Abstract
In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of nonautonomous linear parabolic initial and finalboundary 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 onedimensional bounded interval and in a twodimensional 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 ETHForschungsinstitut für Mathematik in Zurich where parts of this work were completed for their financial support and warm hospitality.
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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≤s≤t≤T } 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 matrixvalued 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∈ℝ^{d}. Finally, there exist finite constants c _{∗}>0, \(\beta\in( \frac{1}{2},1 ] \) such that the Hölder continuity estimate
$$\max_{i,j\in\{ 1,\ldots,d \} }\bigl k_{i,j}(x,t)k_{i,j}(x,s)\bigr \leq c_{\ast} ts^{\beta}$$is valid for every x∈D and all s,t∈[0,T].
As for the lowerorder 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 vectorfield l:D×[0,T]↦ℝ^{d} satisfies l _{ i }∈L ^{∞}(D×(0,T)). Moreover, the Hölder continuity estimate
$$\max_{i\in\{ 1,\ldots,d \} }\bigl l_{i}(x,t)l_{i} (x,s)\bigr \leq c_{\ast} ts^{\beta}$$holds for every x∈D and all s,t∈[0,T].
 (V′):

The function V:D×(0,T)↦ℝ is such that V∈L ^{∞}(D×(0,T)) and satisfies
$$\bigl V(x,t)V(x,s)\bigr \leq c_{\ast} ts ^{\beta}$$for every x∈D and all s,t∈[0,T].
Moreover, both the initial condition φ and the final condition ψ are realvalued and the following hypothesis holds:
 (IF′):

We have φ,ψ∈L ^{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≤s≤t≤T } 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,T]×H ^{1}(D)×H ^{1}(D)↦ℂ defined by
satisfies the estimates
for all s,t∈[0,T] and all f,h∈H ^{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 ℂ^{d}. Consequently, the formal elliptic operator
corresponding to the righthand side of (1) can be realized as a regularly accretive operator defined on some timedependent and dense domain \(\mathcal{D(}A(t))\subset L^{2}(D)\), and as such generates an evolution system U _{ A }(t,s)_{0≤s≤t≤T } in L ^{2}(D) given by
for every f∈L ^{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
is associated with the linear operator A ^{∗}(t) adjoint to A(t), which in turn generates the adjoint evolution system
where \(G_{A}^{\ast}\) is the parabolic Green function associated with (2) that satisfies the relation
for all s,t∈[0,T] with t>s.
The important features of (138) and (139) are that they provide the realvalued functions defined by
and
which satisfy
and
for every h∈H ^{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/s1095901204263
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DOI: https://doi.org/10.1007/s1095901204263
Keywords
 Diffusion processes
 Parabolic partial differential equations
Mathematics Subject Classification
 35K20
 60H30
 60K99