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Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

Abstract

In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of non-autonomous linear parabolic initial- and final-boundary value problems defined in bounded convex subsets of Euclidean space of arbitrary dimension. Under certain conditions regarding their joint endpoint distributions, we also prove that such processes become reversible Markov diffusions. Furthermore we show that those diffusions satisfy two Itô equations for some suitably constructed Wiener processes, and from that analysis derive Feynman–Kac representations for the solutions to the given equations. We then illustrate some of our results by considering the heat equation with Neumann boundary conditions both in a one-dimensional bounded interval and in a two-dimensional disk.

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Acknowledgements

The research of both authors was supported by the FCT of the Portuguese government under Grant PTDC/MAT/69635/2006, and by the Mathematical Physics Group of the University of Lisbon under Grant ISFL/1/208. The first author is also indebted to Madalina Deaconu and Elton Hsu for stimulating discussions and correspondence on the theme of reflected diffusions. Last but not least, he wishes to thank the Complexo Interdisciplinar da Universidade de Lisboa and the ETH-Forschungsinstitut für Mathematik in Zurich where parts of this work were completed for their financial support and warm hospitality.

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Correspondence to Pierre A. Vuillermot.

Appendix: A Variational Construction of Weak Solutions in L 2(D)

Appendix: A Variational Construction of Weak Solutions in L 2(D)

The solutions u φ and v ψ we used throughout this article are generated by two evolution systems, U A (t,s)0≤stT and \(U_{A}^{\ast}(t,s)_{0\leq s\leq t\leq T}\) on L 2(D). We show here how to construct these evolution systems by applying the standard methods of [30], under the following hypotheses regarding the coefficients k, l and V in (1) and (2):

(K′):

The function \(k:D\times[ 0,T ] \mapsto \mathbb{R}^{d^{2}}\) is matrix-valued and for every i,j∈{1,…,d} we have k i,j =k j,i L (D×(0,T)); moreover, there exists a finite constant \(\underline{k}>0\) such that the inequality

$$ \bigl( k(x,t)q,q \bigr)_{\mathbb{R}^{d}}\geq\underline{k}| q |^{2} $$
(134)

holds uniformly in (x,t)∈D×[0,T] for all q∈ℝd. Finally, there exist finite constants c >0, \(\beta\in( \frac{1}{2},1 ] \) such that the Hölder continuity estimate

$$\max_{i,j\in\{ 1,\ldots,d \} }\bigl| k_{i,j}(x,t)-k_{i,j}(x,s)\bigr| \leq c_{\ast}| t-s|^{\beta}$$

is valid for every xD and all s,t∈[0,T].

As for the lower-order differential operators we assume that the following hypotheses are valid, where we assume without restricting the generality that the constants c and β are the same as in hypothesis (K′):

(L′):

Each component of the vector-field l:D×[0,T]↦ℝd satisfies l i L (D×(0,T)). Moreover, the Hölder continuity estimate

$$\max_{i\in\{ 1,\ldots,d \} }\bigl| l_{i}(x,t)-l_{i} (x,s)\bigr| \leq c_{\ast}| t-s|^{\beta}$$

holds for every xD and all s,t∈[0,T].

(V′):

The function V:D×(0,T)↦ℝ is such that VL (D×(0,T)) and satisfies

$$\bigl| V(x,t)-V(x,s)\bigr| \leq c_{\ast}| t-s |^{\beta}$$

for every xD and all s,t∈[0,T].

Moreover, both the initial condition φ and the final condition ψ are real-valued and the following hypothesis holds:

(IF′):

We have φ,ψL 2(D).

Remark

In the variational theory we are reviewing here we observe that the Hölder continuity requirement relative to the time variable in hypotheses (K′), (L′) and (V′) is stronger than that of hypotheses (K), (L) and (V), since \(\beta\in( \frac{1}{2},1 ] \) whereas \(\frac{\alpha}{2}\in( 0,\frac{1}{2} ) \). However, it is easy to show by uniqueness arguments that the evolution operators U A (t,s)0≤stT and \(U_{A}^{\ast }(t,s)_{0\leq s\leq t\leq T}\) introduced in Sect. 2 are identical to those constructed below. The reason why \(\beta\in( \frac{1}{2},1 ] \) is required here is intimately tied up with the variational structure of the problem, and is thoroughly discussed in [30].

