Particle representations for stochastic partial differential equations with boundary conditions
In this article, we study a weighted particle representation for a class of stochastic partial differential equations with Dirichlet boundary conditions. The locations and weights of the particles satisfy an infinite system of stochastic differential equations (SDEs). The evolution of the particles is modeled by an infinite system of stochastic differential equations with reflecting boundary condition and driven by independent finite dimensional Brownian motions. The weights of the particles evolve according to an infinite system of stochastic differential equations driven by a common cylindrical noise and interact through , the associated weighted empirical measure. When the particles hit the boundary their corresponding weights are assigned a pre-specified value. We show the existence and uniqueness of a solution of the infinite dimensional system of stochastic differential equations modeling the location and the weights of the particles. We also prove that the associated weighted empirical measure is the unique solution of a nonlinear stochastic partial differential equation driven by with Dirichlet boundary condition. The work is motivated by and applied to the stochastic Allen-Cahn equation and extends the earlier of work of Kurtz and Xiong in [9, 10].
Key words: Stochastic partial differential equations, interacting particle systems, diffusions with reflecting boundary, stochastic Allen-Cahn equation, Euclidean quantum field theory equation with quartic interaction
MSC 2010 Subject Classification: Primary: 60H15, 60H35 Secondary: 60B12, 60F17, 60F25, 60H10, 35R60, 93E11.
In the following, we study particle representations for a class of nonlinear stochastic partial differential equations that includes the stochastic version of the Allen-Cahn equation [1, 2] and that of the equation governing the stochastic quantisation of Euclidean quantum field theory with quartic interaction , that is the equation
where is a (possibly) nonlinear function111In the original work of Allen and Cahn (see [1, 2]), for all whilst in the case of the equation of Euclidean quantum field theory, ( represents the state space dimension). and is a space-time noise222A detailed description of the noise is given below and in the Appendix.. These particle representations lead naturally to the solution of a weak version of a stochastic partial differential equation similar to (1.1). See equation (1.11) below and Section 3.
The approach taken here has its roots in the study of the McKean-Vlasov problem and its stochastic perturbation. In its simplest form, the problem begins with a finite system of stochastic differential equations
where the take values in , is the empirical measure , and the are independent, standard Brownian motions in an appropriate Euclidean space. The goal is to prove that the sequence of empirical measures converges in distribution and to characterize the limit as a measure valued process which solves the following nonlinear partial differential equation, written in weak form, 333In (1.3) and thereafter, we use the notation to express the integral of with respect to , that is,
where belongs to a suitably chosen class of Borel measurable functions and
In (1.3), we use and is the differential operator
There are many approaches to this problem [5, 11, 13]. (See also the recent book .) The approach in which we are interested, introduced in  and developed further in [3, 7, 9], is simply to let the limit be given by the infinite system
To make sense out of this system (in particular, the relationship of to the ), note that we can assume, without loss of generality, that the finite system is exchangeable (randomly permute the index ), so if one shows relative compactness of the sequence, any limit point will be an infinite exchangeable sequence and we can require to be the de Finetti measure for the sequence . (See Lemma 4.4 of .)
Note that while the give a particle approximation of the solution of (1.3), the give a particle representation of the solution, that is, the de Finetti measure of is the desired .
With the results of  in mind, we are interested in solutions of stochastic partial differential equations whose solutions are measures or perhaps signed measures. Our approach will be to represent the state in terms of a sequence of weighted particles , where denotes the location of the th particle at time in a state space and the weight. The sequence is required to be exchangeable, so if, for example, is a bounded measurable function and , we have
If the are nonnegative, then will be a measure, but we do not rule out the possibility that the can be negative and a signed measure. The weights and locations will be solutions of an infinite system of stochastic differential equations that are coupled only through and common noise terms.
We emphasize that we are talking about a representation of the solution of the equation, not a limit or approximation theorem (although these representations can be used to prove limit theorems). To specify the representation another way, let be the de Finetti measure for the exchangeable sequence . Then
The models in the current paper differ from those in  in two primary ways. First, the will be independent, stationary diffusion processes defined on a domain with reflecting boundary.
In the current work, we take the to satisfy the Skorohod equation
where is a vector field defined on the boundary and is a local time on for , that is, is a nondecreasing process that increases only when is in . Then, under appropriate regularity conditions and conditions on the coefficients, is a diffusion process whose infinitesimal generator is the closure of the second order differential operator
Much of what we say will hold under more general conditions on the process and domain. We will always assume that is strictly elliptic and has a stationary distribution denoted by and that the are independent, stationary solutions of the martingale problem for . One immediate consequence of this assumption is that the will be absolutely continuous with respect to , that is, we can write
The second primary difference from the models in  is that we will place boundary conditions on the solution. Essentially, we will require that
We assume that is continuous, although most of our results should also hold for piecewise continuous . To give a precise sense in which this boundary condition holds, let be a continuous function such that and for a nonempty compact , define
Since is strictly elliptic, and . Then under regularity conditions on the time-reversed process, for each nonempty, compact ,
See Section 2.4 for details.
