3.4. By lower semicontinuity we obtain (3.22). Since c^{n} → c strongly in L^{2} we can take the limit as n → ∞ in (3.24) to verify that u is the desired weak solution.

Let us remark once again that there is in general no uniqueness in this context. The final theorem in this section is

Theorem 3.12. Suppose that (A1) holds and in addition that there exists a bounded, non-negative weak solution ρ of the equation ∂_{t}ρ = L_{t}^{∗}ρ = ∇·a∇ρ−∇·σcρ, ρ(0, x) = ρ^{0}(x), satisfying

\{ \begin{matrix}∫_{0}^{T} dt ∫ dx \frac{∇ρ·a∇ρ}{ρ} < ∞, \\ ∫_{0}^{T} dt ∫ dx |c|^{2}ρ < ∞. \end{matrix}

Then, for any 0 ≤ R^{0} ≤ Cρ^{0} for some C < ∞, there exists a unique weak solution R of the equation ∂_{t}R = L_{t}^{∗}R, R(0, x) = R^{0}(x) satisfying 0 ≤ R ≤ Cρ on [0, T] × T^{d} and

∫_{0}^{T} dt ∫ dx \frac{∇R · a∇R}{R} < ∞.

Proof. First of all note that the uniqueness part of the lemma follows immediately from Corollary 3.10. We now prove the existence part.

Step 1. Suppose that ρ ≥ δ for some δ > 0, and

∫_{0}^{T} dt ∫ dx |c|^{2} < ∞.

Let

g = 2σ \frac{∇ρ}{ρ} − c,

Since g ∈ L^{2} ([0, T] × T^{d}), Theorem 3.5 gives us a bounded non-negative weak solution u of the equation

\frac{∂u}{∂t} = ∇ · a∇u + σg · ∇u, u(0, x) = R^{0}/ρ^{0}.

satisfying

∫_{0}^{T} dt ∫ dx \frac{∇u · a∇u}{u} < ∞.

Now,

\begin{matrix}∫_{0}^{T} dt ∫ dx (\frac{∂R}{∂t}ϕ + ∇R · a∇ϕ + σc · R∇ϕ) \\ = ∫_{0}^{T} dt ∫ dx (\frac{∂u}{∂t}ψ + ∇u · a∇ψ − σg · ∇u ψ + \frac{∂ρ}{∂t}ξ + ∇ρ · a∇ξ + σc · ρ∇ξ) \end{matrix} (3.25)

where ψ = ρϕ and ξ = uϕ. If ϕ is a smooth test function, then since both ρ and u are elements of H_{1,a} ∩ L^{∞}, so are ψ and ξ. Therefore the right hand side, and consequently the left hand side of (3.25) vanishes for any smooth ϕ. It follows that R = ρu is a weak solution of the equation

\frac{∂R}{∂t} = ∇ · a∇R − ∇ · σcR, R(0) = R^{0}.

Furthermore, since u and ρ are bounded,

∫_{0}^{T}dt ∫ dx \frac{∇R · a∇R}{R} ≤ ∫_{0}^{T}dt ∫ dx \frac{∇u · a∇u}{u}ρ + ∫_{0}^{T}dt ∫ dx \frac{∇ρ · a∇ρ}{ρ}u < ∞.

This completes the proof in case ∫ dt ∫ dx |c|^{2} < ∞.

Step 2. In the general case let

ρ_{δ} = ρ + δ

and note that ρ_{δ} is a weak solution of the forward equation with c replaced by c_{δ} given by

c_{δ} = cρ/ρ_{δ}.

This ρ_{δ} satisfies the assumptions of the lemma and furthermore ρ_{δ} ≥ δ and

∫_{0}^{T} dt ∫ dx c_{δ}^{2} < ∞.

Let R^{0} ≤ Cρ^{0}. By Step 1 there exists a weak solution R_{δ} of the forward equation

\frac{∂R_{δ}}{∂t} = ∇ · a∇R_{δ} − ∇ · σc_{δ}R_{δ}, R_{δ}(0) = R^{0}.

satisfying

∫_{0}^{T} dt ∫ dx \frac{∇R_{δ} · a∇R_{δ}}{R_{δ}} < ∞.