Under the preceding three conditions, it is easily verified that the quadratic form a:[0,TH 1(DH 1(D)↦ℂ defined by

satisfies the estimates

(135)
(136)
(137)

for all s,t∈[0,T] and all f,hH 1(D), where ∥.∥2 and ∥.∥1,2 stand for the usual norms in L 2(D) and H 1(D), respectively, and where \(( .\,,. )_{\mathbb{C}^{d}}\) denotes the Hermitian inner product in ℂd. Consequently, the formal elliptic operator

$$A(t):=-\frac{1}{2}\operatorname{div} \bigl( k(.\,,t)\nabla\bigr) + \bigl( l(.,t),\nabla\bigr)_{\mathbb{C}^{d}}+V(.\,,t) $$

corresponding to the right-hand side of (1) can be realized as a regularly accretive operator defined on some time-dependent and dense domain \(\mathcal{D(}A(t))\subset L^{2}(D)\), and as such generates an evolution system U A (t,s)0≤stT in L 2(D) given by

$$ U_{A}(t,s)f(x)= \begin{cases} f(x)&\mbox{if}\ t=s,\\ \int_{D}dy\,g_{A}(x,t;y,s)f(y)&\mbox{if}\ t>s \end{cases} $$
(138)

for every fL 2(D), where g A denotes the parabolic Green function associated with (1). Indeed, all these assertions follow directly from estimates (135)–(137) and the general theory developed in Sect. 5.4 of [30], together with Schwartz’s kernel theorem which guarantees the existence of g A (see [28] for a summary of the many possible applications of that theorem).

In a similar way, the Hermitian conjugate form

$$a^{\ast} ( t,f,h ) :=\overline{a ( t,h,f ) }$$

is associated with the linear operator A (t) adjoint to A(t), which in turn generates the adjoint evolution system

$$ U_{A}^{\ast}(t,s)f(x)= \begin{cases} f(x)&\mbox{if}\ t=s,\\ \int_{D}dy\,g_{A}^{\ast}(x,s;y,t)f(y)&\mbox{if}\ t>s, \end{cases} $$
(139)

where \(G_{A}^{\ast}\) is the parabolic Green function associated with (2) that satisfies the relation

$$g_{A}^{\ast}(x,s;y,t)=g_{A}(y,t;x,s) $$

for all s,t∈[0,T] with t>s.

The important features of (138) and (139) are that they provide the real-valued functions defined by

$$ u_{\varphi}(x,t):=U_{A}(t,0)\varphi(x)=\int_{D}dy\,g_{A}(x,t;y,0) \varphi(y),\quad t\in( 0,T ] $$
(140)

and

$$ v_{\psi}(x,t):=U_{A}^{\ast}(T,t)\psi(x)=\int _{D}dy\,g_{A}^{\ast}(x,t;y,T)\psi(y),\quad t\in[ 0,T ) , $$
(141)

which satisfy

$$\biggl( \frac{\partial}{\partial t}u_{\varphi}(.\,,t),h \biggr)_{2} +a\bigl(t,u_{\varphi}(.\,,t),h\bigr)=0,\quad t\in( 0,T ] $$

and

$$- \biggl( \frac{\partial}{\partial t}v_{\psi}(.\,,t),h \biggr )_{2}+a^{\ast } \bigl(t,v_{\psi}(.\,,t),h\bigr)=0,\quad t\in[ 0,T ) $$

for every hH 1(D), respectively, where (. ,.)2 stands for the usual inner product in L 2(D). Moreover, we have u φ ,v ψ L 2(D×(0,T)), so that (140) and (141) provide weak solutions to (1) and (2), respectively (see e.g. Sect. 5.5 in [30]).

These solutions are those which ultimately possess the properties listed in Lemma 1 of Sect. 2, according to the above remark regarding the Hölder regularity in time.

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Vuillermot, P.A., Zambrini, J.C. Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations. J Theor Probab 27, 449–492 (2014). https://doi.org/10.1007/s10959-012-0426-3

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Keywords

  • Diffusion processes
  • Parabolic partial differential equations

Mathematics Subject Classification

  • 35K20
  • 60H30
  • 60K99