In the same vein as equation (1.3), we will consider a class of nonlinear SPDE written in weak form
To obtain, for example, the stochastic Allen-Cahn equation (1.1), we can choose to be the generator for the normally reflecting Brownian motion, and for all and . In this case, is normalized Lebesgue measure on and, as above, is a signed-measure-valued process, is the density of with respect to , and satisfies (1.10). In equation (1.11), the test functions are chosen to belong to , the twice continuously differentiable functions with compact support in the interior of . Whilst the set of test functions is included in the domain of it is not sufficient to ensure the uniqueness of solutions of (1.11). Extension of (1.11) to larger classes of test functions are given in Section 3.
Throughout, we will assume that is a complete separable metric space, is a -finite Borel measure on , and is Lebesgue measure on . is Gaussian white noise on with covariance measure , that is, has expectation zero for all and with , and . (See Appendix A.1.)
Formally, equation (1.11) is the weak form of
where is the formal adjoint of the operator . For this equation, assuming that is sufficiently smooth and , we can take the locations to satisfy the Skorohod equation (1.6) with setting , then
To see that these weights should give the desired representation, let be in and apply Itô’s formula to obtain
Since vanishes in a neighborhood of the ,
the next to the last term is a martingale and the
martingales are orthogonal for different values of .
Assuming exchangeability, which will follow from the
exchangeability (that is, the independence) of the
provided we can show uniqueness for the system
(1), averaging gives (1.11). The following is the first main result of the paper:
Theorem 1 Under certain assumptions (see Condition 2.1 below) , there exists a unique solution of the system
Let be the space of processes compatible with taking values in such that for each , and some , satisfies
Theorem 1 tells us that there exists a (weak) solution of the equation (1.12). More it tells us that there exists a measure valued process that satisfies (1.11) for any . In general, uniqueness will not hold for (1.11) using test functions .444For example, consider reflecting Brownian motion with differing directions of reflection but whose stationary distribution is still normalized Lebesgue measure. Consequently, to obtain an equation that has a unique solution, we must enlarge the class of test functions. We do that in two different ways, first by taking the test functions to be , the space of twice continuously differentiable functions that vanish on the boundary (Section 3.1). The second extension is obtained by taking the test functions to be , the domain of the generator (Section 3.2). We summarize the second main result of the paper through the following:
Theorem 2 There exist extensions of the stochastic partial differential equation (1.11) with a unique solution in given by the above particle representation.
2 Particle representation
With the example in the previous section in mind, let be independent, stationary reflecting diffusions in with generator . For our purposes in this section, it is enough to define the generator to be the collection of pairs of bounded measurable functions such that
is a martingale, although in Section 3, we will need to define more precisely as the generator of the semigroup of operators corresponding to the . Note that for satisfying (1.6), the domain will contain given in (1.8) with given by (1.7).
Let , and consider the system
for . Then, assuming that a solution exists, we have, for
which is the weak form of
The system of SDEs (2) must be considered in conjunction with the existence of a empirical distribution
required to have a density with respect to . It is by no means clear that a solution satisfying all these constraints exists. We assume the following:
and are bounded with sup norms and .
2.1 Preliminary results
First we explore the properties that a solution must have by replacing by an arbitrary, measurable -valued stochastic process that is independent of and compatible with , that is, for each , is independent of
Define to be the solution of
Then is a martingale with respect to the filtration and there exists a standard Brownian motion such that is independent of and
For all defined as in (2.1) and , , and as above,
For each , there exists such that
Proof. Let and . Then letting
and letting ,
so we have a similar bound on . Together the bounds give the first two inequalities in (2.3).
Note that is a continuous martingale with quadratic variation . Define
and . Then is a continuous martingale with respect to the filtration and , so is a standard Brownian motion. Since , is independent of .
The first inequality in (2.7) follows by the monotonicity of and the finiteness by standard estimates on the distribution of the supremum of Brownian motion.
The will be exchangeable, so we can define to be the density with respect to of the signed measure determined by
Suppose that is -adapted. Then is -adapted and for each ,
Proof. By exchangeability,
where the second equality follows by the independence of and . The lemma then follows by the definition of conditional expectation.
Let . Then there exists a version of such that
where we interpret the right side as the optional projection, and for this version
Moreover the identity (2.9) holds with replaced by any nonnegative -measurable random variable .
It follows that is a version of .
With (2.9) in mind, given an exchangeable family such that is adapted to , define taking to be given by
For all and , there exists a constant so that for all
Proof. Recall that we have the bound
Notice that Jensen’s inequality gives
and that implies . It follows that
Fix with , so that is an -martingale measure under . By the Burkholder-Davis-Gundy inequality, we find