By (?),

\begin{aligned}∫_{0}^{T} dt ∫ dx\frac{∇R_{δ} · a∇R_{δ}}{R_{δ}} ≤ & 4 [h(R_{δ}(T)/ρ_{δ}(T)) − h(R^{0}/ρ_{δ}^{0})] \\ & + 2 ∫_{0}^{T} dt ∫ dx\frac{∇ρ · a∇ρ}{ρ}. \end{aligned}  (3.26)

The right hand side is bounded independent of δ.

Since R_{δ} and u_{δ} are uniformly bounded we can extract a subsequences R_{δ} ⇀ R and u_{δ} ⇀ u weak^{∗}-L^{∞}. In other words, for any ϕ ∈ L^{1} ([0, T] × T^{d}), ∫_{0}^{T} dt ∫ dx ϕR_{δ} → ∫_{0}^{T} dt ∫ dx ϕR and similarly for u_{δ}. But then since ρ is bounded ∫_{0}^{T} dt ∫ dx ϕρ_{δ}u_{δ} → ∫_{0}^{T} dt ∫ dx ϕρu. Therefore R = ρu. Also by (3.26), as in the proof of Lemma 3.3 we can find a subsequence σ∇R_{δ} ⇀ σ∇R in L^{2}. For any ϕ ∈ C_{0}^{∞} ([0, T) × T^{d}),

\begin{aligned}0 = & − ∫_{0}^{T} dt ∫ dx R_{δ} \frac{∂ϕ}{∂t} − ∫ dx R^{0}ϕ(0, x) \\ & + ∫_{0}^{T} dt ∫ dx ∇R_{δ} · a∇ϕ − ∫_{0}^{T} dt ∫ dx R_{δ}c_{δ} · σ∇ϕ. \end{aligned}  (3.27)

Write

∫_{0}^{T} dt ∫ dx R_{δ}c_{δ} · σ∇ϕ = ∫_{0}^{T} dt ∫ dx u_{δ}ρc · σ∇ϕ.

Note that ρc · σ∇ϕ ∈ L^{1} ([0, T] × T^{d}). Therefore we can let δ → 0 to get

∫_{0}^{T} dt ∫ dx uρc · σ∇ϕ = ∫_{0}^{T} dt ∫ dx Rc · σ∇ϕ.

We can take the limit of the other terms in (3.27) as in the proof of the previous theorems and find that R is the desired weak solution. Remarks. We make a few comments regarding some earlier work.

1. The special case of a ≡ I has been studied in detail by Eric Carlen in [C]

2. In the time homogeneous case the energy inequality holds for weak solutions of (3.41) using only smooth test functions by the following lemma which is a special case of a general theorem of Lions and Magenes [LM], and therefore such solutions are unique.

Lemma. Let V ⊂ H ⊂ V′ be separable Hilbert spaces with V and V′ being in duality relative to the inner product of H. Suppose we have a curve u(t) : 0 ≤ t ≤ T satisfying u ∈ L^{2}(0, T; V) and u′ ∈ L^{2}(0, T; V′). Then u is almost everywhere equal to a function continuous from [0, T] into H and for 0 ≤ t_{1} ≤ t_{2} ≤ T,

∥u(t_{2})∥_{H}^{2} − ∥u(t_{1})∥_{H}^{2} = 2 ∫_{t_{1}}^{t_{2}}⟨u(t), u′(t)⟩dt

3. There are a few previous results of existence in the time homogeneous case. Let

E(u, v) = ∫ dx∇ϕ · a∇ψ.

If a ∈ W^{1,2} then L_{0} with b = 0 maps C^{∞} into L^{2} so E comes from an operator and therefore it has a minimal self-adjoint extension, the Freidrichs extension. If only σ ∈ W^{1,2} this is no longer the case, however Albeverio, Hoegh-Krohn and Streit [AHS] proved that if σ ∈ W^{1,2}, E with domain C_{0}^{∞} is closable on L^{2}. The closure Ē is then the quadratic form of a self-adjoint operator L_{0}. In particular, for each u_{0} ∈ L^{2} there exists a weak solution u of

\frac{∂u}{∂t} − ∇ · a∇u = 0, in [0, T] × T^{d}, u(0, x) = u^{0}(x).

The closure is a regular Dirichlet form, so for almost every x ∈ T^{d} there exists a diffusion process P with generator L_{0} and P (x(0) = x) = 1. Detailed properties of P can be found in [F].

The argument in [AHS] helps explain the condition (A1). For ϕ ∈ C_{0}^{∞}(T^{d}; R) and ψ ∈ C_{0}^{∞}(T^{d}; R^{d}) let

λ(ϕ) = ∫ dx (σ∇ϕ) · ψ.

Integrating by parts, and applying Schwarz’s inequality,

|λ| ≤ (∥ψ∥_{∞} + ∥∇ψ∥_{∞}) ∫ dx (|∇ · σ|^{2} + ∥σ∥^{2}) ,

so if ∫ dx (|∇ · σ|^{2} + ∥σ∥^{2}) < ∞, then λ is bounded on L^{2}. Therefore (σ∇)^{∗} is densely defined, and consequently σ∇ is closable. Finally if σ∇ is closable then the quadratic form E(ϕ, ψ) = ∫ dx σ∇ϕ · σ∇ψ is closable.

4. The difference in difficulty between existence and uniqueness is made very apparent by the situation in one dimension. As we saw already, uniqueness fails for the operator L = D_{x}|x|^{α}D_{x} if α < 1. On the other hand, E is closable under much more general conditions. A point x is called regular for a(x) if a^{−1} is locally integrable at x and nonregular points are called singular. The following theorem is well known: A proof can be found in [F].

Theorem. In one dimension, the form E is closable if and only if a(x) vanishes almost everywhere on its singular set.

5. Let m ∈ L^{1} (T^{d}). One can consider the Dirichlet form

E(u, v) = ∫ m(x)dx ∇ϕ · a∇ψ.

The corresponding operator

Lu = ∇ · a∇u + \frac{∇m}{m} · a∇u  (3.28)

is reversible on L^{2}(m(x)dx) instead of L^{2}(dx). Certainly there is no reason to prefer one to the other. However, since our methods are not based on Dirichlet forms, operators of the form (3.28) fit easily into the theory. This is a case in which the conditions of Theorem 3.12 are very easy to check. If m is non-negative, bounded and

∫ dx\frac{∇m · a∇m}{m} < ∞.

then m itself is an apriori weak solution of the forward equation ∂_{t}m − L^{∗}m = 0 satisfying the conditions of the Theorem. So if σ_{ij} ∈ W^{1,2}, i, j = 1, …, d then L^{∗} generates a nice semigroup on L^{∞}(m(x)dx).

4. Very weak solutions.

A very weak solution of the forward equation

\{ \begin{matrix}\frac{∂u}{∂t} = ∇ · a∇u − ∇ · σcu, in [0, T] × T^{d} \\ u(0) = u^{0} \end{matrix}  (4.1)

is a function u ∈ L^{∞} satisfying

\begin{aligned}0 = & − ∫_{0}^{T} dt ∫ dxu\frac{∂ϕ}{∂t} − ∫ dxu^{0}ϕ(0, x) \\ & − ∫_{0}^{T} dt ∫ dxu (∇ · a∇ϕ) + ∫_{0}^{T} dt ∫ dxuσc · ∇ϕ \end{aligned}  (4.2)

for all ϕ ∈ C_{0}^{∞} ([0, T) × T^{d}). In order for the last term in (4.2) to make sense we will assume that

∫_{0}^{T} dt ∫ dx |c|^{2}u < ∞.  (4.3)

If (A1) holds then the rest of (4.2) makes sense for any smooth ϕ since

∇ · a∇ϕ = (∇ · a) ∇ϕ + a∇ · ∇ϕ,

and (A1) implies in particular that ∇ · a and a are in L^{1} ([0, T] × T^{d}).

Such questions come up naturally when we study the question of existence and uniqueness of solutions to the martingale problem corresponding to the operator

L_{t} = ∇ · a∇u + σc · ∇u.

It is natural to write the operator in nondivergence form and ask questions of uniqueness for weak solutions of the adjoint equation, having defined the weak solution by using only smooth test functions. In this context it is not natural to assume a priori that u has any regularity.

Theorem 4.1. Suppose that (A2) holds. Assume that u ∈ L^{∞} ([0, T] × T^{d}) is a very weak solution of the forward equation (4.1), and (4.3) holds. Then (i) u ∈ H_{1,a}, and u is a weak solution of (4.1) (ii) If, in addition, u is non-negative, then we also have

∫_{0}^{T} dt ∫ dx\frac{⟨∇u, a∇u⟩}{u} < ∞.

Proof. Suppose we have a very weak solution of the forward equation (4.1). Let G_{ϵ} be the fundamental solution of the heat equation at time t = ϵ^{2} and u_{ϵ} = u ∗ G_{ϵ}. Then it follows that

\frac{∂u_{ϵ}}{∂t} = ∇ · (a∇u)_{ϵ} − ∇ · (σcu)_{ϵ}  (4.4)

Multiply equation (4.4) by u_{ϵ}, integrate over space and time and integrate by parts to obtain,

\begin{matrix}∫ dx u_{ϵ}^{2}(T) − ∫ dx u_{ϵ}^{2}(0) \\ = − ∫_{0}^{T} dt ∫ dx ∇u_{ϵ} · (a∇u)_{ϵ} + ∫_{0}^{T} dt ∫ dx (σcu)_{ϵ} · ∇u_{ϵ}. \end{matrix}  (4.5)

Consider the principal term

\begin{aligned}∫ dx ∇u_{ϵ} · (a∇u)_{ϵ} = & ∫ dx (∇u_{ϵ}) · a(∇u_{ϵ}) \\ & + ∫ dx ∇u_{ϵ}(x) · ∫ dy (a(x + y) − a(x)) ∇u(x + y)G_{ϵ}(y). \end{aligned} (4.6)

An integration by parts gives

∫ dx ∇u_{ϵ}(x) ∫ dy (a(x + y) − a(x)) ∇u(x + y)G_{ϵ}(y)
= ∫ dx ∇u_{ϵ}(x) ∫ dy ∇ · a(x + y) u(x + y)G_{ϵ}(y)
+ ∫ dx ∇u_{ϵ}(x) ∫ dy (a(x + y) − a(x)) u(x + y)∇G_{ϵ}(y)

We use Schwarz’s inequality to bound the first term on the right hand side by

\frac{1}{4} ∫ dx ∇u_{ϵ} · a_{ϵ}∇u_{ϵ} + ∥u∥_{L^{∞}}^{2} ∫ dx (∇ · a)a^{−1}(∇ · a).

The second term above is bounded by assumption, by some constant B_{1} < ∞. By Lemma 2.4,

∫ dx ∇u_{ϵ} · a_{ϵ}∇u_{ϵ} = ∫ dx |(σ∇u)_{ϵ}|^{2} + o(1).

The second term on the right hand side of (4.6) can be handled similarly. We prefer to do the convolution on R^{d} rather than on T^{d}.

∫ dx ∇u_{ϵ}(x) ∫_{T^{d}} dy (a(x + y) − a(x)) u(x + y)∇G_{ϵ}(y)
∫ dx ∇u_{ϵ}(x) ∫_{R^{d}} dy (a(x + y) − a(x)) u(x + y)∇G_{ϵ}(y)

= ∫ dx ∇u_{ϵ}(x) ∫_{R^{d}} dy ∫_{0}^{1} y · ∇a(x + ty)dt u(x + y)∇G_{ϵ}(y)

We use Schwarz’s inequality to dominate this by

\begin{matrix}\frac{1}{4} ∫_{0}^{1}dt∫ dx∫_{R^{d}}dy ∇u_{ϵ}(x) · a(x + ty)∇u_{ϵ}(x)G_{ϵ}(y) \\ + ∥u∥_{L^{∞}}^{2} ∫_{0}^{1}dt∫ dx∫_{R^{d}}dy y · ∇a(x+yt)∇G_{ϵ}a^{−1}(x + ty)y · ∇a(x + ty)∇G_{ϵ}(y)G_{ϵ}^{−1}(y) \end{matrix}

(4.7)

By Lemma 2.4,

∫_{0}^{1}dt∫ dx∫_{R^{d}}dy ∇u_{ϵ}(x) · a(x + ty)∇u_{ϵ}(x)G_{ϵ}(y) = ∫ dx |(σ∇u)_{ϵ}|^{2} + o(1).

The integrand in the last term in (4.11) is dominated by a dimensional constant times

∑_{i,j,k,ℓ} (a_{jk,i}a_{jℓ}^{−1}a_{ℓk,i}) (x + ty) (y_{i}G_{ϵ,k}(y))^{2} G_{ϵ}^{−1}(y).

For any f ∈ L^{1} it is straightforward to show that ∫_{0}^{1} dt ∫ dx ∫ dyf(x+ty) (y_{i} \frac{G_{ϵ,k}}{G_{ϵ}} (y))^{2} G_{ϵ}(y) is bounded uniformly in ϵ. Therefore by (A2) the last term in (4.7) is bounded by B_{2} < ∞ independent of ϵ. For the lower order part of (4.5),

\begin{aligned}∫ dx (σcu)_{ϵ} · ∇u_{ϵ} & ≤ \frac{1}{4} ∫ dx ∇u_{ϵ}a_{ϵ}∇u_{ϵ} + ∫ dx |c|^{2}u \\ & = \frac{1}{4} ∫ dx |(σ∇u)_{ϵ}|^{2} + o(1) + B_{3}, \end{aligned}

where B_{3} < ∞. From (4.5) we conclude that

∫ dx u_{ϵ}^{2}(T) − ∫ dx u_{ϵ}^{2}(0) = −\frac{1}{4} ∫ dx |(σ∇u)_{ϵ}|^{2} + B_{1} + B_{2} + B_{3} + o(1).

Since u is uniformly bounded, we conclude that

∫ dx |(σ∇u)_{ϵ}|^{2} ≤ C

for some C independent of ϵ. Letting ϵ → 0 and using the lower semicontinuity of the norm, the theorem is proved.

Suppose in addition u is non-negative, we want to show that ∫ dt ∫ dx\frac{⟨∇u,a∇u⟩}{u} < ∞. We have

\frac{∂u}{∂t} = ∇ · σF

with

F = σ∇u − cu ∈ L^{2}([0, T] × T^{d})

We can apply Lemma 3.1 to conclude that

∫ dxh(u(T)) − ∫ dxh(u(0)) = ∫_{0}^{T} dt ∫ dxh′(u(t, x))∇ · σF

= − ∫_{0}^{T} dt ∫ dxh′′(u(t, x)) ⟨F, σ∇u⟩

= − ∫_{0}^{T} dt ∫ dxh′′(u(t, x))[⟨∇u, a∇u⟩ − ⟨cu, σ∇u⟩]

We can take h_{δ}(u) = (u + δ) log(u + δ) to obtain

∫ dxh_{δ}(u(T)) − ∫ dxh_{δ}(u(0)) = − ∫_{0}^{T} dt ∫ dx\frac{⟨∇u, a∇u⟩}{u + δ} + ∫_{0}^{T} dt ∫ dx \frac{⟨c, σ∇u⟩ u}{u + δ}

We can estimate by Schwarz’s inequlity

| ⟨c, σ∇u⟩ u| ≤ |c|^{2}u^{2} + \frac{1}{4} ⟨∇u, a∇u⟩

and therefore

|∫_{0}^{T} dt ∫ dx \frac{c · σ∇u}{u + δ} | ≤ ∫_{0}^{T} dt ∫ dx\frac{|c|^{2}u^{2}}{u + δ} + \frac{1}{4} ∫_{0}^{T} dt ∫ dx\frac{⟨∇u, a∇u⟩}{u + δ}

This yields the bound

∫_{0}^{T} dt ∫ dx\frac{⟨∇u, a∇u⟩}{u + δ} ≤ C[ ∫_{0}^{T} dt ∫ dx\frac{|c|^{2}u^{2}}{u + δ} + ∫ dxh_{δ}(u(0)) − ∫ dxh_{δ}(u(T))] (4.8)

with a constant C not depending on δ. We can let δ → 0 in (4.8) to establish

∫_{0}^{T} dt ∫ dx\frac{⟨∇u, a∇u⟩}{u} < ∞.

We can now state the uniqueness theorem for very weak solutions. Theorem 4.2. Suppose ρ_{i}, i = 1, 2 are two bounded, non-negative, very weak solutions satisfying

\{ \begin{matrix}\frac{∂ρ_{i}}{∂t} = ∇ · a∇ρ_{i} − ∇ · σcρ_{i} \\ ρ_{1}(0, x) = ρ_{2}(0, x) on T^{d} \end{matrix}

with

∫_{0}^{T} ∫ dx|c|^{2}ρ_{i} < ∞

then ρ_{1} ≡ ρ_{2}.

Proof. Since we know now that very weak solutions are really weak solutions we can use Corollary 3.10.

5. Existence and uniqueness for a class of degenerate diffusion processes. In this section we will construct diffusion processes corresponding to the operator L_{t}

with diffusion matrix satisfying (A1) and obtain uniqueness if (A2) holds as well. Now for such wild coefficients, existence of a process for any starting point x is just not in the cards. So we give the following slightly non-standard definition: Let Ω = C ([0, T]; T^{d}), F the Borel σ-field, and F_{t} = σ(x(s) : 0 ≤ s ≤ t), 0 ≤ t ≤ T. Given a bounded density function ρ^{0}, a probability measure P on (Ω, F) is a solution of the martingale problem for L_{t} with initial distribution f if for every ϕ ∈ C^{∞}(T^{d}),

(i) E^{P} [ϕ(x_{0})] = ∫ dx ϕf,
(ii) ϕ(x_{t}) − ∫_{0}^{t}[L_{u}ϕ](x_{u})du is a martingale with respect to P and F_{t}.

Lemma 5.1. Let a^{α} : [0, T] × T^{d} → R^{d} ⊗ R^{d} be non-negative definite symmetric matrix valued and c^{α} : [0, T] × T^{d} → R^{d} be a family depending on a parameter α ∈ A. Let σ^{α} be the symmetric non-negative definite square root of a^{α}. For each α ∈ A let P^{α} be a solution of the martingale problem with initial distribution π^{α} and generator

L^{α} = ∇ · a^{α}∇ + σ^{α}c^{α} · ∇.

Let π^{α,t} be the marginals of P^{α}. Suppose that for each i, j = 1, …, d, (σ_{ij}^{α})^{2} and [∇ · a^{α} + σ^{α}c^{α}]_{ij} are uniformly integrable with respect to dt π^{α,t}(dx) on [0, T] × T^{d}. Then the family P^{α}, α ∈ A is tight.

Proof. To show tightness we will prove that for every ϵ > 0,

lim_{δ→0} sup_{α∈A} P^{α}( sup_{{0≤s<t≤T \atop |t−s|<δ}} |x(t) − x(s)| > ϵ) = 0.  (5.1)

Under P^{α} we have

x(t) − x(s) = ∫_{s}^{t} σ^{α}dβ + ∫_{s}^{t} (∇ · a^{α} + σ^{α}c^{α}) du  (5.2)

where β is a standard d-dimensional Brownian motion and the right hand side is computed on R^{d}, then projected onto the torus. Let

y_{ij}(t) = ∫_{0}^{t} σ_{ij}^{α}dβ_{j}.

For each i, j we can write y_{ij}(t) as a random time change of another Brownian motion β̂(·) in R^{1},

y_{ij}(t) = β̂ (∫_{0}^{t}(σ_{ij}^{α})^{2}du) .

Since Brownian motions are almost surely continuous the desired estimate (5.1) for the first term on the right hand side of (5.2) will follow if we can show

lim_{δ→0} sup_{α∈A} P^{α} ( sup_{{0≤s<t≤T \atop |t−s|<δ}} |∫_{s}^{t}(σ_{ij}^{α})^{2}du| > ϵ) = 0.  (5.3)

If we replace (σ_{ij}^{α})^{2} by (σ_{ij}^{α})^{2} ∧ ℓ for some ℓ > 0 then (5.3) is immediate. But then

P^{α} ( sup_{{0≤s<t≤T \atop |t−s|<δ}} |∫_{s}^{t} [(σ_{ij}^{α})^{2} − (σ_{ij}^{α})^{2} ∧ ℓ] du| > ϵ) ≤ ϵ^{−1}E^{P^{α}} [∫_{0}^{T} [(σ_{ij}^{α})^{2} − (σ_{ij}^{α})^{2} ∧ ℓ] du] .

The right hand side vanishes in the limit of large ℓ, uniformly in α, by the uniform integrability of (σ_{ij}^{α})^{2} with respect to dt π^{α,t}(dx). The second term on the right hand side of (5.2) can be handled in a similar fashion. In fact if we examine the preceding argument we see that the key point is the uniform integrability of each coordinate of (∇ · a^{α} + σ^{α}c^{α}) with respect to dt π^{α,t}(dx).

Lemma 5.2. Suppose that (A1) holds. Then there exists a solution P_{0} of the martingale problem for

L_{t} = ∇ · a∇

with initial distribution f ≡ 1.

Proof. For each ϵ > 0 let P_{ϵ} be the solution of the martingale problem for

L_{t}^{ϵ} = ∇ · a_{ϵ}∇

with initial distribution f ≡ 1 where a_{ϵ} is chosen according to Lemma 3.1. Note that all marginals are given by ρ_{ϵ} ≡ 1 and the conditions of Lemma 5.1 then follow from (A1). By Lemma 5.1 there is a weak limit P_{0}. We have to show that P_{0} solves the martingale problem for L_{t}. This amounts to the statement that for any ϕ ∈ C^{∞}(T^{d}) and any bounded continuous F ∈ F_{s},

E^{P_{ϵ}}[(ϕ(x_{t})−ϕ(x_{s})−∫_{s}^{t}[L_{u}^{ϵ}]ϕ(x_{u})du) F ] → E^{P_{0}}[(ϕ(x_{t})−ϕ(x_{s})−∫_{s}^{t}[L_{u}ϕ](x_{u})du) F ] .

By the Markov property it suffices to take F (x(·)) of the form f(x_{s}). The boundary terms present no problem and therefore one only has to verify that for bounded continuous f

E^{P_{ϵ}} [∫_{s}^{t}[(∇ · a_{ϵ}∇)ϕ](x_{u})du f(x_{s})] → E^{P_{0}} [∫_{s}^{t}[∇ · a∇ϕ](x_{u})du f(x_{s})] .

which is not difficult.

Theorem 5.3. Suppose that (A1) holds and that there exists a bounded non-negative weak solution ρ of the forward equation ∂_{t}ρ − L_{t}^{∗}ρ = 0, 0 ≤ t ≤ T, satisfying

\{ \begin{matrix}∫_{0}^{T} dt ∫ dx\frac{∇ρ·a∇ρ}{ρ} < ∞, \\ ∫_{0}^{T} dt ∫ dx |c|^{2}ρ < ∞. \end{matrix}

Then for any 0 ≤ f(x) ≤ Cρ(0, x) with ∫ dx f = 1 there exists a solution P of the martingale problem for L_{t} with initial density f and having marginal densities R(t, x), the unique weak solution of ∂_{t}R − L_{t}^{∗}R = 0, R(0) = f satisfying 0 ≤ R ≤ Cρ and ∫_{0}^{T} dt ∫ dx \frac{∇R·a∇R}{R} < ∞.

Proof. By Theorem ?, for any 0 ≤ f(x) ≤ Cρ(s, x) there exists a weak solution u(t, x) = [P_{s,t}^{∗}f] (x) of the forward equation after time s with ∫_{s}^{T} dt ∫ dx \frac{∇u·a∇u}{u} < ∞ and

P_{s,t}^{∗} : L^{∞}(ρ(s)) → L^{∞}(ρ(t))

is bounded. For any n and any partition 0 ≤ t_{1} < t_{2} < ⋯ < t_{n} ≤ T, let P^{t_{1},...,t_{n}} be the probability measure on (T^{d})^{n} given as follows: For any bounded functions, f_{i} : T^{d} → R, i = 1, …, n,

∫ f_{1}(x_{1}) ⋯ f_{n}(x_{n})dP^{t_{1},...,t_{n}} = ∫ dx f_{n}(x)[P_{t_{n−1},t_{n}}^{∗}f_{n−1}P_{t_{n−2},t_{n−1}}^{∗} ⋯ P_{t_{1},t_{2}}^{∗}f_{1}P_{0,t_{1}}^{∗}f](x).

Let P be the unique probability measure on (T^{d})^{[0,T]} such that

E^{P} [f_{1}(x_{t_{1}}) ⋯ f_{n}(x_{t_{n}})] = ∫ f_{1}(x_{1}) ⋯ f_{n}(x_{n})dP^{t_{1},...,t_{n}}.

Let P_{0} be a solution of the martingale problem for ∇ · a∇ with initial density f ≡ 1. Such a solution exists by the previous theorem. We will show that the relative entropy satisfies

H(P/P_{0}) ≤ ∫ dx f log f + \frac{C}{4} ∫_{0}^{T} dt ∫ dx|c|^{2}ρ.  (5.4)

Then P is a probability measure on C ([0, T]; T^{d}) and by construction P solves the martingale problem. To prove (5.4) we note that, because

H(P/P_{0}) = sup h (P^{t_{1},...,t_{n}}/P_{0}^{t_{1},...,t_{n}}) .

where the supremum runs over all partitions of [0, T] it is sufficient to show that

h (P^{t_{1},...,t_{n}}/P_{0}^{t_{1},...,t_{n}}) ≤ ∫ dx f log f + \frac{C}{4} ∫_{0}^{T} dt ∫ dx|c|^{2}ρ.  (5.5)

for all partitions. By Lemma 3.5,

h (P^{t_{1},...,t_{n}}/P_{0}^{t_{1},...,t_{n}}) ≤ h (P^{t_{1},...,t_{n−1}}/P_{0}^{t_{1},...,t_{n−1}}) + \frac{1}{4} ∫_{t_{n−1}}^{t_{n}} dt ∫ dx |c|^{2}R

from which we establish (5.5) by induction.

Remark. If one is interested in the more standard concept of solutions of the martingale problem starting from a point x at time s, then Theorem 5.3 provides us with the result that for each s, there exists a solution starting from x at time s for almost every x with respect to ρ(s, x)dx. This is the most that can be expected under our conditions.

Now we consider the problem of uniqueness of our process. While we have an existence and uniqueness theorem for weak solutions under assumptiom (A1), this is not of much use in proving uniqueness of solutions to the martingale problem. Even if we start with a solution to the martingale problem with marginals having densities in class H_{1,a} the conditionals are not necessarily in the same regularity class. Assumption (A2) will be important in this context.

Suppose we have an operator of the form

L_{t} = ∇ · a∇ + σc · ∇

with a satisfying (A2). We will limit overselves to the class P of processes P that satisfy the following conditions:

(i) There exist a constant C such that all the one dimensional marginals of P have densities bounded by C.

(ii) If ρ(t, x) is the marginal density of P then,

∫_{0}^{T} ∫ dx|c|^{2}ρ(t, x) = E^{P} [ ∫_{0}^{T} |c(t, x(t))|^{2}dt] < ∞.

We can now state and prove our main theorem.

Theorem 5.4. Assume that (A2) is valid. For any bounded initial density ρ_{0}(x) the solution to the martingale problem for

L_{t} = ∇ · a∇ + σc · ∇

is unique in the class P. If we assume that we have a non-negative solution ρ(t, x) that satisfies

\{ \begin{matrix}\frac{∂ρ}{∂t} = L_{t}^{∗}ρ = ∇ · a∇ρ − ∇ · σcρ \\ ∫_{0}^{T} dt ∫ dx∥c∥^{2}ρ < ∞ \end{matrix}

then a solution to the martingale problem for L_{t} exists (uniquely) in the class P and has the solution ρ(t, x) given above for its marginal densities.

Proof. We need only uniqueness to complete the proof. We will prove by induction the following statement. If P_{1} and P_{2} are two solutions to the martingale problem in class P and if 0 = t_{1} < t_{2} < ⋯ < t_{n} ≤ T are n time points the distribution of P_{1} and P_{2} agree on the sigma field σ{x(t_{i}), i = 1, …, n}. Assume that the statement is true for 0 = t_{1} < t_{2} < ⋯ < t_{n} ≤ T and we want to prove it for 0 = t_{1} < t_{2} < ⋯ < t_{n} < t_{n+1} ≤ T. In other words for any A ∈ σ{x(t_{i}), i = 1, …, n} and Borel set B ⊂ T^{d} we would like to show that

P_{1}[A ∩ {x(t_{n+1}) ∈ B}] = P_{2}[A ∩ {x(t_{n+1}) ∈ B}].  (5.6)

Clearly we can assume without loss of generality that P_{i}(A) > 0. Define Q_{1} and Q_{2} by

Q_{i}(E) = \frac{P_{i}(A ∩ E)}{P_{i}(A)} for i = 1, 2.

Then one can check that on the interval [t_{n}, T], Q_{1} and Q_{2} are in class P and are again solutions to the martingale problem. From the uniqueness of the marginals we conclude that

Q_{1}[x(t_{n+1}) ∈ B] = Q_{2}[x(t_{n+1}) ∈ B]

which is equivalent to (5.6). This establishes the induction.

Acknowledgement. J.Q. would like to thank S.R.S. Varadhan, H.-T. Yau and the Courant Institute for their warm hospitality while this work was completed.

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